CBSE Maths CBSE for 12th Standard CBSE Question paper & Study Materials

12th Maths Vector Algebra Chapter Case Study Question with Answers CBSE - by users_admin View & Read

12th Maths Three Dimensional Geometry Chapter Case Study Question with Answers CBSE - by users_admin View & Read

12th Maths Probability Chapter Case Study Question with Answers CBSE - by users_admin View & Read

12th Maths Linear Programming Chapter Case Study Question with Answers CBSE - by users_admin View & Read

12th Maths Differential Equations Chapter Case Study Question with Answers CBSE - by users_admin View & Read

12th Maths Continuity and Differentiability Chapter Case Study Question with Answers CBSE - by users_admin View & Read

12th Maths Application of Integrals Chapter Case Study Question with Answers CBSE - by users_admin View & Read

CBSE 12th Standard Maths Subject Differential Equations Value Based Questions 4 Marks Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    the doctor took the temperature of a dead body at 11.30 P.M., which was 94.60 F. He took the temperature of the body again after one hour, which was 93.40 F. If the temperature of the room was 700 F, estimate the time of death. taking normal temperature of human body = 98.60 F.

  • 2)

    The velocity v of mass, of a rocket at time t, is given by the equation: \(m\frac { dv }{ dt } +V\frac { dm }{ dt } =0,\)Where 'V' is the constant velocity of emission. If the rocket starts from when t = 0 with mass m, prove that : \(v=Vlog\left( \frac { { m }_{ 0 } }{ m } \right) .\) Should we encourage rocket technology, why?

  • 3)

    It is given that the rate at which some bacteria multiply is proportional to the instantaneous number present. If the original number of bacteria doubles in two hours, in how will it be five times?

  • 4)

    In a culture,the bacteria count is 1,00,000. The number is increased by 10% in 2 hours. In how many hours will the count reach 2,00,000, if the rate of growth of bacteria proportional to the number present?

  • 5)

    The rate of increases in the number of bacteria in a certain culture is proportional to the number present. Given that the number triples in 5 hours, find how many bacteria will be present after 10 hours. Also, find the time necessary of bacteria to be 10 times the number initially present.\(\left( { log }_{ e }3=1.0986,{ e }^{ 2.1972 }=9 \right) \)  (approx.)

CBSE 12th Standard Maths Subject Differential Equations Value Based Questions 4 Marks Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    \(\frac { dy }{ dx } +\frac { y }{ x } =0,\) where 'x' denotes the percentage population in a city and 'y' denotes the area for living healthy life of population. Find the particular solution when \(x=100,y=1.\) Is higher density of population harmful? Justify your answer

  • 2)

    Suppose the growth of a population is proportional to the number present. If the population of a colony doubles in 50 months, in how many months will the population becomes triple?

  • 3)

    It is given that the rate at which some bacteria multiply is proportional to the instantaneous number present. If the original number of bacteria doubles in two hours, in how will it be five times?

  • 4)

    In a culture,the bacteria count is 1,00,000. The number is increased by 10% in 2 hours. In how many hours will the count reach 2,00,000, if the rate of growth of bacteria proportional to the number present?

  • 5)

    The rate of increases in the number of bacteria in a certain culture is proportional to the number present. Given that the number triples in 5 hours, find how many bacteria will be present after 10 hours. Also, find the time necessary of bacteria to be 10 times the number initially present.\(\left( { log }_{ e }3=1.0986,{ e }^{ 2.1972 }=9 \right) \)  (approx.)

CBSE 12th Standard Maths Subject Determinants Value Based Questions 6 Marks Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    A total amount of Rs. 7,000 is deposited in three different savings bank accounts with annual interest rates of 5%, 8% and 81/2% respectively. The total annual interest from these three accounts is Rs. 550. Equal amounts have been deposited in 5% and 8% savings accounts. Find the amount deposited in each of the three accounts, with the help of matrices. Write the value.

  • 2)

    Two schools A and B want to award their selected students on the values of Sincerity, Truthfulness and helpfulness. The school A wants to award Rs. x each, Rs. y each and Rs. z each for the three respective values to its 3, 2 and 1 students with a total award money of Rs. 1,600. School B wants to spend Rs. 2,300 to award its 4, 1 and 3 students on the respective values (by giving the same award money for the three values as before). If the total amount of awards for one prize on each value is Rs. 900. Using matrices, find the award money for each value. Apart from these three values, suggest one more value which should be considered for award.

  • 3)

    Two factories decided to award their employees for three values of
    (a) adaptable new techniques,
    (b) careful and alert in difficult situations and
    (c) keeping calm in tense situations, at the rate of Rs. x, Rs. y and Rs. z per person respectively 2, 4 and 3 employees with a total prize money of Rs. 29,000. The second factory decided to honour respectively 5, 2 and 3 employees with the prize money of Rs. 30,500. If the three prizes per person together cost Rs. 9,500; then
    (i) Represent the above situation by a matrix equation and form linear equations using matrix multiplication.
    (ii) Solve these equations using matrices
    (iii) Which values are reflected in this question?

  • 4)

    A school wants to award its students for the value of Honesty, Regularity and Hard work with a total cash award of Rs. 6,000. Three times the award money for hard work added to that given for honesty amounts to Rs. 11,000.The award money given for honesty and hard work together is double the one given for Regularity. Represent the above situation algebraically and find the award money for each value, using matrix method. Apart from these values namely, honesty, regularity and hard work, suggest one more value which the school must include for awards.

  • 5)

    Find the inverse of the matrix : \(A=\left[ \begin{matrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{matrix} \right] \)
    Can you find the inverse for all values of a, b, c? If same rule is applied to the progress of a person, what value is most essential in?

CBSE 12th Standard Maths Subject Determinants Value Based Questions 6 Marks Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    A total amount of Rs. 7,000 is deposited in three different savings bank accounts with annual interest rates of 5%, 8% and 81/2% respectively. The total annual interest from these three accounts is Rs. 550. Equal amounts have been deposited in 5% and 8% savings accounts. Find the amount deposited in each of the three accounts, with the help of matrices. Write the value.

  • 2)

    Two schools P and Q want to award their selected students on the values of Tolerance, Kindness and Leadership. The school P wants to award Rs. x each, Rs. y each and Rs. z each for the three respective values to its 3, 2 and 1 students, respectively with a total award money of Rs. 2,200. School Q wants to spend  Rs. 3,100 to award its 4, 1 and 3 students on the respective values (by giving the same award money for the three values as school P). If the total amount of award for one prize on each value is Rs. 1,200. Using matrices, find the award money for each value. Apart from the above these three values, suggest one more value which should be considered for award.

  • 3)

    Two schools P and Q want to award their selected students on the values of Discipline, Politeness and Punctuality. The school P wants to award Rs. x each, Rs. y each and Rs. z each for the three respective values to its 3, 2 and 1 students with a total award money of Rs. 1,000. School Q wants to spend Rs. 1,500 to award its 4, 1 and 3 students on the respective values (by giving the same award money for the three values as before). If the total amount of awards for one prize on each value is Rs. 600. Using matrices, find the award money for each value. Apart from the above three values, suggest one more value for awards.

  • 4)

    Two schools A and B want to award their selected students on the values of Sincerity, Truthfulness and helpfulness. The school A wants to award Rs. x each, Rs. y each and Rs. z each for the three respective values to its 3, 2 and 1 students with a total award money of Rs. 1,600. School B wants to spend Rs. 2,300 to award its 4, 1 and 3 students on the respective values (by giving the same award money for the three values as before). If the total amount of awards for one prize on each value is Rs. 900. Using matrices, find the award money for each value. Apart from these three values, suggest one more value which should be considered for award.

  • 5)

    Two schools A and B want to award their selected teachers on the values of Honesty, Hard work and Regularity. School A wants to award Rs. x each, Rs. y each and Rs. z each for the three respective values to 3, 2 and 1 teachers with a total award money of Rs.1.28 lakh. School B wants to spend Rs. 1·54 lakh to award its 4, 1 and 3 teachers on the respective values (by giving the same award money for the three values as before). If the total amount of award for one prize on each value is Rs. 57,000, using matrices, find the award money for each value.

CBSE 12th Standard Maths Subject Determinants Value Based Questions 4 Marks Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    A typsit charges Rs. 145 for typing 10 English and 3 Hindi pages, while charges for typing 3 English and 10 Hindi pages are Rs. 180. Using matrices, find the charges oof typing one English and one Hindi page respectively.
    However, typist charged only Rs. 2 per page from a poor student Shyam for 5 Hindi pages.How much less was charged from this poor boy? Which values are reflected in this problem

  • 2)

    An amount of Rs. 6500 is invested in three investments at the rate of 6%, 8% and 9% per annum respectively.The total annual income is Rs. 4800.The income from the third instalment is Rs.600 more than the income from the second investment.
    (i) Represent the above situation by matrix equation and form linear equations using matrix multiplication.
    (ii) Is it possible to solve the system of equations, so obtained, using matrices?
    (iii) A company invites investments.It promises to return double the money after a period of 3 years.Will you like to invest in the company

  • 3)

    Three shopkeepers A, B and C are using polythene, hand made bags (prepared by prisoners) and newspaper's envelope as carry bags.It is found that shopkeepers A, B and C are using (20, 30, 40), (30, 40, 20) and (40, 20, 30) polythene, hand-made bags and newspaper envelopes respectively. The shopkeepers A, B and C spent Rs. 250, Rs. 270 and Rs. 200 on these carry bags respectively. Find the cost of each carry bag using matrices. Keeping in mind the social and environmental conditions, which shopkeeper is better and why?

  • 4)

    For keeping fit X people believe in morning walk, Y people believe in Yoga and Z people join GYM. Total number of people are 70. Further 20%, 30% and 40% people are suffering from any disease who believe in morning walk, Yoga and GYM respectively. Total number of such people is 21. If morning walk cost Rs. 0, Yoga cost Rs. 500/month and Gym Rs. 400/month and total expenditure is Rs. 23,000.
    (i) Formulate a matrix problem.
    (ii) Calculate the number of each type of people.
    (iii) why exercise is important for health.

  • 5)

    An amount of Rs. 600 crores is spent by the government in three schemes.Scheme A is for saving girl child from the cruel parents who don't want girl child and get the abortion belore her birth.Scheme B is for saving of newlywed girls from death due to dowry.Scheme C is planning for good health for senior citizens.Now twice the amount spent on Scheme C together with amount spent on Scheme A is Rs. 500 crores.And three times the amount spent on Scheme A together with amount spent on Scheme B and Scheme C is Rs. 1200 crores.Find the amount spent on each Scheme, using matrices.What is the importance of saving girl child from the cruel parents who don't want girl child and get the abortion before her birth?

CBSE 12th Standard Maths Subject Determinants Value Based Questions 4 Marks Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    A typsit charges Rs. 145 for typing 10 English and 3 Hindi pages, while charges for typing 3 English and 10 Hindi pages are Rs. 180. Using matrices, find the charges oof typing one English and one Hindi page respectively.
    However, typist charged only Rs. 2 per page from a poor student Shyam for 5 Hindi pages.How much less was charged from this poor boy? Which values are reflected in this problem

  • 2)

    Two schools A and B decides to award prizes to their students for three values honesty(x), punctuality (y) and obedience (z).School A decides to award a total of Rs. 11000 for three value to 5, 4 and 3 students respectively while school B decided to award Rs. 10700 for teh three values to 4,3 and 5 students respectively.If all the three prizes together amount to Rs. 2700 then:
    (i) Represent the above situation by a matrix equation and form linear equations, using matrix multiplication.
    (ii) Is it possible to solve the system of equations so obtained using matrix multiplication?
    (iii) Which value you prefer to be awarded most and why?

  • 3)

    Using matrix method solve the following system of equations:
    x + 2y + z = 7, x - y + z = 4, x + 3y + 2z = 10
    If 'x' represents the number of persons who take food at home 'y' represents the number of persons who take junk food in the market and 'z' represents the number of persons who take food at hotel. Which way of taking food you prefer and why? 

  • 4)

    An amount of Rs. 6500 is invested in three investments at the rate of 6%, 8% and 9% per annum respectively.The total annual income is Rs. 4800.The income from the third instalment is Rs.600 more than the income from the second investment.
    (i) Represent the above situation by matrix equation and form linear equations using matrix multiplication.
    (ii) Is it possible to solve the system of equations, so obtained, using matrices?
    (iii) A company invites investments.It promises to return double the money after a period of 3 years.Will you like to invest in the company

  • 5)

    Three shopkeepers A, B and C are using polythene, hand made bags (prepared by prisoners) and newspaper's envelope as carry bags.It is found that shopkeepers A, B and C are using (20, 30, 40), (30, 40, 20) and (40, 20, 30) polythene, hand-made bags and newspaper envelopes respectively. The shopkeepers A, B and C spent Rs. 250, Rs. 270 and Rs. 200 on these carry bags respectively. Find the cost of each carry bag using matrices. Keeping in mind the social and environmental conditions, which shopkeeper is better and why?

CBSE 12th Standard Maths Subject Value Based Questions 4 Marks Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    In a group of 400 people, 160 are smokers and non-vegetarian, 100 are smokers and vegetarian and the remaining are non-smokers and vegetarian. The probabilities of getting a special chest disease are 35%, 20% and 10% respectively. A person is chosen from the group at random and is found to be suffering from the disease. What is the probability that the selected person is a smoker and non-vegetarian? What value is reflected in this question?

  • 2)

    If \(A=\{ (a_{ 1 },a_{ 2 },a_{ 3 },a_{ 4 },a_{ 5 }\} \) and \(B=\{ (b_{ 1 },b_{ 2 },b_{ 3 },b_{ 4 },\} \) , where \(a_{ i }'s\) and \(b_{ i }'s\) are school going students. Define a relation from a set A to a set B by x R y iff y is true friend of X.
    If \(R=\{ (a_{ 1 },b_{ 1 }),(a_{ 2 },b_{ 1 }),(a_{ 3 },b_{ 2 }),(a_{ 4 },b_{ 2 }),(a_{ 5 },b_{ 2 })\} \).
    Is R a bijective function?
    Do you think true friendship is important in life? How?

  • 3)

    Two schools A and B decides to award prizes to their students for three values honesty(x), punctuality (y) and obedience (z).School A decides to award a total of Rs. 11000 for three value to 5, 4 and 3 students respectively while school B decided to award Rs. 10700 for teh three values to 4,3 and 5 students respectively.If all the three prizes together amount to Rs. 2700 then:
    (i) Represent the above situation by a matrix equation and form linear equations, using matrix multiplication.
    (ii) Is it possible to solve the system of equations so obtained using matrix multiplication?
    (iii) Which value you prefer to be awarded most and why?

  • 4)

    Using matrix method solve the following system of equations:
    x + 2y + z = 7, x - y + z = 4, x + 3y + 2z = 10
    If 'x' represents the number of persons who take food at home 'y' represents the number of persons who take junk food in the market and 'z' represents the number of persons who take food at hotel. Which way of taking food you prefer and why? 

  • 5)

    An amount of Rs. 6500 is invested in three investments at the rate of 6%, 8% and 9% per annum respectively.The total annual income is Rs. 4800.The income from the third instalment is Rs.600 more than the income from the second investment.
    (i) Represent the above situation by matrix equation and form linear equations using matrix multiplication.
    (ii) Is it possible to solve the system of equations, so obtained, using matrices?
    (iii) A company invites investments.It promises to return double the money after a period of 3 years.Will you like to invest in the company

CBSE 12th Standard Maths Subject Value Based Questions 4 Marks Questions 2021 Part - II - by Shalini Sharma - Udaipur View & Read

  • 1)

    If \(A=\{ (a_{ 1 },a_{ 2 },a_{ 3 },a_{ 4 },a_{ 5 }\} \) and \(B=\{ (b_{ 1 },b_{ 2 },b_{ 3 },b_{ 4 },\} \) , where \(a_{ i }'s\) and \(b_{ i }'s\) are school going students. Define a relation from a set A to a set B by x R y iff y is true friend of X.
    If \(R=\{ (a_{ 1 },b_{ 1 }),(a_{ 2 },b_{ 1 }),(a_{ 3 },b_{ 2 }),(a_{ 4 },b_{ 2 }),(a_{ 5 },b_{ 2 })\} \).
    Is R a bijective function?
    Do you think true friendship is important in life? How?

  • 2)

    A typsit charges Rs. 145 for typing 10 English and 3 Hindi pages, while charges for typing 3 English and 10 Hindi pages are Rs. 180. Using matrices, find the charges oof typing one English and one Hindi page respectively.
    However, typist charged only Rs. 2 per page from a poor student Shyam for 5 Hindi pages.How much less was charged from this poor boy? Which values are reflected in this problem

  • 3)

    Two schools A and B decides to award prizes to their students for three values honesty(x), punctuality (y) and obedience (z).School A decides to award a total of Rs. 11000 for three value to 5, 4 and 3 students respectively while school B decided to award Rs. 10700 for teh three values to 4,3 and 5 students respectively.If all the three prizes together amount to Rs. 2700 then:
    (i) Represent the above situation by a matrix equation and form linear equations, using matrix multiplication.
    (ii) Is it possible to solve the system of equations so obtained using matrix multiplication?
    (iii) Which value you prefer to be awarded most and why?

  • 4)

    Using matrix method solve the following system of equations:
    x + 2y + z = 7, x - y + z = 4, x + 3y + 2z = 10
    If 'x' represents the number of persons who take food at home 'y' represents the number of persons who take junk food in the market and 'z' represents the number of persons who take food at hotel. Which way of taking food you prefer and why? 

  • 5)

    An amount of Rs. 6500 is invested in three investments at the rate of 6%, 8% and 9% per annum respectively.The total annual income is Rs. 4800.The income from the third instalment is Rs.600 more than the income from the second investment.
    (i) Represent the above situation by matrix equation and form linear equations using matrix multiplication.
    (ii) Is it possible to solve the system of equations, so obtained, using matrices?
    (iii) A company invites investments.It promises to return double the money after a period of 3 years.Will you like to invest in the company

CBSE 12th Standard Maths Subject Value Based Questions 4 Marks Questions 2021 Part - I - by Shalini Sharma - Udaipur View & Read

  • 1)

    In a group of 400 people, 160 are smokers and non-vegetarian, 100 are smokers and vegetarian and the remaining are non-smokers and vegetarian. The probabilities of getting a special chest disease are 35%, 20% and 10% respectively. A person is chosen from the group at random and is found to be suffering from the disease. What is the probability that the selected person is a smoker and non-vegetarian? What value is reflected in this question?

  • 2)

    If \(A=\{ (a_{ 1 },a_{ 2 },a_{ 3 },a_{ 4 },a_{ 5 }\} \) and \(B=\{ (b_{ 1 },b_{ 2 },b_{ 3 },b_{ 4 },\} \) , where \(a_{ i }'s\) and \(b_{ i }'s\) are school going students. Define a relation from a set A to a set B by x R y iff y is true friend of X.
    If \(R=\{ (a_{ 1 },b_{ 1 }),(a_{ 2 },b_{ 1 }),(a_{ 3 },b_{ 2 }),(a_{ 4 },b_{ 2 }),(a_{ 5 },b_{ 2 })\} \).
    Is R a bijective function?
    Do you think true friendship is important in life? How?

  • 3)

    A typsit charges Rs. 145 for typing 10 English and 3 Hindi pages, while charges for typing 3 English and 10 Hindi pages are Rs. 180. Using matrices, find the charges oof typing one English and one Hindi page respectively.
    However, typist charged only Rs. 2 per page from a poor student Shyam for 5 Hindi pages.How much less was charged from this poor boy? Which values are reflected in this problem

  • 4)

    Two schools A and B decides to award prizes to their students for three values honesty(x), punctuality (y) and obedience (z).School A decides to award a total of Rs. 11000 for three value to 5, 4 and 3 students respectively while school B decided to award Rs. 10700 for teh three values to 4,3 and 5 students respectively.If all the three prizes together amount to Rs. 2700 then:
    (i) Represent the above situation by a matrix equation and form linear equations, using matrix multiplication.
    (ii) Is it possible to solve the system of equations so obtained using matrix multiplication?
    (iii) Which value you prefer to be awarded most and why?

  • 5)

    Using matrix method solve the following system of equations:
    x + 2y + z = 7, x - y + z = 4, x + 3y + 2z = 10
    If 'x' represents the number of persons who take food at home 'y' represents the number of persons who take junk food in the market and 'z' represents the number of persons who take food at hotel. Which way of taking food you prefer and why? 

CBSE 12th Standard Maths Subject Determinants HOT Questions 6 Mark Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    If a \(\neq \)p b\(\neq \)q c\(\neq \)r  and \(\left| \begin{matrix} p & b & c \\ a & q & c \\ a & b & r \end{matrix} \right| \) = 0 find the value of \(\frac { p }{ p-a } +\frac { q }{ q-b } +\frac { r }{ r-c } \)

  • 2)

    For the matrix A = \(\left[ \begin{matrix} 3 & 2 \\ 1 & 1 \end{matrix} \right] \)find the numbers a and b such that A2 +aA + bI = O. Hence find A-1.

  • 3)

    Solve the equation if a \(\neq \) 0 and  \(\left| \begin{matrix} x+a & x & x \\ x & x+a & x \\ x & x & x+a \end{matrix} \right| \)

  • 4)

    Using matrix method, determine whether the following system of equation is consisten or inconsistent
    3x - y -2z =2
    2y -z = -1
    3x - 5y = 3

CBSE 12th Standard Maths Subject Matrices HOT Questions 6 Mark Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    If A = \(\left[ \begin{matrix} 3 & 1 \\ 7 & 5 \end{matrix} \right] \) find x, y such that A2 +xI = yA Hence find A-1

  • 2)

    If  \(A=\left[ \begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix} \right] \) Prove that , A =\(\left[ \begin{matrix} { 3 }^{ x-1 } & { 3 }^{ x-1 } & { 3 }^{ x-1 } \\ { 3 }^{ x-1 } & { 3 }^{ x-1 } & { 3 }^{ x-1 } \\ { 3 }^{ x-1 } & { 3 }^{ x-1 } & { 3 }^{ x-1 } \end{matrix} \right] \) for every positive integer n.

  • 3)

    The sum of three numbers is -1. If we multiply the second number by 2 , third number by 3 and add them we get 5. If we subtract the third number from the sum of first and second numbers we get -1. Represent it by a system of equations . Find the three numbers using inverse of a matrix

CBSE 12th Standard Maths Subject Inverse Trigonometric Functions HOT Questions 6 Mark Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Solve tan-1 2x + tan-13x = \(\frac { \pi }{ 4 } \)

CBSE 12th Standard Maths Subject Relations and Functions HOT Questions 6 Mark Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Define a binary operation * on the set {0, 1, 2, 3, 4, 5} as \(a * b=\left\{\begin{array}{ll} a+b, & \text { if } a+b<6 \\ a+b-6 & \text { if } a+b \geq 6 \end{array}\right.\) show that 0 is the identity for this operation and each element of the set is invertible with 6 - a being the inverse of  a.

CBSE 12th Standard Maths Subject Continuity and Differentiability HOT Questions 4 Mark Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    If 2x + 2y = 2x+y, show that \(\frac { dy }{ dx } \) -2y-x

  • 2)

    Differentiare tan-1 \(\left( \frac { acosx-bsinx }{ bcosx+asinx } \right) \)

  • 3)

    Differentiate xx +xa + ax + aa 

  • 4)

    Differentiate \(\left( x+\frac { 1 }{ x } \right) ^{ x }+x^{ \left( x+\frac { 1 }{ x } \right) }\)

CBSE 12th Standard Maths Subject Continuity and Differentiability HOT Questions 4 Mark Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    If x sin (a+y) + sin a cos (a+y)=0, prove that \(\frac { dy }{ dx } =\frac { { sin }^{ 2 }(a+y) }{ sin\quad a } \)

  • 2)

    If \(y={ \left\{ x+\sqrt { { x }^{ 2 }+{ a }^{ 2 } } \right\} }^{ n }\)prove that \(\frac { dy }{ dx } =\frac { ny }{ \sqrt { { x }^{ 2 }+{ a }^{ 2 } } } \)

  • 3)

    If \(y={ \left\{ x+\sqrt { { x }^{ 2 }+{ a }^{ 2 } } \right\} }^{ n }\)prove that \(\frac { dy }{ dx } =\frac { ny }{ \sqrt { { x }^{ 2 }+{ a }^{ 2 } } } \)

  • 4)

    If \({ x }^{ 2 }+{ y }^{ 2 }=t-\frac { 1 }{ t } \)and \({ x }^{ 4 }+{ y }^{ 4 }={ t }^{ 2 }+\frac { 1 }{ { t }^{ 2 } } \) show that \(\frac { dy }{ dx } =\frac { 1 }{ { x }^{ 3 }y } \)

  • 5)

    If \({ x }^{ 2 }+{ y }^{ 2 }=t-\frac { 1 }{ t } \)and \({ x }^{ 4 }+{ y }^{ 4 }={ t }^{ 2 }+\frac { 1 }{ { t }^{ 2 } } \)show that \(\frac { dy }{ dx } =\frac { 1 }{ { x }^{ 3 }y } \)

CBSE 12th Standard Maths Subject Inverse Trigonometric Functions HOT Questions 4 Mark Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

CBSE 12th Standard Maths Subject Inverse Trigonometric Functions HOT Questions 4 Mark Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Find the value of \(\tan \left(\sin ^{-1} \frac{3}{5}+\cot ^{-1} \frac{3}{2}\right)\)

CBSE 12th Standard Maths Subject Relations and Functions HOT Questions 4 Mark Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Let X = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Let R1 be a relation in X given by R1 = {(x, y)}: x - y is divisible by 3} and R2 be another relation on X given by R2 = {(x, y): (x, y)} {1, 4, 7} or {x, y} {2, 5, 8} or {x, y} {3, 6, 9}} show that R1 = R

CBSE 12th Standard Maths Subject Relations and Functions HOT Questions 4 Mark Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Consider identity function \(I_{ N }+I_{ N }:N\rightarrow N\) defined as:
    \(I_{ N }(x)=x\forall x\in N.\)
    Show that although \(I_{ N }\) is onto but \(I_{ N }+I_{ N }:N\rightarrow N\)defined as:
    \((I_{ N }+I_{ N })(x)+I_{ N }(x)=x+x=2x\) is onto.

  • 2)

    Find \(fof^{ -1 }\) and \(f^{ -1 }\) of for the function:
    \(f(x)=\frac { 1 }{ x } ,x\neq 0\). Also prove that \(fof^{ -1 }\)\(f^{ -1 }\) of .

  • 3)

    Show that the relation in the set \(A=\{ x:x\in W,0x\le 12\} \) given by \(R=\{ a,b):(a-b)\) is an multiple of 4} is an equivalence relation. Also find the set of all elements related to 2.

CBSE 12th Standard Maths Subject Determinants HOT Questions 4 Mark Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Find value of k, If area of the triangle with vertices P (k,0),Q (4,0) R(0,2) is 4 square units.

CBSE 12th Standard Maths Subject Determinants HOT Questions 4 Mark Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Without expanding the determinant at any stage, prove that \(\left| \begin{matrix} x+1 & x+2 & x+a \\ x+2 & x+3 & x+b \\ x+3 & x+4 & x+c \end{matrix} \right| =0\), where a, b, c are in A.P.

  • 2)

    If a, b, c are all positive and are pth, qth, rth terms respectively of a G > P, then prove that : \(\left| \begin{matrix} loga & p & 1 \\ logb & q & 1 \\ logc & r & 1 \end{matrix} \right| =0\)

  • 3)

    Prove that : \(\left| \begin{matrix} { a }^{ 2 } & { a }^{ 2 }-{ (b-c) }^{ 2 } & bc \\ { b }^{ 2 } & { b }^{ 2 }-{ (c-a) }^{ 2 } & ca \\ { c }^{ 2 } & { c }^{ 2 }-{ (a-b) }^{ 2 } & ab \end{matrix} \right| =(b-c)(c-a)(a+b+c)({ a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 })\)

  • 4)

    \(\left| \begin{matrix} { yz }-x^{ 2 } & { zx }-y^{ 2 } & { xy-z }^{ 2 } \\ { zx-y }^{ 2 } & { xy }-z^{ 2 } & { yz-x }^{ 2 } \\ { xy-z }^{ 2 } & { yz }-x^{ 2 } & { zx-x }^{ 2 } \end{matrix} \right| \)is divisible by (x+y+z) and here, find the quotient.

  • 5)

    An equilateral triangle has each side equal to a. If the co-ordinates of its vertices are (x1,y1), (x2,y2) and (x3, y3), show that \(\left| \begin{matrix} x_1 &y_1 &1 \\x_2 &y_2 &1 \\x_3 &y_3 &1 \end{matrix} \right| ^2={3\over4}a^4\)

CBSE 12th Standard Maths Subject Continuity and Differentiability HOT Questions Fill Ups Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Write the derivative of the function f(x) = tan-1 \(\sqrt { sinx } \) w.r.to x

  • 2)

    Is it true that log (xsinx+cossinx x)=sinxlogx+sin x logcos x?

  • 3)

    Discuss the continuity of the cosine, cosecant, secant and cotangent functions.

  • 4)

    Discusss the continuity of the function f(x) = sin|x|

CBSE 12th Standard Maths Subject Continuity and Differentiability HOT Questions Fill Ups Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Check the continuity of the function f(x) = {\(\begin{matrix} 1, & if & x\le 0 \\ 2, & if & x>0 \end{matrix}\)

  • 2)

    Check the continuity of the function f (x) = |x| at x = 0

  • 3)

    Check the continuity of the function f(x) = sinx + x2 at x = 0

  • 4)

    Examine the continuity of the function f(x) = |x|

  • 5)

    Write the points of discontinuity of the function f(x) = [x] where [x] denotes the gretest integer function less than or equal to x.

CBSE 12th Standard Maths Subject Determinants HOT Questions Fill Ups Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    For what value of k, the matrix \(\left[ \begin{matrix} 2 & k \\ 3 & 5 \end{matrix} \right] \)has no inverse

  • 2)

    A B C are three non zero matrices of same order, then find the condition on Such that AB = AC \(\Rightarrow\) B =C

  • 3)

    Let Abe a non singular matrix of order 3 x 3 , such that |AdjA| = 100 find |A|

  • 4)

    If A is non singular matrix of order n, then weite the value of Adj(Adj A) and hence write the value of Adj(Adj A) if order of A and |A| = 5

  • 5)

    Let A be a diagonal A = (d1, d2, …, dn ) write the value of | A |

CBSE 12th Standard Maths Subject Determinants HOT Questions Fill Ups Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    If A is square matrix of order 3 and | A | = 5, find the value of |-3,4|

  • 2)

    if \(\omega \)\(\Box\) is cube root of unity find the value of \(\triangle =\left\lfloor \begin{matrix} 1 & \omega & { \omega }^{ 2 } \\ \omega & { \omega }^{ 2 } & \omega \\ { \omega }^{ 2 } & 1 & 1 \end{matrix} \right\rfloor \)

  • 3)

    Find the value of determinant \(\triangle =\left\lfloor \begin{matrix} 1 & 2 & 4 \\ 8 & 16 & 32 \\ 64 & 128 & 256 \end{matrix} \right\rfloor \)

  • 4)

    Find the Value of determinant \(\triangle =\left| \begin{matrix} 2 & 2 & 2 \\ x & y & z \\ y+z & z+x & z+y \end{matrix} \right| \)

  • 5)

    If a, b care in A.P find the value of determinant\(\triangle =\left| \begin{matrix} x+1 & x+2 & x+a \\ x+2 & x+3 & x+b \\ x+3 & x+4 & x+c \end{matrix} \right| \)

CBSE 12th Standard Maths Subject Matrices HOT Questions Fill Ups Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    If A = \(A=\left[ \begin{matrix} \alpha & \beta \\ \gamma & -a \end{matrix} \right] \)and A2 = I, Find the value of \(\alpha\)2 + \(\beta\)\(\gamma \)

  • 2)

    If A = \(\left[ \begin{matrix} sinx & -cosx \\ cosx & sinx \end{matrix} \right] \) 0 < x < \(\frac { \pi }{ 2 } \) and A + A' =I, Where I us unit matrix, find value of x.

  • 3)

    If the following matrix is skew symmetric, find the values of a, b, c
    \(A=\left[ \begin{matrix} 0 & a & 3 \\ 2 & b & -1 \\ c & 1 & 0 \end{matrix} \right] \)

CBSE 12th Standard Maths Subject Matrices HOT Questions Fill Ups Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    If A and B are invertible matrices of the same order, then (AB)–1 = B–1 A–1.

  • 2)

    Let A = [aij]be a matric of order 2 x 3 and aij = \(\frac { i-j }{ i+j } \), write the value of a23

  • 3)

    if \(\left[ \begin{matrix} a+b & 2 \\ 5 & ab \end{matrix} \right] =\left[ \begin{matrix} 6 & 2 \\ 5 & 8 \end{matrix} \right] \)find the relation between a and b 

  • 4)

    If following information regarding the number of men and women workers in three factories I, II and III is written in the form of 3 x 2 matrix. What does the entry in third row and second column represent

      Men Workerts Women Workers
    Factory I 30 25
    Factory II 25 31
    Factory III 27 26
  • 5)

    If, A = |aij| = \(\left[ \begin{matrix} 2 & 3 & -5 \\ 1 & 4 & 9 \\ 0 & 7 & -2 \end{matrix} \right] \)and B =|bij| =\(\left[ \begin{matrix} 2 & -1 \\ -3 & 4 \\ 1 & 2 \end{matrix} \right] \) Write the value of
    (i)a22 + b21
    (ii) a11b11 +a22b22 

CBSE 12th Standard Maths Subject Inverse Trigonometric Functions HOT Questions Fill Ups Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Find the Principal value of \({ cos }^{ -1 }\left( \frac { \sqrt { 3 } }{ 2 } \right) \)

  • 2)

    Find the value of \({ cos }^{ -1 }\left( cos\frac { 13\pi }{ 6 } \right) \)

CBSE 12th Standard Maths Subject Inverse Trigonometric Functions HOT Questions Fill Ups Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    if tan-1 (a/x) + tan-1 (b/x) = \(\pi\) /2, then x = 

  • 2)

    if (a < 0) and x \(\varepsilon \) (-a, a), simplify tan-1 \(\left( \frac { x }{ \sqrt { { a }^{ 2 }-{ x }^{ 2 } } } \right) \)

  • 3)

    If x + y + z = xyz , then the valu of tan-1x + tan-1y + tan-1 z =

CBSE 12th Standard Maths Subject Relations and Functions HOT Questions Fill Ups Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

CBSE 12th Standard Maths Subject Relations and Functions HOT Questions Fill Ups Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Let A = {x : -1\(\le\) x \(\le\) 1} and S be the subset of Aׂ defined b S = {(x,y)}:x2+y2 = 1}m is it a function

  • 2)

    A mappping f : N \(\longrightarrow\) N is defined by f(x) = 2 x \(\forall \)\(\varepsilon \) N, then is f a bijection?

CBSE 12th Standard Maths Subject HOT Questions 6 Mark Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Solve tan-1 2x + tan-13x = \(\frac { \pi }{ 4 } \)

  • 2)

    If  \(A=\left[ \begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix} \right] \) Prove that , A =\(\left[ \begin{matrix} { 3 }^{ x-1 } & { 3 }^{ x-1 } & { 3 }^{ x-1 } \\ { 3 }^{ x-1 } & { 3 }^{ x-1 } & { 3 }^{ x-1 } \\ { 3 }^{ x-1 } & { 3 }^{ x-1 } & { 3 }^{ x-1 } \end{matrix} \right] \) for every positive integer n.

CBSE 12th Standard Maths Subject HOT Questions 6 Mark Questions 2021 Part - II - by Shalini Sharma - Udaipur View & Read

  • 1)

    Prove that \(\tan ^{-1}\left(\frac{\sqrt{1+x^{2}}+\sqrt{1-x^{2}}}{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}\right)=\frac{\pi}{4}+\frac{1}{2} \cos ^{-1} x^{2}\)

CBSE 12th Standard Maths Subject HOT Questions 6 Mark Questions 2021 Part - I - by Shalini Sharma - Udaipur View & Read

  • 1)

    Define a binary operation * on the set {0, 1, 2, 3, 4, 5} as \(a * b=\left\{\begin{array}{ll} a+b, & \text { if } a+b<6 \\ a+b-6 & \text { if } a+b \geq 6 \end{array}\right.\) show that 0 is the identity for this operation and each element of the set is invertible with 6 - a being the inverse of  a.

  • 2)

    Prove that \(\tan ^{-1}\left(\frac{\sqrt{1+x^{2}}+\sqrt{1-x^{2}}}{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}\right)=\frac{\pi}{4}+\frac{1}{2} \cos ^{-1} x^{2}\)

  • 3)

    Solve tan-1 2x + tan-13x = \(\frac { \pi }{ 4 } \)

CBSE 12th Standard Maths Subject HOT Questions 3 Mark Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    \(\left| \begin{matrix} { yz }-x^{ 2 } & { zx }-y^{ 2 } & { xy-z }^{ 2 } \\ { zx-y }^{ 2 } & { xy }-z^{ 2 } & { yz-x }^{ 2 } \\ { xy-z }^{ 2 } & { yz }-x^{ 2 } & { zx-x }^{ 2 } \end{matrix} \right| \)is divisible by (x+y+z) and here, find the quotient.

  • 2)

    Prove that:\(\left| \begin{matrix} -a({ b }^{ 2 }+{ c }^{ 2 }-{ a }^{ 2 }) & { 2b }^{ 3 } & { 2c }^{ 3 } \\ { 2 }a^{ 3 } & -b({ c }^{ 2 }+{ a }^{ 2 }-{ b }^{ 2 }) & { 2c }^{ 3 } \\ { 2a }^{ 3 } & { 2b }^{ 3 } & -c({ a }^{ 2 }+{ b }^{ 2 }-{ c }^{ 2 }) \end{matrix} \right| =abc({ a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 }{ ) }^{ \\ 3 }\)

  • 3)

    If  \(A=\left[ \begin{matrix} 1 & 2 & 3 \\ -1 & 0 & 2 \\ 1 & -3 & -1 \end{matrix} \right] ,\quad B=\left[ \begin{matrix} 4 & 5 & 6 \\ -1 & 0 & 1 \\ 2 & 1 & 2 \end{matrix} \right] ,\quad C=\left[ \begin{matrix} -1 & -2 & 1 \\ -1 & 2 & 3 \\ -1 & -2 & 2 \end{matrix} \right] \)   
    find \((A+B+C)\prime .\) Is \( (A+B+C)\prime =A\prime +B\prime +C\prime ?\)

  • 4)

    If \(y={ e }^{ m{ sin }^{ -1 }x },-1\le x\le 1\)show that \(\left( 1-{ x }^{ 2 } \right) { y }_{ 2 }-x{ y }_{ 1 }={ m }^{ 2 }y\)

CBSE 12th Standard Maths Subject HOT Questions 4 Mark Questions 2021 Part - II - by Shalini Sharma - Udaipur View & Read

  • 1)

    An equilateral triangle has each side equal to a. If the co-ordinates of its vertices are (x1,y1), (x2,y2) and (x3, y3), show that \(\left| \begin{matrix} x_1 &y_1 &1 \\x_2 &y_2 &1 \\x_3 &y_3 &1 \end{matrix} \right| ^2={3\over4}a^4\)

  • 2)

    Consider identity function \(I_{ N }+I_{ N }:N\rightarrow N\) defined as:
    \(I_{ N }(x)=x\forall x\in N.\)
    Show that although \(I_{ N }\) is onto but \(I_{ N }+I_{ N }:N\rightarrow N\)defined as:
    \((I_{ N }+I_{ N })(x)+I_{ N }(x)=x+x=2x\) is onto.

  • 3)

    If \(y={ \left\{ x+\sqrt { { x }^{ 2 }+{ a }^{ 2 } } \right\} }^{ n }\)prove that \(\frac { dy }{ dx } =\frac { ny }{ \sqrt { { x }^{ 2 }+{ a }^{ 2 } } } \)

  • 4)

    The decay rate of radium at any time is proportional to its mass at that time. The mass is \({m}_{0}\) at \(t=0.\) Find the time when the mass will be halved

CBSE 12th Standard Maths Subject HOT Questions 4 Mark Questions 2021 Part - I - by Shalini Sharma - Udaipur View & Read

  • 1)

    Without expanding the determinant at any stage, prove that \(\left| \begin{matrix} x+1 & x+2 & x+a \\ x+2 & x+3 & x+b \\ x+3 & x+4 & x+c \end{matrix} \right| =0\), where a, b, c are in A.P.

  • 2)

    Prove that:\(\left| \begin{matrix} -a({ b }^{ 2 }+{ c }^{ 2 }-{ a }^{ 2 }) & { 2b }^{ 3 } & { 2c }^{ 3 } \\ { 2 }a^{ 3 } & -b({ c }^{ 2 }+{ a }^{ 2 }-{ b }^{ 2 }) & { 2c }^{ 3 } \\ { 2a }^{ 3 } & { 2b }^{ 3 } & -c({ a }^{ 2 }+{ b }^{ 2 }-{ c }^{ 2 }) \end{matrix} \right| =abc({ a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 }{ ) }^{ \\ 3 }\)

  • 3)

    Find \(fof^{ -1 }\) and \(f^{ -1 }\) of for the function:
    \(f(x)=\frac { 1 }{ x } ,x\neq 0\). Also prove that \(fof^{ -1 }\)\(f^{ -1 }\) of .

  • 4)

    If \({ x }^{ 2 }+{ y }^{ 2 }=t-\frac { 1 }{ t } \)and \({ x }^{ 4 }+{ y }^{ 4 }={ t }^{ 2 }+\frac { 1 }{ { t }^{ 2 } } \) show that \(\frac { dy }{ dx } =\frac { 1 }{ { x }^{ 3 }y } \)

  • 5)

    Use Lagrange's Theorem to determine a point P on the curve \(f\left( x \right) =\sqrt { x-2 } \)defined in the interval [2,3], where the tangent is parallel to the chord joining the end points on the curve.

CBSE 12th Standard Maths Subject HOT Questions 3 Mark Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Show that the function \(f\ :\ N\rightarrow N\) given by f (x) = 2x  is one-one but not onto.

  • 2)

    Find gof and fog, if \(f:R\rightarrow R\) and \(g:R\rightarrow R\) are given by: \(f(x)=cosx\) and \(g(x)=3x^{ 2 }\). Show that gof \(\neq \) fog.

  • 3)

    Let \(f:N\rightarrow R\) be a function defined as \(f(x)=4x^{ 2 }+12x+15\) Show that \(f:N\rightarrow \) Range f is invertible. Find the inverse off.

  • 4)

    Find the number of all one-one functions from the set of A = {1, 2, 3} to itself is a permutation of 1, 2, 3.

  • 5)

    Show that the relation R in the set A = {1, 2, 3, 4, 5} given by is even R = {(a, b) : |a - b| is even} is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.

CBSE 12th Standard Maths Subject HOT Questions Fill Ups Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Let A = [aij]be a matric of order 2 x 3 and aij = \(\frac { i-j }{ i+j } \), write the value of a23

CBSE 12th Standard Maths Subject HOT Questions Fill Ups Questions 2021 Part - II - by Shalini Sharma - Udaipur View & Read

  • 1)

    if tan-1 (a/x) + tan-1 (b/x) = \(\pi\) /2, then x = 

  • 2)

    if A = [-1, 2, -5] B =\(\left[ \begin{matrix} 2 \\ -1 \\ 7 \end{matrix} \right] \) Write the orders of AB and BA

  • 3)

    \(\left[ \begin{matrix} 2 & 3 \\ 1 & 0 \end{matrix} \right] \)= P+Q, where P is symmetric and Q is Skew symmetric matrix, find the matrices P and Q 

CBSE 12th Standard Maths Subject HOT Questions Fill Ups Questions 2021 Part - I - by Shalini Sharma - Udaipur View & Read

  • 1)

    Let A = {x : -1\(\le\) x \(\le\) 1} and S be the subset of Aׂ defined b S = {(x,y)}:x2+y2 = 1}m is it a function

  • 2)

    If following information regarding the number of men and women workers in three factories I, II and III is written in the form of 3 x 2 matrix. What does the entry in third row and second column represent

      Men Workerts Women Workers
    Factory I 30 25
    Factory II 25 31
    Factory III 27 26
  • 3)

    If A = \(\left[ \begin{matrix} sinx & -cosx \\ cosx & sinx \end{matrix} \right] \) 0 < x < \(\frac { \pi }{ 2 } \) and A + A' =I, Where I us unit matrix, find value of x.

CBSE 12th Standard Maths Subject Probability Ncert Exemplar 4 Marks Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    A committee of 4 students is selected at random from a group consisting of 8 boys and 4 girls. Given that there is at least one girl in the committee, calculate the probability that there are exactly 2 girls in the committee.

  • 2)

    A and B throw a pair of die alternately. A wins the game if he gets a total of 6 and B wins if she gets a total of 7. If A starts the game, find the probability of winning the game by A in third throw of pair of dice.

  • 3)

    Bag 1 contains 3 black and 2 white balls, bag II contains 2 black and 4 white balls. A bag and a ball is selected at random. Determine the probability of selecting a black ball.

  • 4)

    A biased die is such that \(P(4)=\frac{1}{10}\) and other scores-being equally likely. The die is tossed twice. If X is the 'number of four seen', then find the variance of the random variable X.

  • 5)

    Suppose 10000 tickets are sold in a lottery each for Rs. 1. First prize is of Rs. 3000 and the second prize is of Rs. 2000. There are three third prizes of Rs. 500 each. If you buy one ticket, then what is your expectation?

CBSE 12th Standard Maths Subject Probability Ncert Exemplar 4 Marks Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    A committee of 4 students is selected at random from a group consisting of 8 boys and 4 girls. Given that there is at least one girl in the committee, calculate the probability that there are exactly 2 girls in the committee.

  • 2)

    A and B throw a pair of die alternately. A wins the game if he gets a total of 6 and B wins if she gets a total of 7. If A starts the game, find the probability of winning the game by A in third throw of pair of dice.

  • 3)

    Bag 1 contains 3 black and 2 white balls, bag II contains 2 black and 4 white balls. A bag and a ball is selected at random. Determine the probability of selecting a black ball.

  • 4)

    A biased die is such that \(P(4)=\frac{1}{10}\) and other scores-being equally likely. The die is tossed twice. If X is the 'number of four seen', then find the variance of the random variable X.

  • 5)

    Suppose 10000 tickets are sold in a lottery each for Rs. 1. First prize is of Rs. 3000 and the second prize is of Rs. 2000. There are three third prizes of Rs. 500 each. If you buy one ticket, then what is your expectation?

CBSE 12th Standard Maths Subject Probability Ncert Exemplar MCQ Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    If 3 A and B are two events such that \(P(B)=\frac{3}{5}\),\(P(A / B)=\frac{1}{2} \text { and } P(A \cup B)=\frac{4}{5}\) then P(A) equals

  • 2)

    A flashlight has 8 batteries out of which 3 are dead. If two batteries are selected without replacement and tested, then probability that both are dead is

CBSE 12th Standard Maths Subject Probability Ncert Exemplar MCQ Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    If 3 A and B are two events such that \(P(B)=\frac{3}{5}\),\(P(A / B)=\frac{1}{2} \text { and } P(A \cup B)=\frac{4}{5}\) then P(A) equals

  • 2)

    A flashlight has 8 batteries out of which 3 are dead. If two batteries are selected without replacement and tested, then probability that both are dead is

  • 3)

    If a die is thrown and a card is selected at random from a deck of 52 playing cards, then the probability of getting an even number on the die and a spade card is

CBSE 12th Standard Maths Subject Differential Equations Ncert Exemplar 4 Mark Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Solve the differential equation: y+ \(\frac {d}{dx}\)(xy) = s (sin x + log x)

  • 2)

    Find the general solution of the following differential equations :
    \(\left( 1+{ y }^{ 2 } \right) +\left( x-{ e }^{ \tan ^{ -1 }{ y } } \right) \frac { dy }{ dx } =0\)

  • 3)

    Find the differential equation of system of concentric circles with centre (1,2).

  • 4)

    Find the equation of a curve whose tangent at any point on it, different from origin, has slope \(y+\frac{y}{x}\).

  • 5)

    Find the equation of a curve passing through the point (1, 1), if the tangent drawn at any point P(x, y) on the curve meets the coordinate axes at A and B such that P is the mid-point of AB.

CBSE 12th Standard Maths Subject Differential Equations Ncert Exemplar 4 Mark Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Solve the differential equation: y+ \(\frac {d}{dx}\)(xy) = s (sin x + log x)

  • 2)

    Solve the differential equation: (x + y) (dx - dy) = dx + dy

  • 3)

    If y(t) is a solution of (1 + t) \(\frac{dy}{dt}\) - ty = 1 and y(0) = -1, then show that y(1) = -\(\frac {1}{2}\)

  • 4)

    Find the general solution of the following differential equations :
    \(\left( 1+{ y }^{ 2 } \right) +\left( x-{ e }^{ \tan ^{ -1 }{ y } } \right) \frac { dy }{ dx } =0\)

  • 5)

    Find the equation of a curve whose tangent at any point on it, different from origin, has slope \(y+\frac{y}{x}\).

CBSE 12th Standard Maths Subject Three Dimensional Geometry Ncert Exemplar 4 Marks Questions With SOlution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Find the vector and the Cartesian equations of the line through the point (5, 2, – 4) and which is parallel to the vector \(3 \hat{i}+2 \hat{j}-8 \hat{k}\)

  • 2)

    Show that the equation of a plane which meets the axes in A,B and C and the given centroid of triangle ABC is the point \((\alpha ,\beta ,\gamma )\) is \(\frac { x }{ \alpha } +\frac { y }{ \beta } +\frac { z }{ \gamma } =3\)

  • 3)

    If a variable line in two adjacent positions has direction cosines l,m,n,\(l+\delta l,m+\delta m,n+\delta n\) and show that the small angle \(\delta \theta \) between two positions is given by \({ (\delta \theta ) }^{ 2 }={ (\delta l) }^{ 2 }+(\delta m)^{ 2 }+{ (\delta n) }^{ 2 }\)

  • 4)

    (a) Find the image of the point (1,6,3) in the line \(\frac { x }{ 1 } =\frac { y-1 }{ 2 } =\frac { z-2 }{ 3 } .\)
    (b) Also write the equation of the line joining the: given point and its image and find the length of the: segment joining the given point and its image

  • 5)

    Show that the lines \(\frac { x-1 }{ 2 } =\frac { y-2 }{ 3 } =\frac { z-3 }{ 4 } \) and \(\frac { x-4 }{ 5 } =\frac { y-1 }{ 2 } =z\) intersect Also, find the point of intersection.

CBSE 12th Standard Maths Subject Three Dimensional Geometry Ncert Exemplar 4 Marks Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Find the vector and the Cartesian equations of the line through the point (5, 2, – 4) and which is parallel to the vector \(3 \hat{i}+2 \hat{j}-8 \hat{k}\)

  • 2)

    Show that the equation of a plane which meets the axes in A,B and C and the given centroid of triangle ABC is the point \((\alpha ,\beta ,\gamma )\) is \(\frac { x }{ \alpha } +\frac { y }{ \beta } +\frac { z }{ \gamma } =3\)

  • 3)

    If a variable line in two adjacent positions has direction cosines l,m,n,\(l+\delta l,m+\delta m,n+\delta n\) and show that the small angle \(\delta \theta \) between two positions is given by \({ (\delta \theta ) }^{ 2 }={ (\delta l) }^{ 2 }+(\delta m)^{ 2 }+{ (\delta n) }^{ 2 }\)

  • 4)

    Show that the lines \(\frac { x-1 }{ 2 } =\frac { y-2 }{ 3 } =\frac { z-3 }{ 4 } \) and \(\frac { x-4 }{ 5 } =\frac { y-1 }{ 2 } =z\) intersect Also, find the point of intersection.

  • 5)

    Find the equation of the plane which is perpendicular to the plane 5x + 3y + 6z + 8 = 0 and which contains the line of intersection of the planes  x + 2y + 3z - 4 = 0 and 2x +y - z + 5 = 0.

CBSE 12th Standard Maths Subject Three Dimensional Geometry Ncert Exemplar MCQ Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    The coordinates of the foot of the perpendicular drawn from the point (2. 5. 7) on the X-axis are given by 

  • 2)

    The locus represented by xy + yz = 0 is 

  • 3)

    If the plane \(2 x-3 y+6 z-11=0\) makes an angle \(\sin ^{-1} \alpha\) with X-axis, then the value of \(\alpha\) is

  • 4)

    The equation of X -axis in space is 

CBSE 12th Standard Maths Subject Three Dimensional Geometry Ncert Exemplar MCQ Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    The distance of the plane \(\vec{r}\left(\frac{2}{7} \hat{i}+\frac{3}{7} \hat{j}-\frac{6}{7} \hat{k}\right)=1\) from the origin is

  • 2)

    The locus represented by xy + yz = 0 is 

  • 3)

    If the plane \(2 x-3 y+6 z-11=0\) makes an angle \(\sin ^{-1} \alpha\) with X-axis, then the value of \(\alpha\) is

  • 4)

    The equation of X -axis in space is 

CBSE 12th Standard Maths Subject Vector Algebra Ncert Exemplar 4 Mark Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    If \(\overrightarrow { a } =\hat { i } -\hat { j } +7\hat { k } \) and \(\overrightarrow { b } =5\hat { i } -\hat { j } +\lambda \hat { k } \) then find the value of \(\lambda\)  so that \(\overrightarrow { a } +\overrightarrow { b } \ and\ \overrightarrow { a } -\overrightarrow { b } \) are perpendicular vectors.

  • 2)

    If \(\overrightarrow { a } \times \overrightarrow { b } =\overrightarrow { a } \times \overrightarrow { c } \ and\ \overrightarrow { a } \times \overrightarrow { c } =\overrightarrow { b } \times \overrightarrow { d } \) prove that \(\overrightarrow { a } -\overrightarrow { d } \) is parallel to \(\overrightarrow { b } -\overrightarrow { c } \) provided \(\overrightarrow { a } \neq \overrightarrow { d } \ and\ \overrightarrow { b } \neq \overrightarrow { c } \)

  • 3)

    It is given that:\(\overset { \rightarrow }{ x } =\frac { \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } ,\overset { \rightarrow }{ y } \frac { \overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } and\ \overset { \rightarrow }{ z } =\frac { \overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } \)  where \(\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ c } \) are non-coplanar vectors.

  • 4)

    If a vector \(\vec{r}\) has magnitude 14and direction ratios 2, 3 and - 6. Then, find the direction cosines and components of \(\vec{r}\)given that \(\vec{r}\) makes an acute angle with x-axis.

  • 5)

    If the three vectors \(\vec{a}, \vec{b} \text { and } \vec{c}\) are given as \(a_{1} \hat{i}+a_{2} \hat{j}+a_{3} \hat{k}, b_{1} \hat{i}+b_{2} \hat{j}+b_{3} \hat{k} \text { and } c_{1} \hat{i}+c_{2} \hat{j}+c_{3} \hat{k}\) Then, show that \(\vec{a} \times(\vec{b}+\vec{c})=(\vec{a} \times \vec{b})+(\vec{a} \times \vec{c})\).

CBSE 12th Standard Maths Subject Vector Algebra Ncert Exemplar 4 Mark Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    If \(\overrightarrow { a } =\hat { i } -\hat { j } +7\hat { k } \) and \(\overrightarrow { b } =5\hat { i } -\hat { j } +\lambda \hat { k } \) then find the value of \(\lambda\)  so that \(\overrightarrow { a } +\overrightarrow { b } \ and\ \overrightarrow { a } -\overrightarrow { b } \) are perpendicular vectors.

  • 2)

    Find a vector \(\overrightarrow { r } \) of magnitude 3\(\sqrt2\) units which makes an angle of \(\pi\over4\) and \(\pi\over2\) with y and z-axis respectively. 

  • 3)

    If \(\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } \ and\ \overset { \rightarrow }{ c } \) are perpendicular to each other, show that 

  • 4)

    If a vector \(\vec{r}\) has magnitude 14and direction ratios 2, 3 and - 6. Then, find the direction cosines and components of \(\vec{r}\)given that \(\vec{r}\) makes an acute angle with x-axis.

  • 5)

    If the three vectors \(\vec{a}, \vec{b} \text { and } \vec{c}\) are given as \(a_{1} \hat{i}+a_{2} \hat{j}+a_{3} \hat{k}, b_{1} \hat{i}+b_{2} \hat{j}+b_{3} \hat{k} \text { and } c_{1} \hat{i}+c_{2} \hat{j}+c_{3} \hat{k}\) Then, show that \(\vec{a} \times(\vec{b}+\vec{c})=(\vec{a} \times \vec{b})+(\vec{a} \times \vec{c})\).

CBSE 12th Standard Maths Subject Vector Algebra Ncert Exemplar 3 Mark Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Find all vectors of magnitude 10\(\sqrt { 3 }\)   that are perpendicular to the plane of:
    \(\overset { \wedge }{ i } +2\overset { \wedge }{ j } +\overset { \wedge }{ k } \ and\ -\overset { \wedge }{ i } +3\overset { \wedge }{ j } +4\overset { \wedge }{ k } \) 

  • 2)

    Using vector, find the value of 'k' such that the point: (k, -10, 3), (1, -1, 3) and (3, 5, 3) are collinear.

  • 3)

    If A,B,C are position vectors:  \(\hat{i}+\hat{j}-\hat{k}, 2 \hat{i}-\hat{j}+3 \hat{k}, \hat{i}-2 \hat{j}+\hat{k}\)
    Respectively, find the projection of \(\overset { \rightarrow }{ AB } \) along \(\overset { \rightarrow }{ CD } \).

  • 4)

    If A and B are two points vectors and respectively. write the position vectors of a point of a P, which divides the line segment AB internally in the ratio 1 : 2

  • 5)

    If \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \) are perpendicular vectors, \(|\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } |=13\) and \(\overset { \rightarrow }{ a } \) = 5. Find the value of  \(\overset { \rightarrow }{ b } \)

CBSE 12th Standard Maths Subject Vector Algebra Ncert Exemplar 3 Mark Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Find all vectors of magnitude 10\(\sqrt { 3 }\)   that are perpendicular to the plane of:
    \(\overset { \wedge }{ i } +2\overset { \wedge }{ j } +\overset { \wedge }{ k } \ and\ -\overset { \wedge }{ i } +3\overset { \wedge }{ j } +4\overset { \wedge }{ k } \) 

  • 2)

    Using vector, find the value of 'k' such that the point: (k, -10, 3), (1, -1, 3) and (3, 5, 3) are collinear.

  • 3)

    If A,B,C are position vectors:  \(\hat{i}+\hat{j}-\hat{k}, 2 \hat{i}-\hat{j}+3 \hat{k}, \hat{i}-2 \hat{j}+\hat{k}\)
    Respectively, find the projection of \(\overset { \rightarrow }{ AB } \) along \(\overset { \rightarrow }{ CD } \).

  • 4)

    If \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \) are perpendicular vectors, \(|\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } |=13\) and \(\overset { \rightarrow }{ a } \) = 5. Find the value of  \(\overset { \rightarrow }{ b } \)

  • 5)

    If   \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \) are two unit vectorssuch that \(\overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \)  is also a unit vector, then find the angle between  \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \).

CBSE 12th Standard Maths Subject Vector Algebra Ncert Exemplar MCQ Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    The magnitude of the vector \(6 \hat{i}+2 \hat{j}+3 \hat{k}\) is

  • 2)

    The vector having initial and terminal points as \((2,5,0) \text { and }(-3,7,4)\) respectively is

  • 3)

    If \(\boldsymbol{\theta}\) is the angle between two vectors \(\vec{a} \text { and } \vec{b},\) then \(\vec{a} \cdot \vec{b} \geq 0\) only when

  • 4)

    The value of \(\lambda\) for which the vectors \(\vec{a}=2 \hat{i}+\lambda \hat{j}+\hat{k} \text { and } \vec{b}=\hat{i}+2 \hat{j}+3 \hat{k}\) are orthogonal os

  • 5)

    If \(|\vec{a}|=10,|\vec{b}|=2\) and \(\vec{a} \cdot \vec{b}=12\) 12, then the value of \(|\vec{a} \times \vec{b}|\) is 

CBSE 12th Standard Maths Subject Vector Algebra Ncert Exemplar MCQ Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    The magnitude of the vector \(6 \hat{i}+2 \hat{j}+3 \hat{k}\) is

  • 2)

    The angle between two vectors \(\vec{a} \text { and } \vec{b}\) with magnitude, \(\sqrt{3} \text { and } 4\) respectively and \(\vec{a} \cdot \vec{b}=2 \sqrt{3}\) is 

  • 3)

    If \(\boldsymbol{\theta}\) is the angle between two vectors \(\vec{a} \text { and } \vec{b},\) then \(\vec{a} \cdot \vec{b} \geq 0\) only when

  • 4)

    The value of \(\lambda\) for which the vectors \(\vec{a}=2 \hat{i}+\lambda \hat{j}+\hat{k} \text { and } \vec{b}=\hat{i}+2 \hat{j}+3 \hat{k}\) are orthogonal os

  • 5)

    If \(|\vec{a}|=10,|\vec{b}|=2\) and \(\vec{a} \cdot \vec{b}=12\) 12, then the value of \(|\vec{a} \times \vec{b}|\) is 

CBSE 12th Standard Maths Subject Integrals Ncert Exemplar 4 Mark Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Evaluate the integral: \(\int {x^3\over x^4+3x^2+2}dx.\)

  • 2)

    Evaluate the integral: \(\int{x^2\over x^4+x^2-2}dx\)

  • 3)

    Evaluate the integral: \(\int {sin^6\ x+cos^6\ x\over sin^2\ x\ cos^2\ x}dx\)

  • 4)

    Evaluate the integral: \({\sqrt x\over \sqrt{a^3-x^3}}dx\)

  • 5)

    Evaluate the following integral
    \(\int \frac{x+5}{3 x^{2}+13 x-10} d x\)

CBSE 12th Standard Maths Subject Integrals Ncert Exemplar 4 Mark Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Evaluate the integral: \(\int tan^8\ x\ sec^4\ x\ dx\)

  • 2)

    Evaluate the integral: \(\int{x^2\over x^4+x^2-2}dx\)

  • 3)

    Evaluate the integral: \(\int {\sqrt{1+x^2}\over x^4}dx\)

  • 4)

    Integrate: \(\int^{\pi}_{0}x.sin\ x\ cos^2\ x\ dx\)

  • 5)

    Integrate:\(\int^{1}_{0}{dx\over e^x+e^{-x}}\)

CBSE 12th Standard Maths Subject Integrals Ncert Exemplar 2 Mark Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Evaluate the following integral. \(\int \frac{e^{6 \log x}-e^{5 \log x}}{e^{4 \log x}-e^{3 \log x}} d x\)

  • 2)

    Evaluate the following. 
    \(\int_{0}^{\pi / 4} \sqrt{1+\sin 2 x} d x\)

  • 3)

    Evaluate the following integral.
    \(\int_{0}^{1} \frac{x}{\sqrt{1+x^{2}}} d x\)

  • 4)

    Evaluate the following integral.
    \(\int_{0}^{\pi / 2} \cos x e^{\sin x} d x\)

  • 5)

    Evaluate the following integral.
    \(\int_{0}^{1} \frac{d x}{e^{x}+e^{-x}}\)

CBSE 12th Standard Maths Subject Integrals Ncert Exemplar 2 Mark Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Evaluate the following integral. \(\int \frac{e^{6 \log x}-e^{5 \log x}}{e^{4 \log x}-e^{3 \log x}} d x\)

  • 2)

    Evaluate the following. 
    \(\int_{0}^{\pi / 4} \sqrt{1+\sin 2 x} d x\)

  • 3)

    Evaluate the following integral.
    \(\int_{0}^{1} \frac{x}{\sqrt{1+x^{2}}} d x\)

  • 4)

    Evaluate the following integral.
    \(\int_{0}^{\pi / 2} \cos x e^{\sin x} d x\)

  • 5)

    Evaluate the following integral.
    \(\int_{0}^{1} \frac{d x}{e^{x}+e^{-x}}\)

CBSE 12th Standard Maths Subject Application of Derivatives Ncert Exemplar 4 Marks Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    x and y are the sides of two squares such that y = x - x2. Find the rate of the area of second square with respect to the area of the first quadrant.

  • 2)

    Show that the function f given by f(x) = tan-1 (sin x + cos x), x > 0 is always an increasing function in \((0,{\pi\over 4})\).

  • 3)

    Find the condition for the curves \({x^2 \over a^2}-{y^2\over b^2}=1\)and xy=c2 to intersect orthogonally.

  • 4)

    Find the approximate volume of metal in a hallow spherical shell,where internal and external radii are 3cm and 3.0005cm respectively.

  • 5)

    If the sum of a side and the hypotenuse of a right-angled triangle be given, show that the area of the triangle will be maximum if the angle between the given side and the hypotenuse be 600.

CBSE 12th Standard Maths Subject Application of Derivatives Ncert Exemplar 4 Marks Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    A spherical ball of salt is dissolving in water such a manner that the rate of decrease of the volume at any instant is proportional to the surface .Prove that the radius is decreasing at a constant rate.

  • 2)

    x and y are the sides of two squares such that y = x - x2. Find the rate of the area of second square with respect to the area of the first quadrant.

  • 3)

    Show that \({x\over a}+{y\over b}=1\) touches the curve \(y=be^{-{x\over a}}\) at the point where curve crosses the Y-axis.

  • 4)

    Prove that the curves xy=4 and x2+y2=8 touch each other.

  • 5)

    Find the equation of tangents to the curve y=cos(x+y),-2\(\pi \le x\le 2\pi \)that are parallel to the line x+2y=0.

CBSE 12th Maths Application of Derivatives Ncert Exemplar 2 Marks Questions With Solution - by Shalini Sharma - Udaipur View & Read

  • 1)

    Find an angle \(\theta\) which increases twice as fast as its sine.

  • 2)

    If the area of a circle increase at a uniform rate, then prove that perimeter varies inversely as the radius.

  • 3)

    The volume of a cube increases at a constant rate. Prove that the increase in its surface area varies inversely as the length of the side.

  • 4)

    Aspherical ball of salt is dissolving in water in such a manner that the rate of decreasing of the volume at any instant is proportional to the surface. Prove that the radius is decreasing at a constant rate.

  • 5)

    Show that \(f(x)=2 x+\cot ^{-1} x+\log \left(\sqrt{1}+x^{2}-x\right)\) is increasing in R.

CBSE 12th Standard Maths Subject Application of Derivatives Ncert Exemplar 2 Marks Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Find an angle \(\theta\) which increases twice as fast as its sine.

  • 2)

    If the area of a circle increase at a uniform rate, then prove that perimeter varies inversely as the radius.

  • 3)

    For the curve y = 5x - 2x3, if x increase at therate of 2 units/s, then find the rate of change of the slope of curve changing when x = 3.

  • 4)

    Aspherical ball of salt is dissolving in water in such a manner that the rate of decreasing of the volume at any instant is proportional to the surface. Prove that the radius is decreasing at a constant rate.

  • 5)

    Show that \(f(x)=2 x+\cot ^{-1} x+\log \left(\sqrt{1}+x^{2}-x\right)\) is increasing in R.

CBSE 12th Standard Maths Subject Application of Derivatives Ncert Exemplar MCQ Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    A ladder, 5 m long, standing on a horizontal floor, leans against a vertical wall. If the top of the ladder slides downwards at the rate of 10 cm/s, then the rate at which the angle between the floor and the ladder is decreasing when lower end of ladder is 2 m from the wall is

  • 2)

    If \(y=x(x-3)^{2}\) decreases for the values of x given by 

  • 3)

    The function \(f(x)=\tan x-x\) 

  • 4)

    Which of the following functions is decreasing on \(\left(0, \frac{\pi}{2}\right)\)?

  • 5)

    The tangent to the curve \(y=e^{2 x}\) at the point (0,1) meets X-axis at

CBSE 12th Standard Maths Subject Application of Derivatives Ncert Exemplar MCQ Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    A ladder, 5 m long, standing on a horizontal floor, leans against a vertical wall. If the top of the ladder slides downwards at the rate of 10 cm/s, then the rate at which the angle between the floor and the ladder is decreasing when lower end of ladder is 2 m from the wall is

  • 2)

    The interval on which the function \(f(x)=2 x^{3}+9 x^{2}+12 x-1\) is decreasing is

  • 3)

    If \(y=x(x-3)^{2}\) decreases for the values of x given by 

  • 4)

    The function \(f(x)=\tan x-x\) 

  • 5)

    Which of the following functions is decreasing on \(\left(0, \frac{\pi}{2}\right)\)?

CBSE 12th Standard Maths Subject Continuity and Differentiability Ncert Exemplar 4 Marks Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Find f′(x) if f (x) = (sin x)sin x for all 0 < x < π.

  • 2)

    Find the value of k such that the function
    \(f(x)=\begin{cases} \frac { { 2 }^{ x+2 }-16| }{ { 4 }^{ x }-16 } \ , \ \ if\quad x\neq 2 \\ \quad k\quad \quad , \ \ if\quad x=2 \end{cases}\)is continuous at x=2

  • 3)

    Find the derivative of each of the following function w.r.t. x, or find \(\frac { dy }{ dx } \):
    \(y={ sin }^{ -1 }\left[ x\sqrt { 1-x } -\sqrt { x } \sqrt { 1-{ x }^{ 2 } } \right] \)

  • 4)

    If xpyq = (x + y)p+q, prove that \((i)\frac { dy }{ dx } =\frac { y }{ x } and\quad (ii)\frac { { d }^{ 2 }y }{ { d }x^{ 2 } } =0.\)

  • 5)

    Verify MVT for the following functions:
    f (x) =\(\frac { 1 }{ 4x-1 } \) in [1,4].

CBSE 12th Standard Maths Subject Continuity and Differentiability Ncert Exemplar 4 Marks Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Find the values of a and b such that the function f defined by
    \(f(x)=\begin{cases} \frac { x-4 }{ |x-4|\quad } +a \ \ \ \ , if\quad x<4 \\ \quad a+b\quad \ \ \ \ \ \ \ , \ if\quad x=4\quad is\quad a\quad continuous\quad function\quad function\quad at\quad x=4 \\ \frac { x-4 }{ |x-4| } +b \ \ \ \ \ \ \ \ \ , \ if\quad x>4 \end{cases}\)

  • 2)

    Differentiate w.r.t. x or find \(\frac { dy }{ dx } \)\({ 2 }^{ cos^{ 2 }x }\)

  • 3)

    Differentiate w.r.t. x or find \(\frac { dy }{ dx } \)\(x={ e }^{ \theta }\left( \theta +\frac { 1 }{ \theta } \right) ,y={ e }^{ -\theta }\left( \theta -\frac { 1 }{ \theta } \right) \)

  • 4)

    If y= tan-1 x , find \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \)in terms of y alone.

  • 5)

    Show that the function \(f(x)=\left\{\begin{array}{c} \frac{e^{1 / x}-1}{e^{1 / x}+1}, \text { when } x \neq 0 \\ 0, \quad \text { when } x=0 \end{array}\right.\) is discontinuous at x = 0.

CBSE 12th Standard Maths Subject Matrices Ncert Exemplar 4 Marks Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    If X and Y are \(2\times 2\) matrices, then solve the following matrix equation of X and Y. 
    \(2X+3Y=\begin{bmatrix} 2 & 3 \\ 4 & 0 \end{bmatrix},3X+2Y=\begin{bmatrix} -2 & 2 \\ 1 & -5 \end{bmatrix}\)

  • 2)

    If A is a square matrix such that A2 = A, show that (l + A)3 = 7A + l

  • 3)

    Let \(A=\begin{bmatrix} 2 & 3 \\ -1 & 2 \end{bmatrix}\), then show that \({ A }^{ 2 }-4A+7l=0\). Using this result, calculate \({ A }^{ 5 }\) also.

  • 4)

    Find the matrix A, satisfying the matrix equation

    \(\begin{bmatrix} 2 & 1 \\ 3 & 2 \end{bmatrix}A\begin{bmatrix} -3 & 2 \\ 5 & -3 \end{bmatrix}=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\)

  • 5)

    Find the matrix X so that X \(\left[ \begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix} \right] =\left[ \begin{matrix} -7 & -8 & -9 \\ 2 & 4 & 6 \end{matrix} \right] .\)

CBSE 12th Standard Maths Subject Matrices Ncert Exemplar 4 Marks Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    If A is a square matrix such that A2 = A, show that (l + A)3 = 7A + l

  • 2)

    Let \(A=\begin{bmatrix} 2 & 3 \\ -1 & 2 \end{bmatrix}\), then show that \({ A }^{ 2 }-4A+7l=0\). Using this result, calculate \({ A }^{ 5 }\) also.

  • 3)

    Find the matrix X so that X \(\left[ \begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix} \right] =\left[ \begin{matrix} -7 & -8 & -9 \\ 2 & 4 & 6 \end{matrix} \right] .\)

  • 4)

    Express the matrix \(A=\left[\begin{array}{rrr}2 & 4 & -6 \\ 7 & 3 & 5 \\ 1 & -2 & 4\end{array}\right]\) as the sum of a symmetric and a skew-symmetric matrices.

CBSE 12th Standard Maths Subject Matrices Ncert Exemplar 2 Marks Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    If \(A=\left[ \begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix} \right] \) then prove that \({ A }^{ n }=\left[ \begin{matrix} { 3 }^{ n-1 } & { 3 }^{ n-1 } & { 3 }^{ n-1 } \\ { 3 }^{ n-1 } & { 3 }^{ n-1 } & { 3 }^{ n-1 } \\ { 3 }^{ n-1 } & { 3 }^{ n-1 } & { 3 }^{ n-1 } \end{matrix} \right] ,\) then \(n\epsilon N\).
     

  • 2)

    In the matrix,\(A=\left[\begin{array}{ccc} a & 1 & x \\ 2 & \sqrt{3} & x^{2}-y \\ 0 & 5 & -2 / 5 \end{array}\right]\) 
    (i) the order of the matrix A.
    (ii) the number of elements.
    (iii) the value of elements a23, a31 and a12

  • 3)

    \(\text { If }\left[\begin{array}{ll} 2 x & 3 \end{array}\right]\left[\begin{array}{rr} 1 & 2 \\ -3 & 0 \end{array}\right]\left[\begin{array}{l} x \\ 8 \end{array}\right]=0\) then find the value of x.

  • 4)

     If matrix \(\left[\begin{array}{rrr}0 & a & 3 \\ 2 & b & -1 \\ c & 1 & 0\end{array}\right]\) is a skew-symmetric matrix, then find the values of a, b and c

  • 5)

    If A and B are symmetric matrices, then prove that BA - 2AB is neither a symmetric matrix nor skew-symmetric matrix.

CBSE 12th Standard Maths Subject Matrices Ncert Exemplar 2 Marks Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    If \(A=\left[ \begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix} \right] \) then prove that \({ A }^{ n }=\left[ \begin{matrix} { 3 }^{ n-1 } & { 3 }^{ n-1 } & { 3 }^{ n-1 } \\ { 3 }^{ n-1 } & { 3 }^{ n-1 } & { 3 }^{ n-1 } \\ { 3 }^{ n-1 } & { 3 }^{ n-1 } & { 3 }^{ n-1 } \end{matrix} \right] ,\) then \(n\epsilon N\).
     

  • 2)

    In the matrix,\(A=\left[\begin{array}{ccc} a & 1 & x \\ 2 & \sqrt{3} & x^{2}-y \\ 0 & 5 & -2 / 5 \end{array}\right]\) 
    (i) the order of the matrix A.
    (ii) the number of elements.
    (iii) the value of elements a23, a31 and a12

  • 3)

    Show that if A and B are square matrices such that AB = BA, then (A + B)2 = A2 + 2AB + B2 .

  • 4)

    If A and B are symmetric matrices, then prove that BA - 2AB is neither a symmetric matrix nor skew-symmetric matrix.

CBSE 12th Standard Maths SUbject Matrices Ncert Exemplar MCQ Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    If A and B are square matrices of the sameorder, then (A + B) (A - B) is equal to

  • 2)

    If A is matrix of order m x nand B is a matrix such that AB' and B' A are both defined, then order of matrix B is

  • 3)

    The matrix \(\left[\begin{array}{ccc}0 & -5 & 8 \\ 5 & 0 & 12 \\ -8 & -12 & 0\end{array}\right]\) is a

  • 4)

    If matrix \(A=\left[a_{i j}\right]_{2 \times 2},\ where \ a_{i j}=\left\{\begin{array}{l}1, \text { if } i \neq j \\ 0, \text { if } i=j\end{array}\right.\) Then \(A^{2}\) is equal to 

  • 5)

    For any two matrices A and B, we have

CBSE 12th Standard Maths Subject Matrices Ncert Exemplar MCQ Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    If A and B are two matrices of the order \(3 \times m\) and \(3 \times n\) respectively and m=n, then the order of the matrix \((5 A-2 B)\) is 

  • 2)

    On using elementary column operations \(C_{2} \rightarrow C_{2}-2 C_{1}\) in the following matrix equation \(\left[\begin{array}{cc}1 & -3 \\ 2 & 4\end{array}\right]=\left[\begin{array}{cc}1 & -1 \\ 0 & 1\end{array}\right]\left[\begin{array}{ll}3 & 1 \\ 2 & 4\end{array}\right]\), we have

  • 3)

    On using elementary row operation \(R_{1} \rightarrow R_{1}-3 R_{2}\) in the following matrix equation \(\left[\begin{array}{ll}4 & 2 \\ 3 & 3\end{array}\right]=\left[\begin{array}{ll}1 & 2 \\ 0 & 3\end{array}\right]\left[\begin{array}{ll}2 & 0 \\ 1 & 1\end{array}\right],\) we have

  • 4)

    If matrix \(A=\left[a_{i j}\right]_{2 \times 2},\ where \ a_{i j}=\left\{\begin{array}{l}1, \text { if } i \neq j \\ 0, \text { if } i=j\end{array}\right.\) Then \(A^{2}\) is equal to 

  • 5)

    For any two matrices A and B, we have

CBSE 12th Standard Maths Subject Determinants Ncert Exemplar 6 Marks Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Show that triangle ABC is an isosceles triangle,  if the determinant
    \( \triangle =\left| \begin{matrix} 1 & 1 & 1 \\ 1+cosA & 1+cosB & 1+cosC \\ \cos ^{ 2 }{ A } +cosA & \cos ^{ 2 }{ B } +cosB & \cos ^{ 2 }{ C } +cosC \end{matrix} \right| \)

  • 2)

    If A = \(\left[ \begin{matrix} 1 & -1 & 0 \\ 2 & 3 & 4 \\ 0 & 1 & 2 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 2 & 2 & -4 \\ -4 & 2 & -4 \\ 2 & -1 & 5 \end{matrix} \right] \) are square matrices, find A, B and hence solve the system of equation:
    x - y = 3, 2x + 3y + 4z = 17 and y + 2z = 7

  • 3)

    Using the properties of determinants, prove that 
    \(\left|\begin{array}{ccc} (b+c)^{2} & a^{2} & a^{2} \\ b^{2} & (c+a)^{2} & b^{2} \\ c^{2} & c^{2} & (a+b)^{2} \end{array}\right|=2 a b c(a+b+c)^{3}\)

  • 4)

    If \(A=\left[\begin{array}{rrr} 1 & 2 & 0 \\ -2 & -1 & -2 \\ 0 & -1 & 1 \end{array}\right]\) then findA-1.using A-1,equations x - 2y = 10,,2x - y- z = 8 and - 2y +z = 7.

CBSE 12th Standard Maths Subject Determinants Ncert Exemplar 6 Marks Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Show that triangle ABC is an isosceles triangle,  if the determinant
    \( \triangle =\left| \begin{matrix} 1 & 1 & 1 \\ 1+cosA & 1+cosB & 1+cosC \\ \cos ^{ 2 }{ A } +cosA & \cos ^{ 2 }{ B } +cosB & \cos ^{ 2 }{ C } +cosC \end{matrix} \right| \)

  • 2)

    If a + b + c \(\neq \) 0 and \(\left|\begin{array}{lll} a & b & c \\ b & c & a \\ c & a & b \end{array}\right|=0\) then prove that  a = b = c.

  • 3)

    Using the properties of determinants, prove that 
    \(\left|\begin{array}{ccc} (b+c)^{2} & a^{2} & a^{2} \\ b^{2} & (c+a)^{2} & b^{2} \\ c^{2} & c^{2} & (a+b)^{2} \end{array}\right|=2 a b c(a+b+c)^{3}\)

  • 4)

    If \(A=\left[\begin{array}{rrr} 1 & 2 & 0 \\ -2 & -1 & -2 \\ 0 & -1 & 1 \end{array}\right]\) then findA-1.using A-1,equations x - 2y = 10,,2x - y- z = 8 and - 2y +z = 7.

CBSE 12th Standard Maths Subject Determinants Ncert Exemplar 4 Marks Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Find the value of \(\theta\) satisfying \(\begin{vmatrix} 1 & 1 & sin3\theta \\ -4 & 3 & cos2\theta \\ 7 & -7 & -2 \end{vmatrix}=0\)

  • 2)

    Let \(f(t)=\left| \begin{matrix} \cos { t } & t & 1 \\ 2\sin { t } & t & 2t \\ \sin { t } & t & t \end{matrix} \right| ,\ then\ find\quad \lim _{ t\rightarrow 0 }{ \frac { f(t) }{ { t }^{ 2 } } } .\)

  • 3)

    Using the properties of determinants, prove that 
    \(\left|\begin{array}{ccc} a-b-c & 2 a & 2 a \\ 2 b & b-c-a & 2 b \\ 2 c & 2 c & c-a-b \end{array}\right|=(a+b+c)^{3}\)

  • 4)

    Using properties of determinants, prove that 
    \(\left|\begin{array}{lll} y^{2} z^{2} & y z & y+z \\ z^{2} x^{2} & z x & z+x \\ x^{2} y^{2} & x y & x+y \end{array}\right|=0\)

CBSE 12th Standard Maths Subject Determinants Ncert Exemplar 4 Marks Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Find the value of \(\theta\) satisfying \(\begin{vmatrix} 1 & 1 & sin3\theta \\ -4 & 3 & cos2\theta \\ 7 & -7 & -2 \end{vmatrix}=0\)

  • 2)

    Let \(f(t)=\left| \begin{matrix} \cos { t } & t & 1 \\ 2\sin { t } & t & 2t \\ \sin { t } & t & t \end{matrix} \right| ,\ then\ find\quad \lim _{ t\rightarrow 0 }{ \frac { f(t) }{ { t }^{ 2 } } } .\)

  • 3)

    Using the properties of determinants, prove that 
    \(\left|\begin{array}{ccc} a-b-c & 2 a & 2 a \\ 2 b & b-c-a & 2 b \\ 2 c & 2 c & c-a-b \end{array}\right|=(a+b+c)^{3}\)

  • 4)

    Using properties of determinants, prove that 
    \(\left|\begin{array}{lll} y^{2} z^{2} & y z & y+z \\ z^{2} x^{2} & z x & z+x \\ x^{2} y^{2} & x y & x+y \end{array}\right|=0\)

CBSE 12th Standard Maths Subject Determinants Ncert Exemplar 2 Marks Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    If there are two values of a which makes determinant,\(\begin{equation} \Delta=\left|\begin{array}{rrr} 1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2 a \end{array}\right|=86 \end{equation}\) then find the sum of these numbers.

  • 2)

    If A is a matrix of order 2 x 2, then find the value of (A3)-1.

  • 3)

    If \(f(x)=\left|\begin{array}{lll} (1+x)^{17} & (1+x)^{19} & (1+x)^{23} \\ (1+x)^{23} & (1+x)^{29} & (1+x)^{34} \\ (1+x)^{41} & (1+x)^{43} & (1+x)^{47} \end{array}\right|\) = A + Bx +Cx2 +..., then find the value of A.

CBSE 12th Standard Maths Subject Determinants Ncert Exemplar 2 Marks Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    If there are two values of a which makes determinant,\(\begin{equation} \Delta=\left|\begin{array}{rrr} 1 & -2 & 5 \\ 2 & a & -1 \\ 0 & 4 & 2 a \end{array}\right|=86 \end{equation}\) then find the sum of these numbers.

  • 2)

    If A is a matrix of order 2 x 2, then find the value of (A3)-1.

  • 3)

    If \(f(x)=\left|\begin{array}{lll} (1+x)^{17} & (1+x)^{19} & (1+x)^{23} \\ (1+x)^{23} & (1+x)^{29} & (1+x)^{34} \\ (1+x)^{41} & (1+x)^{43} & (1+x)^{47} \end{array}\right|\) = A + Bx +Cx2 +..., then find the value of A.

CBSE 12th Standard Maths Subject Determinants Ncert Exemplar MCQ Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Let \(\Delta=\left|\begin{array}{lll} A x & x^{2} & 1 \\ B y & y^{2} & 1 \\ C z & z^{2} & 1 \end{array}\right| \text { and } \Delta_{1}=\left|\begin{array}{ccc} A & B & C \\ x & y & z \\ z y & z x & x y \end{array}\right|\) then

  • 2)

    Iff \(f(x)=\left|\begin{array}{ccc} 0 & x-a & x-b \\ x+a & 0 & x-c \\ x+b & x+c & 0 \end{array}\right|\) then

  • 3)

    If \(A=\left|\begin{array}{llr} 2 & \lambda & -3 \\ 0 & 2 & 5 \\ 1 & 1 & 3 \end{array}\right|\) then A-I exists, if

  • 4)

    If \(f(t)=\left[\begin{array}{ccc} \cos t & t & 1 \\ 2 \sin t & t & 2 t \\ \sin t & t & t \end{array}\right] \text { , then } \lim _{t \rightarrow 0} \frac{f(t)}{t^{2}}\) is equal to

  • 5)

    If \(f(t)=\left[\begin{array}{ccc} \cos t & t & 1 \\ 2 \sin t & t & 2 t \\ \sin t & t & t \end{array}\right] \text { , then } \lim _{t \rightarrow 0} \frac{f(t)}{t^{2}}\) is equal to 

CBSE 12th Standard Maths Subject Determinants Ncert Exemplar MCQ Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Let \(\Delta=\left|\begin{array}{lll} A x & x^{2} & 1 \\ B y & y^{2} & 1 \\ C z & z^{2} & 1 \end{array}\right| \text { and } \Delta_{1}=\left|\begin{array}{ccc} A & B & C \\ x & y & z \\ z y & z x & x y \end{array}\right|\) then

  • 2)

    Iff \(f(x)=\left|\begin{array}{ccc} 0 & x-a & x-b \\ x+a & 0 & x-c \\ x+b & x+c & 0 \end{array}\right|\) then

  • 3)

    If \(A=\left|\begin{array}{llr} 2 & \lambda & -3 \\ 0 & 2 & 5 \\ 1 & 1 & 3 \end{array}\right|\) then A-I exists, if

  • 4)

    If \(f(t)=\left[\begin{array}{ccc} \cos t & t & 1 \\ 2 \sin t & t & 2 t \\ \sin t & t & t \end{array}\right] \text { , then } \lim _{t \rightarrow 0} \frac{f(t)}{t^{2}}\) is equal to

  • 5)

    If \(f(t)=\left[\begin{array}{ccc} \cos t & t & 1 \\ 2 \sin t & t & 2 t \\ \sin t & t & t \end{array}\right] \text { , then } \lim _{t \rightarrow 0} \frac{f(t)}{t^{2}}\) is equal to 

CBSE 12th Standard Maths Subject Inverse Trigonometric Functions Ncert Exemplar 4 Marks Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Which is greater tan 1 or tan-1 1? 

  • 2)

    Find the value of the expression \(sin\left( 2{ tan }^{ -1 }\frac { 1 }{ 3 } \right) +cos\left( { tan }^{ -1 }2\sqrt { 2 } \right) \)

  • 3)

    If \({ a }_{ 1 },{ a }_{ 2 },{ a }_{ 3 }.........{ a }_{ n }\) is and arithmetic progressive with common difference d, then evaluate,
     \(tan\left[ { tan }^{ -1 }\left( \frac { d }{ 1+{ a }_{ 1 }{ a }_{ 2 } } \right) +{ tan }^{ -1 }\left( \frac { d }{ 1+{ a }_{ 2 }{ a }_{ 3 } } \right) +{ tan }^{ -1 }\left( \frac { d }{ 1+{ a }_{ 3 }{ a }_{ 4 } } \right) +.....{ tan }^{ -1 }\left( \frac { d }{ 1+{ a }_{ n-1 }{ a }_{ n } } \right) \right] \)

CBSE 12th Standard Maths Subject Inverse Trigonometric Functions Ncert Exemplar 4 Marks Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Which is greater tan 1 or tan-1 1? 

  • 2)

    Find the value of the expression \(sin\left( 2{ tan }^{ -1 }\frac { 1 }{ 3 } \right) +cos\left( { tan }^{ -1 }2\sqrt { 2 } \right) \)

  • 3)

    If \({ a }_{ 1 },{ a }_{ 2 },{ a }_{ 3 }.........{ a }_{ n }\) is and arithmetic progressive with common difference d, then evaluate,
     \(tan\left[ { tan }^{ -1 }\left( \frac { d }{ 1+{ a }_{ 1 }{ a }_{ 2 } } \right) +{ tan }^{ -1 }\left( \frac { d }{ 1+{ a }_{ 2 }{ a }_{ 3 } } \right) +{ tan }^{ -1 }\left( \frac { d }{ 1+{ a }_{ 3 }{ a }_{ 4 } } \right) +.....{ tan }^{ -1 }\left( \frac { d }{ 1+{ a }_{ n-1 }{ a }_{ n } } \right) \right] \)

CBSE 12th Standard Maths Subject Relations and Functions Ncert Exemplar 3 Marks Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Let A = {0,1, 2, 3} and define a relation R on A as follows:
    R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)}. Is R Reflexive? Symmetric? Transitive?

  • 2)

    Let f : \(R\rightarrow R\) be defined by \(f(X)=X^{ 2 }+1\) Find the pre-image of
    (i) 17
    (ii) -3.

  • 3)

    Let the function f:\(R\rightarrow R\) to be defined by:
    \(f(x)=cosx\) for all \(x\in R\).
    Show that 'f' is neither one-one nor onto.

  • 4)

    In the set of natural numbers N, define a relation R as follows:
    \(\forall n,m\in N,\ nRm\) if on division by 5 each of the integers \(n\) leaves the remainder less than 5 i.e. one of the numbers 0,1,2,3 and 4. Show that R is equivalence relation. Also, obtain the pairwise disjoint subsets determined by R.

  • 5)

    Is the binary operation '*' defined on Z (set of integers) by:
    \(m*n=m-n+mn\quad for\quad all\quad m,n\in Z\) commutative?

CBSE 12th Standard Maths Subject Relations and Functions Ncert Exemplar 3 Marks Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    If A = {1,2,3} and f,g are relations corresponding to the subset \(A\times A\) indicated against them, which of f,g is a function? why?
    f = {(1, 3) (2, 3), (3, 2)}; g = {(1, 2), (1, 3), (3, 1)}.

  • 2)

    Let A = {0,1, 2, 3} and define a relation R on A as follows:
    R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)}. Is R Reflexive? Symmetric? Transitive?

  • 3)

    If f = {(5, 2), (6, 3)}, g = {(2, 5), (3, 6)}, write fog.

  • 4)

    Let f : \(R\rightarrow R\) be defined by \(f(X)=X^{ 2 }+1\) Find the pre-image of
    (i) 17
    (ii) -3.

  • 5)

    Let the function f:\(R\rightarrow R\) to be defined by:
    \(f(x)=cosx\) for all \(x\in R\).
    Show that 'f' is neither one-one nor onto.

CBSE 12th Standard Maths Subject Relations and Functions Ncert Exemplar MCQ Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    If a relation R on the set {1,2, 3} be defined by R = {(1, 2)}, then R is

  • 2)

    For the set A = {1, 2, 3}, define a relation R in the set A as follows
    R = {(1, 1), (2,2), (3, 3), (1, 3)}
    Then, the ordered pair to be added to R to make it the smallest equivalence relation is

  • 3)

    If the set A contains 5 elements and the set B contains 6 elements, then the number of one-one and onto mappings from A to B is

  • 4)

     Let \(f: R \rightarrow R\) be the functions defined by \(f(x)=x^{3}+5\). Then, \(f^{-1}(x)\) is 

CBSE 12th Standard Maths Subject Relations and Functions Ncert Exemplar MCQ Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    If a relation R on the set {1,2, 3} be defined by R = {(1, 2)}, then R is

  • 2)

    For the set A = {1, 2, 3}, define a relation R in the set A as follows
    R = {(1, 1), (2,2), (3, 3), (1, 3)}
    Then, the ordered pair to be added to R to make it the smallest equivalence relation is

  • 3)

    If the set A contains 5 elements and the set B contains 6 elements, then the number of one-one and onto mappings from A to B is

  • 4)

     Let \(f: R \rightarrow R\) be the functions defined by \(f(x)=x^{3}+5\). Then, \(f^{-1}(x)\) is 

CBSE 12th Standard Maths Subject Ncert Exemplar 6 Marks Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    AB is a diameter of a circle and C is any point on the circle. Show that the area of ΔABC is maximum when it is isosceles.

  • 2)

    If the sum of the lengths of the hypotenuse and a side of a right-angled triangle is given, show that the area of triangle is maximum when the angle between them is \(60^\circ\)
    \((i.e., \frac{\pi}{3})\)

  • 3)

    Using properties of determinants, show that triangle ABC is isosceles if: 
    \(\left| \begin{matrix} 1 & 1 & 1 \\ 1+cosA & 1+cosB & 1+cosC \\ \cos ^{ 2 }{ A } +cosA & \cos ^{ 2 }{ B } +cosB & \cos ^{ 2 }{ B } +cosC \end{matrix} \right| =0\)

  • 4)

    Show that triangle ABC is an isosceles triangle,  if the determinant
    \( \triangle =\left| \begin{matrix} 1 & 1 & 1 \\ 1+cosA & 1+cosB & 1+cosC \\ \cos ^{ 2 }{ A } +cosA & \cos ^{ 2 }{ B } +cosB & \cos ^{ 2 }{ C } +cosC \end{matrix} \right| \)

  • 5)

    A company manufactures two types of sweaters, type A and B. It costs Rs. 360 to make one unit of type A and Rs. 120 to make a unit of type B. The company can make at most 300 sweaters and can spend Rs. 72,000 a day. The number of sweaters of type A cannot exceed the number of type B by more than 100. The company makes a profit of Rs 200 on each unit of type A. The company charging a nominal profit of Rs. 20 on a unit of type B. Using LPP, solve for max. profit.

CBSE 12th Standard Maths Subject Ncert Exemplar 6 Marks Questions 2021 part - II - by Shalini Sharma - Udaipur View & Read

  • 1)

    A metal box with a square base and vertical sides is to contain 1024 cm3. The material for the top and bottom costs Rs. 5 per cm2 and the material for the sides cost Rs. 2.50/cm2. Find the least cost of the box.

  • 2)

    AB is a diameter of a circle and C is any point on the circle. Show that the area of ΔABC is maximum when it is isosceles.

  • 3)

    If the sum of the lengths of the hypotenuse and a side of a right-angled triangle is given, show that the area of triangle is maximum when the angle between them is \(60^\circ\)
    \((i.e., \frac{\pi}{3})\)

  • 4)

    Using properties of determinants, show that triangle ABC is isosceles if: 
    \(\left| \begin{matrix} 1 & 1 & 1 \\ 1+cosA & 1+cosB & 1+cosC \\ \cos ^{ 2 }{ A } +cosA & \cos ^{ 2 }{ B } +cosB & \cos ^{ 2 }{ B } +cosC \end{matrix} \right| =0\)

  • 5)

    Show that triangle ABC is an isosceles triangle,  if the determinant
    \( \triangle =\left| \begin{matrix} 1 & 1 & 1 \\ 1+cosA & 1+cosB & 1+cosC \\ \cos ^{ 2 }{ A } +cosA & \cos ^{ 2 }{ B } +cosB & \cos ^{ 2 }{ C } +cosC \end{matrix} \right| \)

CBSE 12th Standard Maths Subject Ncert Exemplar 6 Marks Questions 2021 part - I - by Shalini Sharma - Udaipur View & Read

  • 1)

    A metal box with a square base and vertical sides is to contain 1024 cm3. The material for the top and bottom costs Rs. 5 per cm2 and the material for the sides cost Rs. 2.50/cm2. Find the least cost of the box.

  • 2)

    AB is a diameter of a circle and C is any point on the circle. Show that the area of ΔABC is maximum when it is isosceles.

  • 3)

    If the sum of the lengths of the hypotenuse and a side of a right-angled triangle is given, show that the area of triangle is maximum when the angle between them is \(60^\circ\)
    \((i.e., \frac{\pi}{3})\)

  • 4)

    Using properties of determinants, show that triangle ABC is isosceles if: 
    \(\left| \begin{matrix} 1 & 1 & 1 \\ 1+cosA & 1+cosB & 1+cosC \\ \cos ^{ 2 }{ A } +cosA & \cos ^{ 2 }{ B } +cosB & \cos ^{ 2 }{ B } +cosC \end{matrix} \right| =0\)

  • 5)

    Show that triangle ABC is an isosceles triangle,  if the determinant
    \( \triangle =\left| \begin{matrix} 1 & 1 & 1 \\ 1+cosA & 1+cosB & 1+cosC \\ \cos ^{ 2 }{ A } +cosA & \cos ^{ 2 }{ B } +cosB & \cos ^{ 2 }{ C } +cosC \end{matrix} \right| \)

CBSE 12th Standard Maths Subject Ncert Exemplar 4 Marks Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Show that the function f : R \(\rightarrow\)R defined by f(x) = \({x\over{x^2+1}}{\forall \ x\in R}\) is neither one-one nor onto.

  • 2)

    Which is greater tan 1 or tan-1 1? 

  • 3)

    Find the value of the expression \(sin\left( 2{ tan }^{ -1 }\frac { 1 }{ 3 } \right) +cos\left( { tan }^{ -1 }2\sqrt { 2 } \right) \)

  • 4)

    A spherical ball of salt is dissolving in water such a manner that the rate of decrease of the volume at any instant is proportional to the surface .Prove that the radius is decreasing at a constant rate.

CBSE 12th Standard Maths Subject Ncert Exemplar 4 Marks Questions 2021 part - II - by Shalini Sharma - Udaipur View & Read

  • 1)

    Let A = {1, 2, 3,.......,9} and R be the relation in A x A  defined by (a,b)R(c,d) if a+d = b+c for (a,b),(c,d) \(\in\) A x A. Prove that R is an equivalence relation also obtain the equivalence class[(2, 5)]

  • 2)

    Which is greater tan 1 or tan-1 1? 

  • 3)

    Find the value of the expression \(sin\left( 2{ tan }^{ -1 }\frac { 1 }{ 3 } \right) +cos\left( { tan }^{ -1 }2\sqrt { 2 } \right) \)

  • 4)

    If X and Y are \(2\times 2\) matrices, then solve the following matrix equation of X and Y. 
    \(2X+3Y=\begin{bmatrix} 2 & 3 \\ 4 & 0 \end{bmatrix},3X+2Y=\begin{bmatrix} -2 & 2 \\ 1 & -5 \end{bmatrix}\)

  • 5)

    If A is a square matrix such that A2 = A, show that (l + A)3 = 7A + l

CBSE 12th Standard Maths Subject Ncert Exemplar 4 Marks Questions 2021 part - I - by Shalini Sharma - Udaipur View & Read

  • 1)

    Show that the function f : R \(\rightarrow\)R defined by f(x) = \({x\over{x^2+1}}{\forall \ x\in R}\) is neither one-one nor onto.

  • 2)

    Which is greater tan 1 or tan-1 1? 

  • 3)

    Find the value of the expression \(sin\left( 2{ tan }^{ -1 }\frac { 1 }{ 3 } \right) +cos\left( { tan }^{ -1 }2\sqrt { 2 } \right) \)

  • 4)

    A spherical ball of salt is dissolving in water such a manner that the rate of decrease of the volume at any instant is proportional to the surface .Prove that the radius is decreasing at a constant rate.

CBSE 12th Standard Maths Subject Ncert Exemplar 3 Marks Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Show that \(\left| \begin{matrix} p & p & q \\ p & x & q \\ q & q & x \end{matrix} \right| =(x-p)(x^{ 2 }+px-2q^{ 2 })\)

  • 2)

    If A = {1,2,3} and f,g are relations corresponding to the subset \(A\times A\) indicated against them, which of f,g is a function? why?
    f = {(1, 3) (2, 3), (3, 2)}; g = {(1, 2), (1, 3), (3, 1)}.

  • 3)

    Let A = {0,1, 2, 3} and define a relation R on A as follows:
    R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)}. Is R Reflexive? Symmetric? Transitive?

  • 4)

    If f = {(5, 2), (6, 3)}, g = {(2, 5), (3, 6)}, write fog.

  • 5)

    In the set of natural numbers N, define a relation R as follows:
    \(\forall n,m\in N,\ nRm\) if on division by 5 each of the integers \(n\) leaves the remainder less than 5 i.e. one of the numbers 0,1,2,3 and 4. Show that R is equivalence relation. Also, obtain the pairwise disjoint subsets determined by R.

CBSE 12th Standard Maths Subject Ncert Exemplar 3 Marks Questions 2021 part - II - by Shalini Sharma - Udaipur View & Read

  • 1)

    Show that \(\left| \begin{matrix} p & p & q \\ p & x & q \\ q & q & x \end{matrix} \right| =(x-p)(x^{ 2 }+px-2q^{ 2 })\)

  • 2)

    Let A = {0,1, 2, 3} and define a relation R on A as follows:
    R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)}. Is R Reflexive? Symmetric? Transitive?

  • 3)

    If f = {(5, 2), (6, 3)}, g = {(2, 5), (3, 6)}, write fog.

  • 4)

    Let the function f:\(R\rightarrow R\) to be defined by:
    \(f(x)=cosx\) for all \(x\in R\).
    Show that 'f' is neither one-one nor onto.

  • 5)

    Let '*' be the binary operation defined on R by:
    a*b=1+ab   \(\forall a,b,\in R\).
    Then the operation '*' is:
    (i) commutative but not associative
    (ii) associative but not commutative
    (iii) neither commutative nor associative
    (iv) both commutative nor associative

CBSE 12th Standard Maths Subject Ncert Exemplar 3 Marks Questions 2021 part - I - by Shalini Sharma - Udaipur View & Read

  • 1)

    If \(\Delta =\left| \begin{matrix} 1 & x & x^{ 2 } \\ 1 & y & { y }^{ 2 } \\ 1 & z & { z }^{ 2 } \end{matrix} \right| ,{ \Delta }_{ 1 }=\left| \begin{matrix} 1 & 1 & 1 \\ yz & zx & xy \\ x & y & z \end{matrix} \right| \)then prove that \(\Delta+\Delta_1=0\)

  • 2)

    Let A = {0,1, 2, 3} and define a relation R on A as follows:
    R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)}. Is R Reflexive? Symmetric? Transitive?

  • 3)

    If the mappings f and g are given by:
    f = {(1, 2), (3, 5) (4, 1) and g = {(2, 3), (5, 1), (1, 3)}, write fog.

  • 4)

    Is the binary operation '*' defined on Z (set of integers) by:
    \(m*n=m-n+mn\quad for\quad all\quad m,n\in Z\) commutative?

  • 5)

    Find the area of the region bounded by the parabola y2 = 2x and the straight line x - y = 4. 

CBSE 12th Standard Maths Subject Ncert Exemplar 2 Marks Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    If \(A=\left[ \begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix} \right] \) then prove that \({ A }^{ n }=\left[ \begin{matrix} { 3 }^{ n-1 } & { 3 }^{ n-1 } & { 3 }^{ n-1 } \\ { 3 }^{ n-1 } & { 3 }^{ n-1 } & { 3 }^{ n-1 } \\ { 3 }^{ n-1 } & { 3 }^{ n-1 } & { 3 }^{ n-1 } \end{matrix} \right] ,\) then \(n\epsilon N\).
     

  • 2)

    Find an angle \(\theta\) which increases twice as fast as its sine.

  • 3)

    Let A = {0, 1, 2, 3} and define a relation R on A as R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0),(3, 3)}. is R reflexive, symmetric and transitive?

  • 4)

    Find the value of \(2 \sec ^{-1} 2+\sin ^{-1}\left(\frac{1}{2}\right)\)

CBSE 12th Standard Maths Subject Ncert Exemplar 2 Marks Questions 2021 part - II - by Shalini Sharma - Udaipur View & Read

  • 1)

    If \(A=\left[ \begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix} \right] \) then prove that \({ A }^{ n }=\left[ \begin{matrix} { 3 }^{ n-1 } & { 3 }^{ n-1 } & { 3 }^{ n-1 } \\ { 3 }^{ n-1 } & { 3 }^{ n-1 } & { 3 }^{ n-1 } \\ { 3 }^{ n-1 } & { 3 }^{ n-1 } & { 3 }^{ n-1 } \end{matrix} \right] ,\) then \(n\epsilon N\).
     

  • 2)

    Find the value of \(2 \sec ^{-1} 2+\sin ^{-1}\left(\frac{1}{2}\right)\)

  • 3)

    In the matrix,\(A=\left[\begin{array}{ccc} a & 1 & x \\ 2 & \sqrt{3} & x^{2}-y \\ 0 & 5 & -2 / 5 \end{array}\right]\) 
    (i) the order of the matrix A.
    (ii) the number of elements.
    (iii) the value of elements a23, a31 and a12

CBSE 12th Standard Maths Subject Ncert Exemplar 2 Marks Questions 2021 part - I - by Shalini Sharma - Udaipur View & Read

  • 1)

    If \(A=\left[ \begin{matrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{matrix} \right] \) then prove that \({ A }^{ n }=\left[ \begin{matrix} { 3 }^{ n-1 } & { 3 }^{ n-1 } & { 3 }^{ n-1 } \\ { 3 }^{ n-1 } & { 3 }^{ n-1 } & { 3 }^{ n-1 } \\ { 3 }^{ n-1 } & { 3 }^{ n-1 } & { 3 }^{ n-1 } \end{matrix} \right] ,\) then \(n\epsilon N\).
     

  • 2)

    Find an angle \(\theta\) which increases twice as fast as its sine.

  • 3)

    Let A = {0, 1, 2, 3} and define a relation R on A as R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0),(3, 3)}. is R reflexive, symmetric and transitive?

  • 4)

    If \(\tan ^{-1} x+\tan ^{-1} y=\frac{4 \pi}{5}\), then find \(\cot ^{-1} x+\cot ^{-1} y\)

  • 5)

    Find the value of \(2 \sec ^{-1} 2+\sin ^{-1}\left(\frac{1}{2}\right)\)

CBSE 12th Standard Maths Subject Ncert Exemplar Fill Up Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Let the function f : R\(\rightarrow\)R to be defined by f(x) = cos x \(\forall \) x \(\in\)R. Show that  is neither one-one nor onto.

CBSE 12th Standard Maths Subject Ncert Exemplar Fill Up Questions 2021 part - II - by Shalini Sharma - Udaipur View & Read

  • 1)

    Let the function f : R\(\rightarrow\)R to be defined by f(x) = cos x \(\forall \) x \(\in\)R. Show that  is neither one-one nor onto.

CBSE 12th Standard Maths Subject Ncert Exemplar Fill Up Questions 2021 part - I - by Shalini Sharma - Udaipur View & Read

  • 1)

    Let the function f : R\(\rightarrow\)R to be defined by f(x) = cos x \(\forall \) x \(\in\)R. Show that  is neither one-one nor onto.

CBSE 12th Standard Maths Subject Ncert Exemplar 1 Marks Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Find the- domain of the function defined by \(f(x)=\sin ^{-1} \sqrt{x-1} \cdot \)

  • 2)

    Find the domain of the function \(\cos ^{-1}(2 x-1)\)

  • 3)

    If A is a skew-symmetric matrix, then show that A2 is a symmetric matrix.

  • 4)

    If possible, then find the sum of the matrices A and B where \(\text {A models} \left[\begin{array}{cc}\sqrt{3} & 1 \\ 2 & 3\end{array}\right]\)and \(B=\left[\begin{array}{ccc}x & y & z \\ a & b & c\end{array}\right]\)

  • 5)

    If A is a matrix of order 3x 3, then find the number of minors in determinant A.

CBSE 12th Standard Maths Subject Ncert Exemplar 1 Marks Questions 2021 part - II - by Shalini Sharma - Udaipur View & Read

  • 1)

    Find the- domain of the function defined by \(f(x)=\sin ^{-1} \sqrt{x-1} \cdot \)

  • 2)

    If A is a skew-symmetric matrix, then show that A2 is a symmetric matrix.

  • 3)

    If possible, then find the sum of the matrices A and B where \(\text {A models} \left[\begin{array}{cc}\sqrt{3} & 1 \\ 2 & 3\end{array}\right]\)and \(B=\left[\begin{array}{ccc}x & y & z \\ a & b & c\end{array}\right]\)

  • 4)

    If A is a matrix of order 3x 3, then find the number of minors in determinant A.

  • 5)

    If A and B are matrix of order 3 and IAI = 5, IBI = 3, then find the value of |3AB|.

CBSE 12th Standard Maths Subject Ncert Exemplar 1 Marks Questions 2021 part - I - by Shalini Sharma - Udaipur View & Read

  • 1)

    Find the- domain of the function defined by \(f(x)=\sin ^{-1} \sqrt{x-1} \cdot \)

  • 2)

    Find the domain of the function \(\cos ^{-1}(2 x-1)\)

  • 3)

    If A is a skew-symmetric matrix, then show that A2 is a symmetric matrix.

  • 4)

    If possible, then find the sum of the matrices A and B where \(\text {A models} \left[\begin{array}{cc}\sqrt{3} & 1 \\ 2 & 3\end{array}\right]\)and \(B=\left[\begin{array}{ccc}x & y & z \\ a & b & c\end{array}\right]\)

  • 5)

    If A is a matrix of order 3x 3, then find the number of minors in determinant A.

CBSE 12th Standard Maths Subject Ncert Exemplar MCQ Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    If the set A contains 5 elements and the set B contains 6 elements, then the number of one-one and onto mappings from A to B is

  • 2)

     Let \(f: R \rightarrow R\) be the functions defined by \(f(x)=x^{3}+5\). Then, \(f^{-1}(x)\) is 

  • 3)

    If A and B are square matrices of the sameorder, then (A + B) (A - B) is equal to

  • 4)

    If A is matrix of order m x nand B is a matrix such that AB' and B' A are both defined, then order of matrix B is

CBSE 12th Standard Maths Subject Ncert Exemplar MCQ Questions 2021 Part - II - by Shalini Sharma - Udaipur View & Read

  • 1)

    For the set A = {1, 2, 3}, define a relation R in the set A as follows
    R = {(1, 1), (2,2), (3, 3), (1, 3)}
    Then, the ordered pair to be added to R to make it the smallest equivalence relation is

  • 2)

    If the set A contains 5 elements and the set B contains 6 elements, then the number of one-one and onto mappings from A to B is

  • 3)

    The matrix \(\left[\begin{array}{ccc}0 & -5 & 8 \\ 5 & 0 & 12 \\ -8 & -12 & 0\end{array}\right]\) is a

  • 4)

    On using elementary column operations \(C_{2} \rightarrow C_{2}-2 C_{1}\) in the following matrix equation \(\left[\begin{array}{cc}1 & -3 \\ 2 & 4\end{array}\right]=\left[\begin{array}{cc}1 & -1 \\ 0 & 1\end{array}\right]\left[\begin{array}{ll}3 & 1 \\ 2 & 4\end{array}\right]\), we have

CBSE 12th Standard Maths Subject Ncert Exemplar MCQ Questions 2021 Part - I - by Shalini Sharma - Udaipur View & Read

  • 1)

    If a relation R on the set {1,2, 3} be defined by R = {(1, 2)}, then R is

  • 2)

    For the set A = {1, 2, 3}, define a relation R in the set A as follows
    R = {(1, 1), (2,2), (3, 3), (1, 3)}
    Then, the ordered pair to be added to R to make it the smallest equivalence relation is

  • 3)

    If the set A contains 5 elements and the set B contains 6 elements, then the number of one-one and onto mappings from A to B is

  • 4)

     Let \(f: R \rightarrow R\) be the functions defined by \(f(x)=x^{3}+5\). Then, \(f^{-1}(x)\) is 

CBSE 12th Standard Maths Subject Probability Case Study Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    In a play zone, Aastha is playing crane game. It has 12 blue balls, 8 red balls, 10 yellow balls and 5 green balls. If Aastha draws two balls one after the other without replacement, then answer the following questions.

    (i) What is the probability that the first ball is blue and the second ball is green?

    \((a) \ \frac{5}{119}\) \((b) \ \frac{12}{119}\) \((c) \ \frac{6}{119}\) \((d) \ \frac{5}{119}\)

    (ii) What is the probability that the first ball is yellow and the second ball is red?

    \((a) \ \frac{6}{119}\) \((b) \ \frac{8}{119}\) \((c) \ \frac{24}{119}\) (d) None of these

    (iii) What is the probability that both the balls are red?

    \((a) \ \frac{4}{85}\) \((b) \ \frac{24}{595}\) \((c) \ \frac{12}{119}\) \((c) \ \frac{64}{119}\)

    (iv) What is the probability that the first ball is green and the second ball is not yellow?

    \((a) \ \frac{10}{119}\) \((b) \ \frac{6}{85}\) \((c) \ \frac{12}{119}\) (d) None of these

    (v) What is the probability that both the balls are not blue?

    \((a) \ \frac{6}{595}\) \((b) \ \frac{12}{85}\) \((c) \ \frac{15}{17}\) \((d) \ \frac{253}{595}\)
  • 2)

    A doctor is to visit a patient. From the past experience, it is known that the probabilities that he will come by cab, metro, bike or by other means of transport are respectively 0.3, 0.2, 0.1 and 0.4. The probabilities that he will be late are 0.25, 0.3, 0.35 and 0.1 if he comes by cab, metro, bike and other means of transport respectively.

    Based on the above information, answer the following questions.
    (i) When the doctor arrives late, what is the probability that he comes by metro?

    \((a) \ \frac{5}{4}\) \((b) \ \frac{2}{7}\) \((c) \ \frac{5}{21}\) \((d) \ \frac{1}{6}\)

    (ii) When the doctor arrives late, what is the probability that he comes by cab?

    \((a) \ \frac{4}{21}\) \((b) \ \frac{1}{7}\) \((c) \ \frac{5}{14}\) \((d) \ \frac{2}{21}\)

    (iii) When the doctor arrives late, what is the probability that he comes by bike?

    \((a) \ \frac{5}{21}\) \((b) \ \frac{4}{7}\) \((c) \ \frac{5}{6}\) \((d) \ \frac{1}{6}\)

    (iv) When the doctor arrives late, what is the probability that he comes by other means of transport?

    \((a) \ \frac{6}{7}\) \((b) \ \frac{5}{14}\) \((c) \ \frac{4}{21}\) \((d) \ \frac{2}{7}\)

    (v) What is the probability that the doctor is late by any means?

    \((a) \ 1\) \((b) \ 0\) \((c) \ \frac{1}{2}\) \((d) \ \frac{1}{4}\)
  • 3)

    On a holiday, a father gave a puzzle from a newspaper to his son Ravi and his daughter Priya. The probability of solving this specific puzzle independently by Ravi and Priya are \(\frac{1}{4}\) and \(\frac{1}{5}\) respectively.

    Based on the above information, answer the following questions.
    (i) The chance that both Ravi and Priya solved the puzzle, is

    (a) 10% (b) 5% (c) 25% (d) 20%

    (ii) Probability that puzzle is solved by Ravi but not by Priya, is

    (a) \(\frac{1}{2}\) (b) \(\frac{1}{5}\) (c) \(\frac{3}{5}\) (d) \(\frac{1}{3}\)

    (iii) Find the probability that puzzle is solved.

    (a) \(\frac{1}{2}\) (b) \(\frac{1}{5}\) (c) \(\frac{2}{5}\) (d) \(\frac{5}{6}\)

    (iv) Probability that exactly one of them solved the puzzle, is

    (a) \(\frac{1}{30}\) (b)\(\frac{1}{20}\) (c) \(\frac{7}{20}\) (d) \(\frac{3}{20}\)

    (v) Probability that none of them solved the puzzle, is

    (a) \(\frac{1}{5}\) (b) \(\frac{3}{5}\) (c) \(\frac{2}{5}\) (d) None of these
  • 4)

    One day, a sangeet mahotsav is to be organised in an open area of Rajasthan. In recent years, it has rained only 6 days each year. Also, it is given that when it actually rains, the weatherman correctly forecasts rain 80% of the time. When it doesn't rain, he incorrectly forecasts rain 20% of the time.
    If leap year is considered, then answer the following questions

    (i) The probability that it rains on chosen day is

    a) \(\frac{1}{366}\) (b) \(\frac{1}{73}\) (c) \(\frac{1}{60}\) (d) \(\frac{1}{61}\)

    (ii) The probability that it does not rain on chosen day is

    a) \(\frac{1}{366}\) (b) \(\frac{5}{366}\) (c) \(\frac{360}{366}\) (d) None of these

    (iii) The probability that the weatherman predicts correctly is

    a) \(\frac{5}{6}\) (b) \(\frac{7}{8}\) (c) \(\frac{4}{5}\) (d) \(\frac{1}{5}\)

    (iv) The probability that it will rain on the chosen day, if weatherman predict rain for that day, is

    (a) 0.0625 (b) 0.0725 (c) 0.0825 (d) 0.0925

    (v) The probability that it will not rain on the chosen day, if weatherman predict rain for that day, is

    (a) 0.94 (b) 0.84 (c) 0.74 (d) 0.64
  • 5)

    To teach the application of probability a maths teacher arranged a surprise game for 5 of his students namely Archit, Aadya, Mivaan, Deepak and Vrinda. He took a bowl containing tickets numbered 1 to 50 and told the students go one by one and draw two tickets simultaneously from the bowl and replace it after noting the numbers.

    Based on the above information, answer the following questions
    (i) Teacher ask Vrinda, what is the probability that both tickets drawn by Archit shows even number?

    (a) 1/50 (b) 12/49 (c) 13/49 (d) 15/49

    (ii) Teacher ask Mivaan, what is the probability that both tickets drawn by Aadya shows odd number?

    a) 1/50 (b) 2/49 (c) 12/49 (d)  5/49

    (iii) Teacher ask Deepak, what is the probability that tickets drawn by Mivaan, shows a multiple of 4 on one ticket and a multiple 5 on other ticket?

    a) 14/245 (b) 16/245 (c) 24/245 (d) None of these

    (iv) Teacher ask Archit, what is the probability that tickets are drawn by Deepak, shows a prime number on one ticket and a multiple of 4 on other ticket?

    a) 3/245 (b) 17/245 (c) 18/245 (d) 36/245

    (v) Teacher ask Aadya, what is the probability that tickets drawn by Vrinda, shows an even number on first ticket and an odd' number on second ticket?

    a) 15/98 (b) 25/98 (c) 35/98 (d) none of these

CBSE 12th Standard Maths Subject Probability Case Study Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Three friends A, B and C are playing a dice game. The numbers rolled up by them in their first three chances were noted and given by A = {1, 5},B = {2, 4, 5} and C = {1, 2, 5} as A reaches the cell 'SKIP YOUR NEXT TURN' in second throw.

    Based on the above information, answer the following questions.
    (i) P (A I B) =

    \((a) \ \frac{1}{6}\) \((b) \ \frac{1}{3}\) \((c) \ \frac{1}{2}\) \((d) \ \frac{2}{3}\)

    (ii) P (B I C) =

    \((a) \ \frac{2}{3}\) \((b) \ \frac{1}{12}\) \((c) \ \frac{1}{9}\) \((d) \ 0\)

    (iii) P (A ⋂ B I C) =

    \((a) \ \frac{1}{6}\) \((b) \ \frac{1}{2}\) \((c) \ \frac{1}{12}\) \((d) \ \frac{1}{3}\)

    (iv) P (A I C) =

    \((a) \ \frac{1}{4}\) \((b) \ 1\) \((c) \ \frac{2}{3}\) (d) None of these

    (v) P (A ∪ B I C) =

    \((a) \ 0\) \((b) \ \frac{1}{2}\) \((c) \ \frac{2}{3}\) \((d) \ 1\)
  • 2)

    In a play zone, Aastha is playing crane game. It has 12 blue balls, 8 red balls, 10 yellow balls and 5 green balls. If Aastha draws two balls one after the other without replacement, then answer the following questions.

    (i) What is the probability that the first ball is blue and the second ball is green?

    \((a) \ \frac{5}{119}\) \((b) \ \frac{12}{119}\) \((c) \ \frac{6}{119}\) \((d) \ \frac{5}{119}\)

    (ii) What is the probability that the first ball is yellow and the second ball is red?

    \((a) \ \frac{6}{119}\) \((b) \ \frac{8}{119}\) \((c) \ \frac{24}{119}\) (d) None of these

    (iii) What is the probability that both the balls are red?

    \((a) \ \frac{4}{85}\) \((b) \ \frac{24}{595}\) \((c) \ \frac{12}{119}\) \((c) \ \frac{64}{119}\)

    (iv) What is the probability that the first ball is green and the second ball is not yellow?

    \((a) \ \frac{10}{119}\) \((b) \ \frac{6}{85}\) \((c) \ \frac{12}{119}\) (d) None of these

    (v) What is the probability that both the balls are not blue?

    \((a) \ \frac{6}{595}\) \((b) \ \frac{12}{85}\) \((c) \ \frac{15}{17}\) \((d) \ \frac{253}{595}\)
  • 3)

    Ajay enrolled himself in an online practice test portal provided by his school for better practice. Out of 5 questions in a set-I, he was able to solve 4 of them and got stuck in the one which is as shown below.

    If A and B are independent events, P(A) = 0.6 and P(B) = 0.8, then answer the following questions.
    (i) P (A \(\cap\) B) =

    (a) 0.2 (b) 0.9 (c) 0.48 (d) 0.6

    (ii) P (A \(\cup\) B) =

    (a) 0.92 (b) 0.08 (c) 0.48 (d) 0.64

    (iii) P (B | A) =

    (a) 0.14 (b) 0.2 (c) 0.6 (d) 0.8

    (iv) P (A | B) =

    (a) 0.6 (b) 0.9 (c) 0.19 (d) 0.11

    (v) P ( not A and not B ) =

    (a) 0.01 (b) 0.48 (c) 0.08 (d) 0.91
  • 4)

    A doctor is to visit a patient. From the past experience, it is known that the probabilities that he will come by cab, metro, bike or by other means of transport are respectively 0.3, 0.2, 0.1 and 0.4. The probabilities that he will be late are 0.25, 0.3, 0.35 and 0.1 if he comes by cab, metro, bike and other means of transport respectively.

    Based on the above information, answer the following questions.
    (i) When the doctor arrives late, what is the probability that he comes by metro?

    \((a) \ \frac{5}{4}\) \((b) \ \frac{2}{7}\) \((c) \ \frac{5}{21}\) \((d) \ \frac{1}{6}\)

    (ii) When the doctor arrives late, what is the probability that he comes by cab?

    \((a) \ \frac{4}{21}\) \((b) \ \frac{1}{7}\) \((c) \ \frac{5}{14}\) \((d) \ \frac{2}{21}\)

    (iii) When the doctor arrives late, what is the probability that he comes by bike?

    \((a) \ \frac{5}{21}\) \((b) \ \frac{4}{7}\) \((c) \ \frac{5}{6}\) \((d) \ \frac{1}{6}\)

    (iv) When the doctor arrives late, what is the probability that he comes by other means of transport?

    \((a) \ \frac{6}{7}\) \((b) \ \frac{5}{14}\) \((c) \ \frac{4}{21}\) \((d) \ \frac{2}{7}\)

    (v) What is the probability that the doctor is late by any means?

    \((a) \ 1\) \((b) \ 0\) \((c) \ \frac{1}{2}\) \((d) \ \frac{1}{4}\)
  • 5)

    Suman was doing a project on a school survey, on the average number of hours spent on study by students selected at random. At the end of-survey, Suman prepared the following report related to the data. Let X denotes the average number of hours spent on study by students. The probability that X can take the values x, has the following form, where k is some unknown constant.
    \(P(X=x)=\left\{\begin{array}{l} 0.2, \text { if } x=0 \\ k x, \text { if } x=1 \text { or } 2 \\ k(6-x), \text { if } x=3 \text { or } 4 \\ 0, \text { otherwise } \end{array}\right.\)


    Based on the above information, answer the following questions.
    (i) Find the value of k.

    (a) 0.1  (b) 0.2 (c) 0.3 (d) 0.05

    (ii) What is the probability that the average study time of students is not more than 1 hour?

    (a) 0.4  (b) 0.3 (c) 0.5 (d) 0.1

    (iii) What is the probability that the average study time of students is at least 3 hours?

    (a) 0.5 (b) 0.9 (c) 0.8 (d) 0.1

    (iv) What is the probability that the average study time of students is exactly 2 hours?

    (a) 0.4  (b) 0.5 (c) 0.7 (d) 0.2

    (v) What is the probability that the average study time of students is at least 1 hour?

    (a) 0.2 (b) 0.4 (c) 0.8 (d) 0.6

CBSE 12th Standard Maths Subject Linear Programming Case Study Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Linear programming is a method for finding the optimal values (maximum or minimum) of quantities subject to the constraints when relationship is expressed as linear equations or inequations.
    Based on the above information, answer the following questions.
    (i) The optimal value of the objective function is attained at the points

    (a) on X-axis  (b) on Y-axis  (c) which are corner points of the feasible region  (d) none of these

    (ii) The graph of the inequality 3x + 4y < 12 is

    (a) half plane that contains the origin (b) half plane that neither contains the origin nor the points of the line 3x + 4y =12. (c) whole XOY-plane excluding the points on line 3x + 4y = 12 (d) none of these

    (iii) The feasible region for an LPP is shown in the figure. Let Z = 2x + 5y be the objective function. Maximum of Z occurs at

    (a) (7,0)  (b) (6,3) (c) (0,6) (d) (4,5)

    (iv) The corner points of the feasible region determined by the system of linear constraints are (0, 10), (5, 5), (15, 15), (0, 20). Let Z = px + qy, where p, q > 0. Condition on p and q so that the maximum of Z occurs at both the points (15, 15) and (0, 20) is

    (a) p = q (b) p = 2q (c) q=2p (d) q=3p

    (v)  The corner points of the feasible region determined by the system of linear constraints are (0, 0), (0, 40), (20,40), (60,20), (60, 0). The objective function is Z = 4x + 3y. Compare the quantity in Column A and Column B

    Column A Column B
    Maximum of Z 325
    (a) The quantity in column A is greater (b) The quantity in column B is greater (c) The two quantities are equal (d) The relationship cannot be determined on the basis of the information supplied
  • 2)

    Deepa rides her car at 25 km/hr, She has to spend Rs. 2 per km on diesel and if she rides it at a faster speed of 40 km/hr, the diesel cost increases to Rs. 5 per km. She has Rs. 100 to spend on diesel. Let she travels x kms with speed 25 km/hr and y kms with speed 40 km/hr. The feasible region for the LPP is shown below:
    Based on the above information, answer the following questions

    Based on the above information, answer the following questions.
    (i) What is the point of intersection of line l1 and l2,

    \(\text { (a) }\left(\frac{40}{3}, \frac{50}{3}\right)\) \(\text { (b) }\left(\frac{50}{3}, \frac{40}{3}\right)\) \(\text { (c) }\left(\frac{-50}{3}, \frac{40}{3}\right)\) \(\text { (d) }\left(\frac{-50}{3}, \frac{-40}{3}\right)\)

    (ii) The corner points of the feasible region shown in above graph are

    \(\text { (a) }(0,25),(20,0),\left(\frac{40}{3}, \frac{50}{3}\right)\) \(\text { (b) }(0,0),(25,0),(0,20)\) \(\text { (c) }(0,0),\left(\frac{40}{3}, \frac{50}{3}\right),(0,20)\) \(\text { (d) }(0,0),(25,0),\left(\frac{50}{3}, \frac{40}{3}\right),(0,20)\)

    (iii) If Z = x + y be the objective function and max Z = 30. The maximum value occurs at point

    \(\text { (a) }\left(\frac{50}{3}, \frac{40}{3}\right)\) (b) (0, 0) (c) (25, 0) (d) (0, 20)

    (iv) If Z = 6x - 9y be the objective function, then maximum value of Z is

    (a) -20 (b) 150 (c) 180 (d) 20

    (v) If Z = 6x + 3y be the objective function, then what is the minimum value of Z?

    (a) 120 (b) 130 (c) 0 (d) 150
  • 3)

    Corner points of the feasible region for an LPP are (0, 3), (5, 0), (6, 8), (0, 8). Let Z = 4x - 6y be the objective function.
    Based on the above information, answer the following questions.
    (i) The minimum value of Z occurs at

    (a) (6, 8) (b) (5, 0) (c) (0, 3) (d) (0, 8)

    (ii) Maximum value of Z occurs at

    (a) (5, 0) (b) (0, 8) (c) (0, 3) (d) (6, 8)

    (iii) Maximum of Z - Minimumof Z =

    (a) 58 (b) 68 (c) 78 (d) 88

    (iv) The corner points of the feasible region determined by the system of linear inequalities are

    (a) (0, 0), (-3, 0), (3, 2), (2, 3) (b) (3, 0), (3, 2), (2, 3), (0, -3) (c) (0, 0), (3, 0), (3, 2), (2, 3), (0, 3) (d) None of these

    (v) The feasible solution of LPP belongs to

    (a) first and second quadrant (b) first and third quadrant (c) only second quadrant (d) only first quadrant
  • 4)

    Suppose a dealer in rural area wishes to purpose a number of sewing machines. He has only Rs. 5760 to invest and has space for at most 20 items for storage. An electronic sewing machine costs him Rs. 360 and a manually operated sewing machine Rs. 240. He can sell an electronic sewing machine at a profit of Rs. 22 and a manually operated sewing machine at a profit of Rs. 18.
    Based on the above information, answer the following questions.

    (i) Let x and y denotes the number of electronic sewing machines and manually operated sewing machines purchased by the dealer. If it is assume that the dealer purchased atleast one of the given machines, then

    (a) x + y ≥  0 (b) x + y < 0 (c) x + y > 0 (d) x + y ≤ 0

    (ii) Let the constraints in the given problem is represented by the following inequalities
    x + y ≤ 20
    360x + 240y ≤ 5760
    x, y ≥ 0
    Then which of the following point lie in its feasible region.

    (a) (0, 24) (b) (8, 12) (c) (20, 2) (d) None of these

    (iii) If the objective function of the given problem is maximise z = 22x + 18y, then its optimal value occur at

    (a) (0, 0) (b) (16, 0) (c) (8, 12) (d) (0, 20)

    (iv) Suppose the following shaded region APDO, represent the feasible region corresponding to mathematical formulation. of given problem. Then which of the following represent the coordinates of one of its corner points.

    (a) (0, 24) (b) (12, 8) (c) (8, 12) (d) (6, 14)

    (v) If an LPP admits optimal solution at two consecutive vertices of a feasible region, then

    (a) the required optimal solution is at the midpoint of the line joining two points. (b) the optimal solution occurs at every point on the line joining these two points.
    (c) the LPP under consideration is not solvable. (d) the LPP under consideration must be reconstructed.
  • 5)

    Let R be the feasible region (convex polygon) for a linear programming problem and let Z = ax + by be the objective function. When Z has an optimal value (maximum or minimum), where the variables x and y are subject to, constraints described by linear inequalities, this optimal value must occur at a corner point (vertex) of the feasible region.
    Based on the above information, answer the following questions.
    (i) Objective function of a L.P.P. is

    (a) a constant (b) a function to be optimised (c) a relation between the variables (d) none of these

    (ii) Which of the following statement is correct?

    (a) Every LPP has at least one optimal solution. (b) Every LPP has a unique optimal solution. (c) If an LPP has two optimal solutions, then it has infinitely many solutions (d) none of these

    (iii) In solving the LPP : "minimize f = 6x + 10y subject to constraints x ≥  6, Y ≥ 2, 2x + y  ≥ 10,x  ≥ 0,y ≥ 0" redundant constraints are

    (a) x ≥ 6, y ≥ 2 (b) 2x + y ≥ 10, x ≥ 0, y ≥ 0 (c) x ≥ 6 (d) none of these

    (iv) The feasible region for a LPP is shown shaded in the figure. Let Z = 3x - 4y be the objective function. Minimum of Z occurs at

    (a) (0, 0) (b) (0, 8) (c) (5, 0) (d) (4, 10)

    (v) The feasible region for a LPP is shown shaded in the figure. Let F = 3x - 4y be the objective function. Maximum value of F is

    (a) 0 (b) 8 (c) 12 (d) -18

CBSE 12th Standard Maths Subject Linear Programming Case Study Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Linear programming is a method for finding the optimal values (maximum or minimum) of quantities subject to the constraints when relationship is expressed as linear equations or inequations.
    Based on the above information, answer the following questions.
    (i) The optimal value of the objective function is attained at the points

    (a) on X-axis  (b) on Y-axis  (c) which are corner points of the feasible region  (d) none of these

    (ii) The graph of the inequality 3x + 4y < 12 is

    (a) half plane that contains the origin (b) half plane that neither contains the origin nor the points of the line 3x + 4y =12. (c) whole XOY-plane excluding the points on line 3x + 4y = 12 (d) none of these

    (iii) The feasible region for an LPP is shown in the figure. Let Z = 2x + 5y be the objective function. Maximum of Z occurs at

    (a) (7,0)  (b) (6,3) (c) (0,6) (d) (4,5)

    (iv) The corner points of the feasible region determined by the system of linear constraints are (0, 10), (5, 5), (15, 15), (0, 20). Let Z = px + qy, where p, q > 0. Condition on p and q so that the maximum of Z occurs at both the points (15, 15) and (0, 20) is

    (a) p = q (b) p = 2q (c) q=2p (d) q=3p

    (v)  The corner points of the feasible region determined by the system of linear constraints are (0, 0), (0, 40), (20,40), (60,20), (60, 0). The objective function is Z = 4x + 3y. Compare the quantity in Column A and Column B

    Column A Column B
    Maximum of Z 325
    (a) The quantity in column A is greater (b) The quantity in column B is greater (c) The two quantities are equal (d) The relationship cannot be determined on the basis of the information supplied
  • 2)

    Deepa rides her car at 25 km/hr, She has to spend Rs. 2 per km on diesel and if she rides it at a faster speed of 40 km/hr, the diesel cost increases to Rs. 5 per km. She has Rs. 100 to spend on diesel. Let she travels x kms with speed 25 km/hr and y kms with speed 40 km/hr. The feasible region for the LPP is shown below:
    Based on the above information, answer the following questions

    Based on the above information, answer the following questions.
    (i) What is the point of intersection of line l1 and l2,

    \(\text { (a) }\left(\frac{40}{3}, \frac{50}{3}\right)\) \(\text { (b) }\left(\frac{50}{3}, \frac{40}{3}\right)\) \(\text { (c) }\left(\frac{-50}{3}, \frac{40}{3}\right)\) \(\text { (d) }\left(\frac{-50}{3}, \frac{-40}{3}\right)\)

    (ii) The corner points of the feasible region shown in above graph are

    \(\text { (a) }(0,25),(20,0),\left(\frac{40}{3}, \frac{50}{3}\right)\) \(\text { (b) }(0,0),(25,0),(0,20)\) \(\text { (c) }(0,0),\left(\frac{40}{3}, \frac{50}{3}\right),(0,20)\) \(\text { (d) }(0,0),(25,0),\left(\frac{50}{3}, \frac{40}{3}\right),(0,20)\)

    (iii) If Z = x + y be the objective function and max Z = 30. The maximum value occurs at point

    \(\text { (a) }\left(\frac{50}{3}, \frac{40}{3}\right)\) (b) (0, 0) (c) (25, 0) (d) (0, 20)

    (iv) If Z = 6x - 9y be the objective function, then maximum value of Z is

    (a) -20 (b) 150 (c) 180 (d) 20

    (v) If Z = 6x + 3y be the objective function, then what is the minimum value of Z?

    (a) 120 (b) 130 (c) 0 (d) 150
  • 3)

    Corner points of the feasible region for an LPP are (0, 3), (5, 0), (6, 8), (0, 8). Let Z = 4x - 6y be the objective function.
    Based on the above information, answer the following questions.
    (i) The minimum value of Z occurs at

    (a) (6, 8) (b) (5, 0) (c) (0, 3) (d) (0, 8)

    (ii) Maximum value of Z occurs at

    (a) (5, 0) (b) (0, 8) (c) (0, 3) (d) (6, 8)

    (iii) Maximum of Z - Minimumof Z =

    (a) 58 (b) 68 (c) 78 (d) 88

    (iv) The corner points of the feasible region determined by the system of linear inequalities are

    (a) (0, 0), (-3, 0), (3, 2), (2, 3) (b) (3, 0), (3, 2), (2, 3), (0, -3) (c) (0, 0), (3, 0), (3, 2), (2, 3), (0, 3) (d) None of these

    (v) The feasible solution of LPP belongs to

    (a) first and second quadrant (b) first and third quadrant (c) only second quadrant (d) only first quadrant
  • 4)

    Suppose a dealer in rural area wishes to purpose a number of sewing machines. He has only Rs. 5760 to invest and has space for at most 20 items for storage. An electronic sewing machine costs him Rs. 360 and a manually operated sewing machine Rs. 240. He can sell an electronic sewing machine at a profit of Rs. 22 and a manually operated sewing machine at a profit of Rs. 18.
    Based on the above information, answer the following questions.

    (i) Let x and y denotes the number of electronic sewing machines and manually operated sewing machines purchased by the dealer. If it is assume that the dealer purchased atleast one of the given machines, then

    (a) x + y ≥  0 (b) x + y < 0 (c) x + y > 0 (d) x + y ≤ 0

    (ii) Let the constraints in the given problem is represented by the following inequalities
    x + y ≤ 20
    360x + 240y ≤ 5760
    x, y ≥ 0
    Then which of the following point lie in its feasible region.

    (a) (0, 24) (b) (8, 12) (c) (20, 2) (d) None of these

    (iii) If the objective function of the given problem is maximise z = 22x + 18y, then its optimal value occur at

    (a) (0, 0) (b) (16, 0) (c) (8, 12) (d) (0, 20)

    (iv) Suppose the following shaded region APDO, represent the feasible region corresponding to mathematical formulation. of given problem. Then which of the following represent the coordinates of one of its corner points.

    (a) (0, 24) (b) (12, 8) (c) (8, 12) (d) (6, 14)

    (v) If an LPP admits optimal solution at two consecutive vertices of a feasible region, then

    (a) the required optimal solution is at the midpoint of the line joining two points. (b) the optimal solution occurs at every point on the line joining these two points.
    (c) the LPP under consideration is not solvable. (d) the LPP under consideration must be reconstructed.
  • 5)

    Let R be the feasible region (convex polygon) for a linear programming problem and let Z = ax + by be the objective function. When Z has an optimal value (maximum or minimum), where the variables x and y are subject to, constraints described by linear inequalities, this optimal value must occur at a corner point (vertex) of the feasible region.
    Based on the above information, answer the following questions.
    (i) Objective function of a L.P.P. is

    (a) a constant (b) a function to be optimised (c) a relation between the variables (d) none of these

    (ii) Which of the following statement is correct?

    (a) Every LPP has at least one optimal solution. (b) Every LPP has a unique optimal solution. (c) If an LPP has two optimal solutions, then it has infinitely many solutions (d) none of these

    (iii) In solving the LPP : "minimize f = 6x + 10y subject to constraints x ≥  6, Y ≥ 2, 2x + y  ≥ 10,x  ≥ 0,y ≥ 0" redundant constraints are

    (a) x ≥ 6, y ≥ 2 (b) 2x + y ≥ 10, x ≥ 0, y ≥ 0 (c) x ≥ 6 (d) none of these

    (iv) The feasible region for a LPP is shown shaded in the figure. Let Z = 3x - 4y be the objective function. Minimum of Z occurs at

    (a) (0, 0) (b) (0, 8) (c) (5, 0) (d) (4, 10)

    (v) The feasible region for a LPP is shown shaded in the figure. Let F = 3x - 4y be the objective function. Maximum value of F is

    (a) 0 (b) 8 (c) 12 (d) -18

CBSE 12th Standard Maths Subject Differential Equations Case Study Questions With Solution - by Shalini Sharma - Udaipur View & Read

  • 1)

    A thermometer reading 800P is taken outside. Five minutes later the thermometer reads 60°F. After another 5 minutes the thermometer reads 50of At any time t the thermometer reading be TOP and the outside temperature be SoF.
    Based on the above information, answer the following questions.
    (i) If \(\lambda\) is positive constant of proportionality, then \(\frac{d T}{d t}\) is

    (a) \(\lambda(T-S)\) (b) \(\lambda(T+S)\) (c) \(\lambda T S\) (d) \(-\lambda(T-S)\)

    (ii) The value of T(S) is

    (a) 300F (b) 40oF (c) 50oF (d) 60oF

    (iii) The value of T(10) is

    (a) 50oF (b) 40oF (c) 50oF (d) 60oF

    (iv) Find the general solution of differential equation formed in given situation.

    (a) logT=St+c (b) \(\log (T-S)=-\lambda t+c\) (c) log S = tT + c (d) \(\log (T+S)=\lambda t+c\)

    (v) Find the valiie of constant of integration c in the solution of differential equation formed in given situation.

    (a) log (60 -S) (b) log (80 + S) (c) log (80 - S) (d) log (60 + S)
  • 2)

    In a murder investigation, a corpse was found by a detective at exactly 8 p.m. Being alert, the detective measured the body temperature and found it to be 70°F. Two hours later, the detective measured the body temperature again and found it to be 60°F, where the room temperature is 50°F. Also, it is given the body temperature at the time of death was normal, i.e., 98.6°F.
    Let T be the temperature of the body at any time t and initial time is taken to be 8 p.m.
     
    Based on the above information, answer the following questions.
    (i) By Newton's law of cooling,\(\frac{d T}{d t}\) is proportional to

    (a) T - 60 (b) T - 50 (c) T - 70 (d) T - 98.6

    (ii) When t = 0, then body temperature is equal to

    (a) 50°F (b) 60°F (c) 70oF (d) 98.6°F

    (iii) When t = 2, then body temperature is equal to

    (a) 50°F (b) 60°F (c) 70oF (d) 98.6°F

    (iv) The value of T at any time tis

    (a) \(50+20\left(\frac{1}{2}\right)^{t}\) (b) \(50+20\left(\frac{1}{2}\right)^{t-1}\) (c) \(50+20\left(\frac{1}{2}\right)^{t / 2}\) (d) None of these

    (v) If it is given that loge(2.43) = 0.88789 and loge(0.5) = -0.69315, then the time at which the murder occur is

    (a) 7:30 p.m. (b) 5:30 p.m. (c) 6:00 p.m. (d) 5:00 p.m.
  • 3)

    Order: The order of a differential equation is the order of the highest order derivative appearing in the differential equation.
    Degree : The degree of differential equation is the power of the highest order derivative, when differential coefficients are made free from radicals and fractions. Also, differential equation must be a polynomial equation
    in derivatives for the degree to be defined.
    Based on the above information, answer the following questions.
    (i) Find the degree of the differential equation \(2 \frac{d^{2} y}{d x^{2}}+3 \sqrt{1-\left(\frac{d y}{d x}\right)^{2}-y}=0\) 

    (a) 3 (b) 4 (c) 2 (d) 1

    (ii) Order and degree of the differential equation \(y \frac{d y}{d x}=\frac{x}{\frac{d y}{d x}+\left(\frac{d y}{d x}\right)^{3}}\) are respectively

    (a) 1,1 (b) 1,2 (c) 1,3 (d) 1,4

    (iii) Find order and degree of the equation \(y^{\prime \prime \prime}+y^{2}+e^{y^{\prime}}=0\) 

    (a) order = 3, degree = undefined (b) order = 1, degree = 3 (c) order = 2, degree = undefined (d) order = 1, degree = 2

    (iv) Determine degree of the differential equation \((\sqrt{a+x}) \cdot\left(\frac{d y}{d x}\right)+x=0\) 

    (a) 3 (b) not defined (c) 1 (d) 2

    (v) Order and degree of the differential equation \(\left(1+\left(\frac{d y}{d x}\right)^{3}\right)^{\frac{7}{3}}=7 \frac{d^{2} y}{d x^{2}}\) are respectively

    (a) 2, 1 (b) 2,3 (c) 1,3 (d) \(1, \frac{7}{3}\)
  • 4)

    A differential equation is said to be in the variable separable form if it is expressible in the form j(x) dx = g(y) dy. The solution of this equation is given by \(\int f(x) d x=\int g(y) d y+c\) where c is the constant of integration. 
    Based on the above information, answer the following questions.
    (i) If the solunon of the differential equation \(\frac{d y}{d x}=\frac{a x+3}{2 y+f}\) represents a circle, then the value of' a 'is

    (a) 2 (b) - 2 (c) 3 (d) - 4

    (ii) The diiftfterential equation \(\frac{d y}{d x}=\frac{\sqrt{1-y^{2}}}{y}\) deterrnines a family of circle with 

    (a) variable radii and fixed centre (0,1) (b) variable radii and fixed centre (0,-1)
    (c) fixed radius 1 and variable centre on x-axis (d) fixed radius 1 and variable centre on y-axis

    (iii) If y' = y + 1, y (0) = 1, theny (In 2) =

    (a) 1 (b) 2 (c) 3 (d) 4

    (iv) The solution of the differential equation \(\frac{d y}{d x}=e^{x-y}+x^{2} e^{-y}\) is

    (a) \(e^{x}=\frac{y^{3}}{3}+e^{y}+c\) (b) \(e^{y}=\frac{x^{2}}{3}+e^{x}+c\) (c) \(e^{y}=\frac{x^{3}}{3}+e^{x}+c\) (d) none of these

    (v) If \(\frac{d y}{d x}=y \sin 2 x, y(0)=1\) then its solution is

    (a) y = esin2x (b) y = sin2 (c) y = cos2x (d) y = cos2x
  • 5)

    If an equation is of the form \(\frac{d y}{d x}+P y=Q\) ,where P, Q are functions of x, then such equation is known as linear differential equation. Its solu~:n is given by \(y \cdot(\mathrm{I} . \mathrm{F} .)=\int \mathrm{Q} \cdot(\mathrm{I} . \mathrm{F} .) d x+c\) where \(\text { I.F. }=e^{\int P d x}\) .
    Now, suppose the given equation is \((1+\sin x) \frac{d y}{d x}+y \cos x+x=0\) 
    Based on the above information, answer the following questions
    (i) The value of P and Q respectively are

    (a) \(\frac{\sin x}{1+\cos x}, \frac{x}{1+\sin x}\) (b) \(\frac{\cos x}{1+\sin x}, \frac{-x}{1+\sin x}\) (c) \(\frac{-\cos x}{1+\sin x}, \frac{x}{1+\sin x}\) (d) \(\frac{\cos x}{1+\sin x}, \frac{x}{1+\sin x}\)

    (ii) The value of I.F. is

    (a) 1 - sin x (b) cos x (c) 1 + sin x (d) 1- cosx

    (iii) Solution of given equation is

    (a) y{1-sinx)=x+c (b) y(l + sin x) = x2+c (c) \(y(1-\sin x)=\frac{-x^{2}}{2}+c\) (d) \(y(1+\sin x)=\frac{-x^{2}}{2}+c\)

    (iv) If y(0) = 1, then y,equals

    (a) \(\frac{2-x^{2}}{2(1+\sin x)}\) (b) \(\frac{2+x^{2}}{2(1+\sin x)}\) (c) \(\frac{2-x^{2}}{2(1-\sin x)}\) (d) \(\frac{2+x^{2}}{2(1-\sin x)}\)

    (v) Value of \(y\left(\frac{\pi}{2}\right)\) is

    (a) \(\frac{4-\pi^{2}}{2}\) (b) \(\frac{8-\pi^{2}}{16}\) (c) \(\frac{8-\pi^{2}}{4}\) (d)  \(\frac{4+\pi^{2}}{2}\)

CBSE 12th Standard Maths Subject Differential Equations Case Study Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    It is known that, if the interest is compounded continuously, the principal changes at the rate equal to the produd' of the rate of bank interest per annum and the principal. Let P denotes the principal at any time t and rate of interest be r % per annum.

    Based on the above information, answer the following questions.
    (i) Find the value of \(\frac{d P}{d t}\) .

    (a) \(\frac{\operatorname{Pr}}{1000}\) (b) \(\frac{P r}{100}\) (c) \(\frac{\operatorname{Pr}}{10}\) (d) Pr

    (ii) fPo be the initial principal, then find the solution of differential equation formed in given situation.

    (a) \(\log \left(\frac{P}{P_{0}}\right)=\frac{r t}{100}\) (b) \(\log \left(\frac{P}{P_{0}}\right)=\frac{r t}{10}\) (c) \(\log \left(\frac{P}{P_{0}}\right)=r t\) (d) \(\log \left(\frac{P}{P_{0}}\right)=100 r t\)

    (iii) If the interest is compounded continuously at 5% per annum, in how many years will Rs. 100 double itself?

    (a) 12.728 years (b) 14.789 years (c) 13.862 years (d) 15.872 years

    (iv) At what interest rate will Rs.100 double itself in 10 years? (log e2 = 0.6931).

    (a) 9.66% (b) 8.239% (c) 7.341% (d) 6.931%

    (v) How much will Rs. 1000 be worth at 5% interest after 10 years? (e0.5 = 1.648).

    (a) Rs. 1648 (b) Rs. 1500 (c) Rs. 1664 (d) Rs. 1572
  • 2)

    A rumour on whatsapp spreads in a population of 5000 people at a rate proportional to the product of the number of people who have heard it and the number of people who have not. Also, it is given that 100 people initiate the rumour and a total of 500 people know the rumour after 2 days.

    Based on the above information, answer the following questions
    (i) If yet) denote the number of people who know the rumour at an instant t, then maximum value of yet) is

    (a) 500 (b) 100 (c) 5000 (d) none of these

    (ii) \(\frac{d y}{d t}\) is proptional to 

    (a) (y - 5000) (b) y(y - 500) (c) y(500 - y) (d) y(5000 - y)

    (iii) The value of y(0) is

    (a) 100 (b) 500 (c) 600 (d) 200

    (iv) The value of y(2) is

    (a) 100 (b) 500 (c) 600 (d) 200

    (v) The value of y at any time t is given by

    (a) \(y=\frac{5000}{e^{-5000 k t}+1}\) (b) \(y=\frac{5000}{1+e^{5000 k t}}\) (c) \(y=\frac{5000}{49 e^{-5000 k t}+1}\) (d) \(y=\frac{5000}{49\left(1+e^{-5000 k t}\right)}\)
  • 3)

    Order: The order of a differential equation is the order of the highest order derivative appearing in the differential equation.
    Degree : The degree of differential equation is the power of the highest order derivative, when differential coefficients are made free from radicals and fractions. Also, differential equation must be a polynomial equation
    in derivatives for the degree to be defined.
    Based on the above information, answer the following questions.
    (i) Find the degree of the differential equation \(2 \frac{d^{2} y}{d x^{2}}+3 \sqrt{1-\left(\frac{d y}{d x}\right)^{2}-y}=0\) 

    (a) 3 (b) 4 (c) 2 (d) 1

    (ii) Order and degree of the differential equation \(y \frac{d y}{d x}=\frac{x}{\frac{d y}{d x}+\left(\frac{d y}{d x}\right)^{3}}\) are respectively

    (a) 1,1 (b) 1,2 (c) 1,3 (d) 1,4

    (iii) Find order and degree of the equation \(y^{\prime \prime \prime}+y^{2}+e^{y^{\prime}}=0\) 

    (a) order = 3, degree = undefined (b) order = 1, degree = 3 (c) order = 2, degree = undefined (d) order = 1, degree = 2

    (iv) Determine degree of the differential equation \((\sqrt{a+x}) \cdot\left(\frac{d y}{d x}\right)+x=0\) 

    (a) 3 (b) not defined (c) 1 (d) 2

    (v) Order and degree of the differential equation \(\left(1+\left(\frac{d y}{d x}\right)^{3}\right)^{\frac{7}{3}}=7 \frac{d^{2} y}{d x^{2}}\) are respectively

    (a) 2, 1 (b) 2,3 (c) 1,3 (d) \(1, \frac{7}{3}\)
  • 4)

    If the equation is of the form \(\frac{d y}{d x}=\frac{f(x, y)}{g(x, y)} \text { or } \frac{d y}{d x}=F\left(\frac{y}{x}\right)\) ,wheref (x, y), g(x, y) are homogeneous functions of the same degree in x and y, then put y = vx and \(\frac{d y}{d x}=v+x \frac{d v}{d x}\), so that the dependent variable y is changed to another variable v and then apply variable separable method. Based on the above information, answer the following questions.
    (i) The general solution of \(x^{2} \frac{d y}{d x}=x^{2}+x y+y^{2}\) is

    (a) \(\tan ^{-1} \frac{x}{y}=\log |x|+c \)  (b) \( \tan ^{-1} \frac{y}{x}=\log |x|+c\) (c) \(y=x \log |x|+c\) (d) \(x=y \log |y|+c\)

    (ii) Solution of the differential equation \(2 x y \frac{d y}{d x}=x^{2}+3 y^{2} \) is

    (a) \( x^{3}+y^{2}=c x^{2}\) (b) \( \frac{x^{2}}{2}+\frac{y^{3}}{3}=y^{2}+c\) (c) \(x^{2}+y^{3}=c x^{2}\) (d) \( x^{2}+y^{2}=c x^{3}\)

    (iii) Solution of the differential equation \(\left(x^{2}+3 x y+y^{2}\right) d x-x^{2} d y=0\) is 

    (a) \(\frac{x+y}{x}-\log x=c\) (b) \( \frac{x+y}{x}+\log x=c\) (c) \(\frac{x}{x+y}-\log x=c\) (d) \(\frac{x}{x+y}+\log x=c\)

    (iv) General solution ofthe differential equation  \(\frac{d y}{d x}=\frac{y}{x}\left\{\log \left(\frac{y}{x}\right)+1\right\}\) is

    (a) \(\log (x y)=c\) (b) \( \log y=c x\) (c) \(\log \left(\frac{y}{x}\right)=c x\) (d) \(\log x=c y\)

    (v) Solution ofthe differential equation  \(\left(x \frac{d y}{d x}-y\right) e^{\frac{y}{x}}=x^{2} \cos x\) is

    (a) \(e^{\frac{y}{x}}-\sin x=c\) (b) \(e^{\frac{y}{x}}+\sin x=c\) (c) \(e^{\frac{-y}{x}}-\sin x=c \) (d) \( e^{\frac{-y}{x}}+\sin x=c\)

     

  • 5)

    If the equation is of the form \(\frac{d y}{d x}+P y=Q\) , where P, Q are functions of x, then the solution of the differential  equation is given by \(y e^{\int P d x}=\int Q e^{\int P d x} d x+c\), where  \(e^{\int P d x}\) is called the integrating factor (I.F.).
    Based on the above information, answer the following questions.
    (i) The integrating factor of the differential equation \(\sin x \frac{d y}{d x}+2 y \cos x=1 \text { is }(\sin x)^{\lambda}, \text { where } \lambda=\)

    (a) 0 (b) 1 (c) 2 (d) 3

    (ii) Integrating  factor of the differential equation \(\left(1-x^{2}\right) \frac{d y}{d x}-x y=1 \) is

    (a) -x (b) \(\frac{x}{1+x^{2}}\) (c) \(\sqrt{1-x^{2}}\) (d) \( \frac{1}{2} \log \left(1-x^{2}\right)\)

    (iii) The solution of \(\frac{d y}{d x}+y=e^{-x}, y(0)=0\) is

    (a) \( y=e^{x}(x-1)\) (b)  \( y=x e^{-x}\) (c) \(y=x e^{-x}+1\) (d)  \( y=(x+1) e^{-x}\)

    (iv) General solution of \(\frac{d y}{d x}+y \tan x=\sec x\) is

    (a) y see x = tan x + c (b) y tan x = sec x + c (c) tan x = y tan x + c (d) x see x = tan y + c

    (v) The integrating factor of differential equation \(\frac{d y}{d x}-3 y=\sin 2 x\)  is

    (a) e3x (b) e-2x (c) e-3x (d) xe-3x

CBSE 12th Standard Maths Subject Three Dimensional Geometry Case Study Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Two motorcycles A and B are running at the speed more than allowed speed on the road along the lines \(\vec{r}=\lambda(\hat{i}+2 \hat{j}-\hat{k}) \text { and } \vec{r}=3 \hat{i}+3 \hat{j}+\mu(2 \hat{i}+\hat{j}+\hat{k})\), respectively.

    Based on the above information, answer the following questions.
    (i) The cartesian equation of the line along which motorcycle A is running, is

    (a) \(\frac{x+1}{1}=\frac{y+1}{2}=\frac{z-1}{-1}\) (b) \(\frac{x}{1}=\frac{y}{2}=\frac{z}{-1}\) (c) \(\frac{x}{1}=\frac{y}{2}=\frac{z}{1}\) (d) none of these

    (ii) The direction cosines of line along which motorcycle A is running, are

    (a) < 1, -2, 1 > (b) < 1, 2, -1 > (c) \(<\frac{1}{\sqrt{6}}, \frac{-2}{\sqrt{6}}, \frac{1}{\sqrt{6}}>\) (d) \(<\frac{1}{\sqrt{6}}, \frac{2}{\sqrt{6}}, \frac{-1}{\sqrt{6}}>\)

    (iii) The direction ratios of line along which motorcycle B is running, are

    (a) <  1, 0, 2 > (b) <  2, 1, 0 > (c) < 1, 1, 2  > (d) <  2, 1, 1  >

    (iv) The shortest distance between the gives lines is

    (a) 4 units (b) 2.\(\sqrt 3\) units (c) 3.\(\sqrt 2\) units (d) 0 units

    (v) The motorcycles will meet with an accident at the point

    (a) (-1, 1, 2) (b) (2, 1, -1) (c) (1, 2, -1) (d) does not exist
  • 2)

    Consider the following diagram, where the forces in the cable are given.

    Based on the above information, answer the following questions.
    (i) The cartesian equation of line along EA is

    \((a) \ \frac{x}{-4}=\frac{y}{3}=\frac{z}{12}\) \((b) \ \frac{x}{-4}=\frac{y}{3}=\frac{z-24}{12}\) \((c) \ \frac{x}{-3}=\frac{y}{4}=\frac{z-12}{12}\) \((d) \ \frac{x}{3}=\frac{y}{4}=\frac{z-24}{12}\)

    (ii) The vector \(\overline{E D}\) is

    (a) \(8 \hat{i}-6 \hat{j}+24 \hat{k}\) (b) \(-8 \hat{i}-6 \hat{j}+24 \hat{k}\) (c) \(-8 \hat{i}-6 \hat{j}-24 \hat{k}\)  (d) \(8 \hat{i}+6 \hat{j}+24 \hat{k}\)

    (iii) The length of the cable EB is

    (a) 24 units (b) 26 units (c) 27 units (d) 25 units

    (iv) The length of cable EC is equal to the length of

    (a) EA (b) EB (c) ED (d) All of these

    (v) The sum of all vectors along the cables is

    (a) \(96 \hat{i}\) (b) \(96 \hat{j}\) (c) \(-96 \hat{k}\) (d) \(96 \hat{k}\)
  • 3)

    Suppose the floor of a hotel is made up of mirror polished Kota stone. Also, there is a large crystal chandelier attached at the ceiling of the hotel. Consider the floor of the hotel as a plane having equation x - 2y + 2z = 3 and crystal chandelier at the point (3, -2, 1).

    Based on the above information, answer the following questions.
    (i) The d.r's of the perpendicular from the point (3, -2, 1) to the plane x - 2y + 2z = 3, is

    (a) < 1,2,2 > (b) < 1, - 2, 2  > (c) < 2,1,2 > (d) < 2, -1, 2 >

    (ii) The length of the perpendicular from the point (3, -2, 1) to the plane x - 2y + 2z = 3, is

    (a) \(\frac{2}{3}\)units (b) 3 units (c) 2 units (d) none of these

    (iii) The equation of the perpendicular from the point (3, -2, 1) to the plane x - 2y + 2z = 3, is

    (a) \(\frac{x-3}{1}=\frac{y-2}{-2}=\frac{z-1}{2}\) (b) \(\frac{x-3}{1}=\frac{y+2}{-2}=\frac{z-1}{2}\) (c) \(\frac{x+3}{1}=\frac{y+2}{-2}=\frac{z-1}{2}\) (d) none of these

    (iv) The equation of plane parallel to the plane x - 2y + 2z = 3, which is at a unit distance from the point (3, -2, 1) is

    (a) x - 2y + 2z = 0 (b) x - 2y + 2z = 6 (c) x - 2y + 2z = 12 (d) Both (b) and (c)

    (v) The image of the point (3, -2, 1) in the given plane is

    (a) \(\left(\frac{5}{3}, \frac{2}{3}, \frac{-5}{3}\right)\) (b) \(\left(\frac{-5}{3}, \frac{-2}{3}, \frac{5}{3}\right)\) (c) \(\left(\frac{-5}{3}, \frac{2}{3}, \frac{5}{3}\right)\) (d) none of these
  • 4)

    In a diamond exhibition, a diamond is covered in cubical glass box having coordinates 0(0, 0, 0), A(1, 0, 0), B(1, 2, 0), C(0, 2, 0), O'(0,0,3), A'(1, 0, 3), B'(1, 2, 3) and C(0, 2, 3).

    Based on the above information, answer the following questions.
    (i) Direction ratios of OA are

    (a) < 0, 1, 0 > (b) <1, 0, 0> (c) < 0, 0, 1 > (d) none of these

    (ii) Equation of diagonal OB' is

    (a) \(\frac{x}{1}=\frac{y}{2}=\frac{z}{3}\) (b) \(\frac{x}{0}=\frac{y}{1}=\frac{z}{2}\) (c) \(\frac{x}{1}=\frac{y}{0}=\frac{z}{2}\) (d) none of these

    (iii) Equation of plane OABC is

    (a) x = 0 (b) y = 0 (c) z = 0 (d) none of these

    (iv) Equation of plane O' A' B' C is

    (a) x = 3 (b) y = 3 (c) z = 3 (d) z = 2

    (v) Equation of plane ABB' A' is

    (a) x = 1 (b) y = 1 (c) z = 2 (d) x = 3
  • 5)

    The equation of motion of a rocket are: x = 2t, y = -4t, z = 4t, where the time 't' is given in seconds, and the distance measured is in kilometres.

    Based on the above information, answer the following questions.
    (i) What is the path of the rocket?

    (a) Straight line (b) Circle (c) Parabola (d) none of these

    (ii) Which of the following points lie on the path of the rocket?

    (a) (0, 1, 2) (b) (1, -2, 2) (c) (2, -2, 2) (d) none of these

    (iii) At what distance will the rocket be from the starting point (0, 0, 0) in 10 seconds?

    (a) 40 km (b) 60 km (c) 30 km (d) 80 km

    (iv) If the position of rocket at certain instant of time is (3, -6, 6), then what will be the height of the rocket from the ground, which is along the xy-plane?

    (a) 3km (b) 2km (c) 4km (d) 6km

    (v) At certain instant of time, if the rocket is above sea level, where equation of surface of sea is given by 3x - y + 4z = 2 and position of rocket at that instant of time is (1, -2,2), then the image of position of rocket in the sea is 

    (a) \(\left(\frac{20}{13}, \frac{15}{13}, \frac{18}{13}\right)\) (b) \(\left(\frac{-20}{13}, \frac{-15}{13}, \frac{-18}{13}\right)\) (c) \(\left(\frac{20}{13}, \frac{-15}{13}, \frac{18}{13}\right)\) (d) none of these

CBSE 12th Standard Maths Subject Three Dimensional Geometry Case Study Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    A football match is organised between students of class XII of two schools, say school A and school B. For which a team from each school is chosen. Remaining students of class XII of school A and B are respectively sitting on the plane represented by the equation \(\vec{r} \cdot(\hat{i}+\hat{j}+2 \hat{k})=5 \text { and } \vec{r} \cdot(\hat{i}-\hat{j}+\hat{k})=6\) ,to cheer up the team of their respective schools.

    Based on the above information, answer the following questions.
    (i) The cartesian equation of the plane on which students of school A are seated is

    (a) 2x - y +z = 8 (b) 2x + y + z = 8 (c) x + y + 2z = 5 (d) x + y + z = 5

    (ii) The magnitude of the normal to the plane on which students of school B are seated, is

    (a) \(\sqrt 5\) (b) \(\sqrt 6\) (c) \(\sqrt 3 \) (d) \(\sqrt 2\)

    (iii) The intercept form of the equation of the plane on which students of school B are seated, is

    (a) \(\frac{x}{6}+\frac{y}{6}+\frac{z}{6}=1\) (b)  \(\frac{x}{3}+\frac{y}{(-6)}+\frac{z}{6}=1\) (c) \(\frac{x}{3}+\frac{y}{6}+\frac{z}{6}=1\) (d) \(\frac{x}{3}+\frac{y}{6}+\frac{z}{3}=1\)

    (iv) Which of the following is a student of school B?

    (a) Mohit sitting at (1, 2, 1) (b) Ravi sitting at (0,1,2) (c) Khushi sitting at (3, 1, 1) (d) Shewta sitting at (2, -1, 2)

    (v) The distance of the plane, on which students of school B are seated, from the origin is

    (a) 6 units (b) \(\frac{1}{\sqrt{6}}\) units (c) \(\frac{5}{\sqrt{6}}\) units (d) \(\sqrt 6\) units
  • 2)

    Consider the following diagram, where the forces in the cable are given.

    Based on the above information, answer the following questions.
    (i) The cartesian equation of line along EA is

    \((a) \ \frac{x}{-4}=\frac{y}{3}=\frac{z}{12}\) \((b) \ \frac{x}{-4}=\frac{y}{3}=\frac{z-24}{12}\) \((c) \ \frac{x}{-3}=\frac{y}{4}=\frac{z-12}{12}\) \((d) \ \frac{x}{3}=\frac{y}{4}=\frac{z-24}{12}\)

    (ii) The vector \(\overline{E D}\) is

    (a) \(8 \hat{i}-6 \hat{j}+24 \hat{k}\) (b) \(-8 \hat{i}-6 \hat{j}+24 \hat{k}\) (c) \(-8 \hat{i}-6 \hat{j}-24 \hat{k}\)  (d) \(8 \hat{i}+6 \hat{j}+24 \hat{k}\)

    (iii) The length of the cable EB is

    (a) 24 units (b) 26 units (c) 27 units (d) 25 units

    (iv) The length of cable EC is equal to the length of

    (a) EA (b) EB (c) ED (d) All of these

    (v) The sum of all vectors along the cables is

    (a) \(96 \hat{i}\) (b) \(96 \hat{j}\) (c) \(-96 \hat{k}\) (d) \(96 \hat{k}\)
  • 3)

    The Indian Coast Guard (lCG) while patrolling, saw a suspicious boat with four men. They were nowhere looking like fishermen. The soldiers were closely observing the movement of the boat for an opportunity to seize the boat. They observe that the boat is moving along a planar surface. At an instant of time, the coordinates of the position of coast guard helicopter and boat are (2, 3, 5) and (1, 4, 2) respectively.

    Based on the above information, answer the following questions.
    (i) If the line joining the positions of the helicopter and boat is perpendicular to the plane in which boat moves, then equation of plane is

    (a) x-y+3z  = 2 (b) x+y+3z = 2 (c) x - y + 3z = 3 (d) x + y + 3z = 3

    (ii) If the soldier decides to shoot the boat at given instant of time, where the distance measured in metres, then what is the distance that bullet has to travel?

    (a) \(\sqrt 5\) m (b) \(\sqrt 8\) m (c) \(\sqrt 10\) m (d) \(\sqrt 11\) m

    (iii) If the speed of bullet is 30 m/sec, then how much time will the bullet take to hit the boat after the shot is fired?

    (a) 30 seconds (b) 1 second (c) \(\frac{1}{2}\) second (d) \(\frac{\sqrt{11}}{30}\) m

    (iv) At the given instant of time, the equation of line passing through the positions of helicopter and boat is

    (a) \(\frac{x}{1}=\frac{y}{-1}=\frac{z}{3}\) (b) \(\frac{x-1}{1}=\frac{y-4}{-1}=\frac{z-2}{3}\) (c) \(\frac{x}{1}=\frac{y}{1}=\frac{z}{-3}\) (d) \(\frac{x-1}{1}=\frac{y-4}{1}=\frac{z-2}{-3}\)

    (v) At a different instant of time, the boat moves to a different position along the planar surface. What should be the coordinates of the location of the boat for the bullet to hit the boat if soldier shoots the bullet along the line whose equation is \(\frac{x-1}{1}=\frac{y-1}{-2}=\frac{z-2}{3} ?\)

    (a) \(\left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right)\) (b) \(\left(\frac{3}{4}, \frac{3}{2}, \frac{5}{4}\right)\) (c) \(\left(\frac{1}{3}, \frac{1}{4}, \frac{1}{5}\right)\) (d)  none of these
  • 4)

    The equation of motion of a rocket are: x = 2t, y = -4t, z = 4t, where the time 't' is given in seconds, and the distance measured is in kilometres.

    Based on the above information, answer the following questions.
    (i) What is the path of the rocket?

    (a) Straight line (b) Circle (c) Parabola (d) none of these

    (ii) Which of the following points lie on the path of the rocket?

    (a) (0, 1, 2) (b) (1, -2, 2) (c) (2, -2, 2) (d) none of these

    (iii) At what distance will the rocket be from the starting point (0, 0, 0) in 10 seconds?

    (a) 40 km (b) 60 km (c) 30 km (d) 80 km

    (iv) If the position of rocket at certain instant of time is (3, -6, 6), then what will be the height of the rocket from the ground, which is along the xy-plane?

    (a) 3km (b) 2km (c) 4km (d) 6km

    (v) At certain instant of time, if the rocket is above sea level, where equation of surface of sea is given by 3x - y + 4z = 2 and position of rocket at that instant of time is (1, -2,2), then the image of position of rocket in the sea is 

    (a) \(\left(\frac{20}{13}, \frac{15}{13}, \frac{18}{13}\right)\) (b) \(\left(\frac{-20}{13}, \frac{-15}{13}, \frac{-18}{13}\right)\) (c) \(\left(\frac{20}{13}, \frac{-15}{13}, \frac{18}{13}\right)\) (d) none of these
  • 5)

    A mobile tower stands at the top of a hill. Consider the surface on which tower stand as a plane having points A(0, 1,2), B(3, 4, -1) and C(2, 4, 2) on it. The mobile tower is tied with 3 cables from the point A, Band C such that it stand vertically on the ground. The peak of the tower is at the point (6, 5, 9), as shown in the figure.

    Based on the above information, answer the following questions
    (i) The equation of plane passing through the points A, Band C is

    (a) 3x - 4y + z = 0 (b) 3x - 2y + z = 0 (c)  4x - 3y + z = 0 (d) 4x - 3y + 3z = 0

    (ii) The height of the tower from the ground is

    (a) 6 units (b) 5 units (c) \( \frac{17}{\sqrt{14}} units\) (d) \((d) \frac{5}{\sqrt{14}} units\)

    (iii) The equation of line of perpendicular drawn from the peak of tower to the ground is

    (a) \( \frac{x-6}{3}=\frac{y-4}{-2}=\frac{z-9}{1}\) (b) \( \frac{x-6}{3}=\frac{y-5}{-2}=\frac{z-9}{1}\) (c) \(\frac{x-6}{3}=\frac{y-4}{2}=\frac{z-9}{1}\) (d) none of these

    (iv) The coordinates of foot of perpendicular drawn from the peak of tower to the ground are

    (a)\( \left(\frac{33}{14}, \frac{104}{14}, \frac{109}{14}\right) \) (b) \(\left(\frac{33}{14}, \frac{109}{14}, \frac{104}{14}\right)\) (c)\(\left(\frac{33}{14}, \frac{105}{14}, \frac{109}{14}\right)\) (d) none of these

    (v) The area of \(\Delta\)ABC is

    (a) \(\frac{1}{2} \sqrt{14} \text { sq. units }\) (b) \(\frac{3}{2} \sqrt{14} \mathrm{sq} \text { units }\) (c) \(\sqrt{14} \mathrm\ {sq}.\ units\) (d)\(2\sqrt{14} \mathrm\ {sq}.\ units\)

CBSE 12th Standard Maths Subject Vector Algebra Case Study Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Teams A, B, C went for playing a tug of war game. Teams A, B, C have attached a rope to a metal ring and is trying to pull the ring into their own area (team areas shown below).
    Team A pulls with force F1\(\hat{4}+\hat{0} \hat{j}\) KN
    Team B ⟶ F2\(-2 \hat{i}+4 \hat{j}\) KN
    Team C ⟶ F3 = \(-3 \hat{i}-3 \hat{j}\) KN

    Based on the above information, answer the following questions.
    (i) Which team will win the game ?

    (a) Team B (b) Team A (c) Team C (d) No one

    (ii) What is the magnitude of the teams combined force ?

    (a) 7 KN (b) 1.4 KN (c) 1.5 KN (d) 2 KN

    (iii) In what direction is the ring getting pulled?

    (a) 2.0 radian (b) 2.5 radian (c) 2.4 radian (d) 3 radian

    (iv) What is the magnitude of the force of Team B?

    (a) 2\(\sqrt 5\) KN (b) 6 KN (c) 2 KN (d) \(\sqrt 6\) KN

    (v) How many KN force is applied by Team A?

    (a) 5 KN (b) 4 KN (c) 2 KN (d) 16 KN
  • 2)

    Ishaan left from his village on weekend. First, he travelled up to temple. After this, he left for the zoo. After this he left for shopping in a mall. The positions of Ishaan at different places is given in the following graph.

    Based on the above information, answer the following questions.
    (i) Position vector of B is

    (a) \(3 \hat{i}+5 \hat{j}\) (b) \(5 \hat{i}+3 \hat{j}\) (c) \(-5 \hat{i}-3 \hat{j}\) (d) \(-5 \hat{i}+3 \hat{j}\)

    (ii) Position vector of D is

    (a) \(5 \hat{i}+3 \hat{j}\) (b) \(3 \hat{i}+5 \hat{j}\) (c) \(8 \hat{i}+9 \hat{j}\) (d) \(9 \hat{i}+8 \hat{j}\)

    (iii) Find the vector \(\overrightarrow{B C}\) in terms of \(\hat{i}, \hat{j}\).

    (a) \( \hat{i}-2 \hat{j}\) (b) \( \hat{i}+2 \hat{j}\) (c) \(2\hat{i}+ \hat{j}\) (d) \(2\hat{i}- \hat{j}\)

    (iv) Length of vector \(\overrightarrow{A D}\)  is

    (a) \(\sqrt 67\) units (b) \(\sqrt 85\)  units (c) 90 units (d) 100 units

    (v) If \(\vec{M}=4 \hat{\jmath}+3 \hat{k}\) , then its unit vector

    (a) \(\frac{4}{5} \hat{j}+\frac{3}{5} \hat{k}\) (b) \(\frac{4}{5} \hat{j}-\frac{3}{5} \hat{k}\) (c) \(-\frac{4}{5} \hat{j}+\frac{3}{5}\hat{k}\) (d) \(-\frac{4}{5} \hat{j}-\frac{3}{5} \hat{k}\)
  • 3)

    If two vectors are represented by the two sides of a triangle taken in order, then their sum is represented by the third side of the triangle taken in opposite order and this is known as triangle law of vector addition. lased on the above information, answer the following questions.
    (i) If \(\vec{p}, \vec{q}, \vec{r}\)are the vectors represented by the sides of a triangle taken in order, then \( \vec{q} +\vec{r}\) =

    (a) \(\vec{p}\) (b) \(2 \vec{p}\) (c) \(-\vec{p}\) (d) None of these

    (ii) If ABCD is a parallelogram and AC and BD are its diagonals, then\( \vec{AC} +\vec{BD}\) =

    (a) \(2 \vec{DA}\) (b) \(2 \vec{AB}\) (c) \(2\overrightarrow{BC}\) (d) \(2\vec{BD}\)

    (iii) If ABCD is a parallelogram, where \(\overrightarrow{A B}\)\(=2\overrightarrow{a}\) and \(\overrightarrow{BC}\) \(=2\overrightarrow{b}\), then \( \vec{AC} -\vec{BD}\) =

    (a) \(3\vec{a}\) (b) \(4\vec{a}\) (c) \(2\vec{b}\) (d) \(4\vec{b}\)

    (iv) If ABCD is a quadrilateral whose diagonals are \( \vec{AC}\) and \(\vec{BD}\), then \( \vec{BA} +\vec{CD}\) =

    (a) \(\overrightarrow{A C}+\overrightarrow{D B}\) (b) \(\overrightarrow{A C}+\overrightarrow{B D}\) (c) \(\overrightarrow{B C}+\overrightarrow{A D}\) (d) \(\overrightarrow{B D}+\overrightarrow{C A}\)

    (v) If T is the mid point of side YZ of \(\triangle\)XYZ, then\(\overrightarrow{XY}\) + \(\overrightarrow{XZ}\) =

    (a) \(2\vec{YT}\) (b) \(2\vec{XT}\) (c) \(2\vec{TZ}\) (d) None of these
  • 4)

    Three slogans on chart papers are to be placed on a school bulletin board at the points A, Band C displaying A (Hub of Learning), B (Creating a better world for tomorrow) and C (Education comes first). The coordinates of these points are (1, 4, 2), (3, -3, -2) and (-2, 2, 6) respectively.

    Based on the above information, answer the following questions.
    (i) Let \(\vec{a}\), \(\vec{b}\)and \(\vec{c}\) be the position vectors of points A, B and C respectively, then \(\vec{a}\) + \(\vec{b}\)+ \(\vec{c}\) is equal to

    (a) \(2 \hat{i}+3 \hat{j}+6 \hat{k} \) (b) \(2 \hat{i}-3 \hat{j}-6 \hat{k}\) (c) \(2 \hat{i}+8 \hat{j}+3 \hat{k} \) (d) \(2(7 \hat{i}+8 \hat{j}+3 \hat{k})\)

    (ii) Which of the following is not true?

    (a) \(\overrightarrow{A B}+\overrightarrow{B C}+\overrightarrow{C A}=\overrightarrow{0}\) (b) \(\overrightarrow{A B}+\overrightarrow{B C}-\overrightarrow{A C}=\overrightarrow{0}\) (c) \(\overrightarrow{A B}+\overrightarrow{ BC}-\overrightarrow{C A}=\overrightarrow{0}\) (d) \(\overrightarrow{A B}-\overrightarrow{C B}+\overrightarrow{C A}=\overrightarrow{0}\)

    (iii) Area of \(\Delta\)ABC is 

    (a) 19 sq. units (b) \(\sqrt 1937 sq. unit\) (c) \(\frac{1}{2}\sqrt 1937 sq. unit\) (d) \(\sqrt 1837 sq. unit\)

    (iv) Suppose, if the given slogans are to be placed on a straight line, then the value of \(|\vec{a} \times \vec{b}+\vec{b} \times \vec{c}+\vec{c} \times \vec{a}|\) will be equal to

    (a) -1 (b) -2  (c) 2 (d) 0

    (v) If \(\vec{a}=2 \hat{i}+3 \hat{j}+6 \hat{k}\) then unit vector in the direction of vector \(\vec{a}\)is

    (a) \(\frac{2}{7} \hat{i}-\frac{3}{7} \hat{j}-\frac{6}{7} \hat{k}\) (b) \(\frac{2}{7} \hat{i}+\frac{3}{7} \hat{j}+\frac{6}{7} \hat{k}\) (c) \(\frac{3}{7} \hat{i}+\frac{2}{7} \hat{j}+\frac{6}{7} \hat{k}\) (d) None of these
  • 5)

    A barge is pulled into harbour by two tug boats as shown in the figure.

    Based on the above information, answer the following questions.
    (i) Position vector of A is

    (a) \(4 \hat{i}+2 \hat{j}\) (b) \(4 \hat{i}+10 \hat{j}\) (c)\(4 \hat{i}-10 \hat{j}\) (d) \(4 \hat{i}-2 \hat{j}\)

    (ii) Position vector of B is

    (a) \(4 \hat{i}+4 \hat{j}\) (b) \(6 \hat{i}+6 \hat{j}\) (c) \( 9 \hat{i}+7 \hat{j}\) (d) \(3 \hat{i}+3 \hat{j}\)

    (iii) Find the vector \(\vec{AC}\) in terms of \(\hat{i}, \hat{j}\)

    (a) \(8 \hat{j}\) (b) \(-8 \hat{j}\) (c) \(8 \hat{i}\) (d) None of  these

    (iv) If \(\vec{A}=\hat{i}+2 \hat{j}+3 \hat{k}\), then its unit vector is

    (a)\(\frac{\hat{i}}{\sqrt{14}}+\frac{2 \hat{j}}{\sqrt{14}}+\frac{3 \hat{k}}{\sqrt{14}}\) (b) \(\frac{3 \hat{i}}{\sqrt{14}}+\frac{2 \hat{j}}{\sqrt{14}}+\frac{\hat{k}}{\sqrt{14}}\) (c) \(\frac{2 \hat{i}}{\sqrt{14}}+\frac{3 \hat{j}}{\sqrt{14}}+\frac{\hat{k}}{\sqrt{14}}\) (d) None of  these

    (v) If  \(\vec{A}=4 \hat{i}+3 \hat{j}\) and \(\vec{B}=3 \hat{i}+4 \hat{j}\), then IAI+ IBI = ___________.

    (a) 12 (b) 13 (c) 14 (d) 10

CBSE 12th Standard Maths Subject Vector Algebra Case Study Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Ginni purchased an air plant holder which is in the shape of a tetrahedron.
    Let A, B, C and D are the coordinates of the air plant holder where A \(\equiv \) (1, 1, 1), B \(\equiv \) (2, 1, 3), C \(\equiv \) (3, 2, 2) and D \(\equiv \)(3, 3, 4).

    Based on the above information, answer the following questions.
    (i) Find the position vector of \(\overrightarrow{A B} \).

    (a) \(-\hat{i}-2 \hat{k}\) (b) \(2 \hat{i}+\hat{k}\) (c) \(\hat{i}+2 \hat{k}\) (d)\(-2 \hat{i}-\hat{k}\)

    (ii) Find the position vector of \(\overrightarrow{A C} \).

    (a) \(2 \hat{i}-\hat{j}-\hat{k}\) (b) \(2 \hat{i}+\hat{j}+\hat{k}\) (c) \(-2 \hat{i}-\hat{j}+\hat{k}\) (d) \(\hat{i}+2 \hat{j}+\hat{k}\)

    (iii) Find the position vector of \(\overrightarrow{AD} .\).

    (a) \( 2 \hat{i}-2 \hat{j}-3 \hat{k}\) (b) \( \hat{i}+\hat{j}-3 \hat{k}\) (c) \(3 \hat{i}+2 \hat{j}+2 \hat{k}\) (d) \(2\hat{i}+2 \hat{j}+3 \hat{k}\)

    (iv) Area of \(\Delta A B C\)

    (a) \(\frac{\sqrt{11}}{2} \mathrm{sq .units}\) (b) \(\frac{\sqrt{14}}{2} sq. units\) (c) \(\frac{\sqrt{13}}{2}\) (d)\(\frac{\sqrt{17}}{2} \mathrm{sq .units}\)

    (v) Find the unit vector along \(\overrightarrow{AD} .\)

    (a) \(\frac{1}{\sqrt{17}}(2 \hat{i}+2 \hat{j}+3 \hat{k})\)  (b)\(\frac{1}{\sqrt{17}}(3 \hat{i}+3 \hat{j}+2 \hat{k})\) (c) \(\frac{1}{\sqrt{11}}(2 \hat{i}+2 \hat{j}+3 \hat{k})\) (d) \((2 \hat{i}+2 \hat{j}+3 \hat{k})\)
  • 2)

    Teams A, B, C went for playing a tug of war game. Teams A, B, C have attached a rope to a metal ring and is trying to pull the ring into their own area (team areas shown below).
    Team A pulls with force F1\(\hat{4}+\hat{0} \hat{j}\) KN
    Team B ⟶ F2\(-2 \hat{i}+4 \hat{j}\) KN
    Team C ⟶ F3 = \(-3 \hat{i}-3 \hat{j}\) KN

    Based on the above information, answer the following questions.
    (i) Which team will win the game ?

    (a) Team B (b) Team A (c) Team C (d) No one

    (ii) What is the magnitude of the teams combined force ?

    (a) 7 KN (b) 1.4 KN (c) 1.5 KN (d) 2 KN

    (iii) In what direction is the ring getting pulled?

    (a) 2.0 radian (b) 2.5 radian (c) 2.4 radian (d) 3 radian

    (iv) What is the magnitude of the force of Team B?

    (a) 2\(\sqrt 5\) KN (b) 6 KN (c) 2 KN (d) \(\sqrt 6\) KN

    (v) How many KN force is applied by Team A?

    (a) 5 KN (b) 4 KN (c) 2 KN (d) 16 KN
  • 3)

    Ishaan left from his village on weekend. First, he travelled up to temple. After this, he left for the zoo. After this he left for shopping in a mall. The positions of Ishaan at different places is given in the following graph.

    Based on the above information, answer the following questions.
    (i) Position vector of B is

    (a) \(3 \hat{i}+5 \hat{j}\) (b) \(5 \hat{i}+3 \hat{j}\) (c) \(-5 \hat{i}-3 \hat{j}\) (d) \(-5 \hat{i}+3 \hat{j}\)

    (ii) Position vector of D is

    (a) \(5 \hat{i}+3 \hat{j}\) (b) \(3 \hat{i}+5 \hat{j}\) (c) \(8 \hat{i}+9 \hat{j}\) (d) \(9 \hat{i}+8 \hat{j}\)

    (iii) Find the vector \(\overrightarrow{B C}\) in terms of \(\hat{i}, \hat{j}\).

    (a) \( \hat{i}-2 \hat{j}\) (b) \( \hat{i}+2 \hat{j}\) (c) \(2\hat{i}+ \hat{j}\) (d) \(2\hat{i}- \hat{j}\)

    (iv) Length of vector \(\overrightarrow{A D}\)  is

    (a) \(\sqrt 67\) units (b) \(\sqrt 85\)  units (c) 90 units (d) 100 units

    (v) If \(\vec{M}=4 \hat{\jmath}+3 \hat{k}\) , then its unit vector

    (a) \(\frac{4}{5} \hat{j}+\frac{3}{5} \hat{k}\) (b) \(\frac{4}{5} \hat{j}-\frac{3}{5} \hat{k}\) (c) \(-\frac{4}{5} \hat{j}+\frac{3}{5}\hat{k}\) (d) \(-\frac{4}{5} \hat{j}-\frac{3}{5} \hat{k}\)
  • 4)

    If two vectors are represented by the two sides of a triangle taken in order, then their sum is represented by the third side of the triangle taken in opposite order and this is known as triangle law of vector addition. lased on the above information, answer the following questions.
    (i) If \(\vec{p}, \vec{q}, \vec{r}\)are the vectors represented by the sides of a triangle taken in order, then \( \vec{q} +\vec{r}\) =

    (a) \(\vec{p}\) (b) \(2 \vec{p}\) (c) \(-\vec{p}\) (d) None of these

    (ii) If ABCD is a parallelogram and AC and BD are its diagonals, then\( \vec{AC} +\vec{BD}\) =

    (a) \(2 \vec{DA}\) (b) \(2 \vec{AB}\) (c) \(2\overrightarrow{BC}\) (d) \(2\vec{BD}\)

    (iii) If ABCD is a parallelogram, where \(\overrightarrow{A B}\)\(=2\overrightarrow{a}\) and \(\overrightarrow{BC}\) \(=2\overrightarrow{b}\), then \( \vec{AC} -\vec{BD}\) =

    (a) \(3\vec{a}\) (b) \(4\vec{a}\) (c) \(2\vec{b}\) (d) \(4\vec{b}\)

    (iv) If ABCD is a quadrilateral whose diagonals are \( \vec{AC}\) and \(\vec{BD}\), then \( \vec{BA} +\vec{CD}\) =

    (a) \(\overrightarrow{A C}+\overrightarrow{D B}\) (b) \(\overrightarrow{A C}+\overrightarrow{B D}\) (c) \(\overrightarrow{B C}+\overrightarrow{A D}\) (d) \(\overrightarrow{B D}+\overrightarrow{C A}\)

    (v) If T is the mid point of side YZ of \(\triangle\)XYZ, then\(\overrightarrow{XY}\) + \(\overrightarrow{XZ}\) =

    (a) \(2\vec{YT}\) (b) \(2\vec{XT}\) (c) \(2\vec{TZ}\) (d) None of these
  • 5)

    A barge is pulled into harbour by two tug boats as shown in the figure.

    Based on the above information, answer the following questions.
    (i) Position vector of A is

    (a) \(4 \hat{i}+2 \hat{j}\) (b) \(4 \hat{i}+10 \hat{j}\) (c)\(4 \hat{i}-10 \hat{j}\) (d) \(4 \hat{i}-2 \hat{j}\)

    (ii) Position vector of B is

    (a) \(4 \hat{i}+4 \hat{j}\) (b) \(6 \hat{i}+6 \hat{j}\) (c) \( 9 \hat{i}+7 \hat{j}\) (d) \(3 \hat{i}+3 \hat{j}\)

    (iii) Find the vector \(\vec{AC}\) in terms of \(\hat{i}, \hat{j}\)

    (a) \(8 \hat{j}\) (b) \(-8 \hat{j}\) (c) \(8 \hat{i}\) (d) None of  these

    (iv) If \(\vec{A}=\hat{i}+2 \hat{j}+3 \hat{k}\), then its unit vector is

    (a)\(\frac{\hat{i}}{\sqrt{14}}+\frac{2 \hat{j}}{\sqrt{14}}+\frac{3 \hat{k}}{\sqrt{14}}\) (b) \(\frac{3 \hat{i}}{\sqrt{14}}+\frac{2 \hat{j}}{\sqrt{14}}+\frac{\hat{k}}{\sqrt{14}}\) (c) \(\frac{2 \hat{i}}{\sqrt{14}}+\frac{3 \hat{j}}{\sqrt{14}}+\frac{\hat{k}}{\sqrt{14}}\) (d) None of  these

    (v) If  \(\vec{A}=4 \hat{i}+3 \hat{j}\) and \(\vec{B}=3 \hat{i}+4 \hat{j}\), then IAI+ IBI = ___________.

    (a) 12 (b) 13 (c) 14 (d) 10

CBSE 12th Standard Maths Subject Application of Integrals Case Study Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Consider the curve x2 +y2 = 16 and line y = x in the first quadrant. Based on the above information, answer the following questions.
    (i) Point of intersection of both the given curves is

    (a) (0, 4) (b) \((0,2 \sqrt{2})\) (c) \((2 \sqrt{2}, 2 \sqrt{2})\) (d) \((2 \sqrt{2}, 4)\)

    (ii) Which of the following shaded portion represent the area bounded by given two curves?
     
    (iii) The value of the integral \(\int_{0}^{2 \sqrt{2}} x d x\) is

    (a) 0 (b) 1 (c) 2 (d).4

    (iv) The value of the integral \(\int_{2 \sqrt{2}}^{4} \sqrt{16-x^{2}} d x\) is

    (a) \(2(\pi-2)\) (b) \(2(\pi-8)\) (c) \(4(\pi-2)\) (d) \(4(\pi+2)\)

    (v) Area bounded by the two given curves is

    (a) \(3 \pi \text { sq. units }\) (b) \(\frac{\pi}{2} \text { sq. units }\) (c) \(\pi \text { sq. units }\) (d) \(2 \pi \text { sq. units }\)
  • 2)

    Consider the following equation of curve I' = 4x and straight line x + y = 3.
    Based on the above information, answer the following questions.
    (i) The line x + y = 3 cuts the x-axis and y-axis respectively at

    (a) (0, 2), (2, 0) (b) (3, 3), (0, 0) (c) (0, 3), (3, 0) (d) (3, 0), (0, 3)

    (ii) Point(s) of intersection of two given curves is (are)

    (a) (1, -2), (-9, 6) (b) (2, 1), (-6, 9) (c) (1, 2), (9, -6) (d) None of these

    (iii) Which of the following shaded portion represent the area bounded by given curves?
     
    (iv) Value of the integral \(\int_{-6}^{2}(3-y) d y\) is

    (a) 10 (b) 20 (c) 30 (d) 40

    (v) Value of area bounded by given curves is

    (a) 56 sq. units (b) \(\frac{63}{5} \text { sq; units }\) (c) \(\frac{64}{3} \text { sq. units }\) (d) 31 sq. units
  • 3)

    Location of three houses of a society is represented by the points A(-1, 0), B(1, 3) and C(3, 2) as shown in figure. Based on the above information, answer the following questions

    (i) Equation of line AB is

    (a) \(y=\frac{3}{2}(x+1)\) (b) \(y=\frac{3}{2}(x-1)\) (c) \(y=\frac{1}{2}(x+1)\) (d) \(y=\frac{1}{2}(x-1)\)

    (ii) Equation of line BC is

    (a) \(y=\frac{1}{2} x-\frac{7}{2}\) (b)  \(y=\frac{3}{2} x-\frac{7}{2}\) (c) \(y=\frac{-1}{2} x+\frac{7}{2}\) (d) \(y=\frac{3}{2} x+\frac{7}{2}\)

    (iii) Area of region ABCD is

    (a) 2 sq. units (b) 4 sq. units (c) 6 sq. units (d) 8 sq. units

    (iv) Area of \(\Delta A D C\) is

    (a) 4 sq. units (b) 8 sq. units (c) 16 sq. units (d) 32 sq. units

    (iv) Area of \(\Delta A B C\) is

    (a) 3 sq. units (b) 4 sq. units (c) 5 sq. units (d) 6 sq. units
  • 4)

    Ajay cut two circular pieces of cardboard and placed one upon other as shown in figure. One of the circle represents the equation (x - 1)2 +1 = 1, while other circle represents the equation x2 +1 = 1.

    Based on the above information, answer the following questions.
    (i) Both the circular pieces of cardboard meet each other at

    (a) x = 1 (b) \(x=\frac{1}{2}\) (c) \(x=\frac{1}{3}\) (d) \(x=\frac{1}{4}\)

    (ii) Graph of given two curves can be drawn as
     
    (iii) Value of \(\int_{0}^{1 / 2} \sqrt{1-(x-1)^{2}} d x\) is

    (a) \(\frac{\pi}{6}-\frac{\sqrt{3}}{8}\) (b) \(\frac{\pi}{6}+\frac{\sqrt{3}}{8}\) (c) \(\frac{\pi}{6}-\frac{\sqrt{3}}{8}\) (d) \(\frac{\pi}{2}-\frac{\sqrt{3}}{4}\)

    (iv) Value of \(\int_{1 / 2}^{1} \sqrt{1-x^{2}} d x\) is

    (a) \(\frac{\pi}{6}-\frac{\sqrt{3}}{8}\) (b) \(\frac{\pi}{6}+\frac{\sqrt{3}}{8}\) (c) \(\frac{\pi}{6}-\frac{\sqrt{3}}{8}\) (d) \(\frac{\pi}{2}-\frac{\sqrt{3}}{4}\)

    (v) Area of hidden portion of lower circle is

    (a) \(\left(\frac{2 \pi}{3}+\frac{\sqrt{3}}{2}\right) \text { sq. units }\) (b) \(\left(\frac{\pi}{3}-\frac{\sqrt{3}}{8}\right) \text { sq. units }\) (c) \(\left(\frac{\pi}{3}+\frac{\sqrt{3}}{8}\right) \text { sq. units }\) (d) \(\left(\frac{2 \pi}{3}-\frac{\sqrt{3}}{2}\right) \text { sq. units }\)
  • 5)

    Consider the following equations of curves y = cos x, y = x + 1 and y = 0. On the basis of above information, answer the following questions.
    (i) The curves y = cos x and y = x + 1 meet at

    (a) (1, 0) (b) (0, 1) (c) (1, 1) (d) (0,0)

    (ii) y = cos x meet the x-axis at

    (a) \(\left(\frac{-\pi}{2}, 0\right)\) (b) \(\left(\frac{\pi}{2}, 0\right)\) (c) both (a) and (b) (d) None of these

    (iii) Value of the integral \(\int_{-1}^{0}(x+1) d x\) is

    (a) \(\frac{1}{2}\) (b) \(\frac{2}{3}\) (c) \(\frac{3}{4}\) (d) \(\frac{1}{3}\)

    (iv) Value of the integral \(\int_{0}^{\pi / 2} \cos x d x\) is

    (a) 0 (b) -1 (c) 2 (d) 1

    (v) Area bounded by the given curves is

    (a) \(\frac{1}{2} \mathrm{sq} . \text { unit }\) (b) \(\frac{3}{2} \text { sq. units }\) (c) \(\frac{3}{4} \text { sq. unit }\) (d) \(\frac{1}{4} \text { sq. unit }\)

CBSE 12th Standard Maths Subject Application of Integrals Case Study Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Consider the curve x2 +y2 = 16 and line y = x in the first quadrant. Based on the above information, answer the following questions.
    (i) Point of intersection of both the given curves is

    (a) (0, 4) (b) \((0,2 \sqrt{2})\) (c) \((2 \sqrt{2}, 2 \sqrt{2})\) (d) \((2 \sqrt{2}, 4)\)

    (ii) Which of the following shaded portion represent the area bounded by given two curves?
     
    (iii) The value of the integral \(\int_{0}^{2 \sqrt{2}} x d x\) is

    (a) 0 (b) 1 (c) 2 (d).4

    (iv) The value of the integral \(\int_{2 \sqrt{2}}^{4} \sqrt{16-x^{2}} d x\) is

    (a) \(2(\pi-2)\) (b) \(2(\pi-8)\) (c) \(4(\pi-2)\) (d) \(4(\pi+2)\)

    (v) Area bounded by the two given curves is

    (a) \(3 \pi \text { sq. units }\) (b) \(\frac{\pi}{2} \text { sq. units }\) (c) \(\pi \text { sq. units }\) (d) \(2 \pi \text { sq. units }\)
  • 2)

    Consider the following equation of curve I' = 4x and straight line x + y = 3.
    Based on the above information, answer the following questions.
    (i) The line x + y = 3 cuts the x-axis and y-axis respectively at

    (a) (0, 2), (2, 0) (b) (3, 3), (0, 0) (c) (0, 3), (3, 0) (d) (3, 0), (0, 3)

    (ii) Point(s) of intersection of two given curves is (are)

    (a) (1, -2), (-9, 6) (b) (2, 1), (-6, 9) (c) (1, 2), (9, -6) (d) None of these

    (iii) Which of the following shaded portion represent the area bounded by given curves?
     
    (iv) Value of the integral \(\int_{-6}^{2}(3-y) d y\) is

    (a) 10 (b) 20 (c) 30 (d) 40

    (v) Value of area bounded by given curves is

    (a) 56 sq. units (b) \(\frac{63}{5} \text { sq; units }\) (c) \(\frac{64}{3} \text { sq. units }\) (d) 31 sq. units
  • 3)

    Location of three houses of a society is represented by the points A(-1, 0), B(1, 3) and C(3, 2) as shown in figure. Based on the above information, answer the following questions

    (i) Equation of line AB is

    (a) \(y=\frac{3}{2}(x+1)\) (b) \(y=\frac{3}{2}(x-1)\) (c) \(y=\frac{1}{2}(x+1)\) (d) \(y=\frac{1}{2}(x-1)\)

    (ii) Equation of line BC is

    (a) \(y=\frac{1}{2} x-\frac{7}{2}\) (b)  \(y=\frac{3}{2} x-\frac{7}{2}\) (c) \(y=\frac{-1}{2} x+\frac{7}{2}\) (d) \(y=\frac{3}{2} x+\frac{7}{2}\)

    (iii) Area of region ABCD is

    (a) 2 sq. units (b) 4 sq. units (c) 6 sq. units (d) 8 sq. units

    (iv) Area of \(\Delta A D C\) is

    (a) 4 sq. units (b) 8 sq. units (c) 16 sq. units (d) 32 sq. units

    (iv) Area of \(\Delta A B C\) is

    (a) 3 sq. units (b) 4 sq. units (c) 5 sq. units (d) 6 sq. units
  • 4)

    A mirror in the shape of an ellipse represented by \(\frac{x^{2}}{9}+\frac{y^{2}}{4}=1\) was hanging on the wall. Arun and his sister were playing with ball inside the house, even their mother refused to do so. All of sudden, ball hit the mirror and got a scratch in the shape of line represented by  \(\frac{x}{3}+\frac{y}{2}=1\) .

    Based on the above information, answer the following questions
    (i) Point(s) of intersection of ellipse and scratch (straight line) is (are)

    (a) (0, 2), (3, 0) (b) (2, 0), (0, 3) (c) (2, 3), (0, 0) (d) (0, 3), (3, 0)

    (ii) Area of smaller region bounded by the ellipse and line is represented by
     
    (iii) The value of \(\frac{2}{3} \int_{0}^{3} \sqrt{9-x^{2}} d x\) is

    (a) \(\frac{\pi}{2}\) (b) \(\pi\) (c) \(\frac{3 \pi}{2}\) (d) \(\frac{\pi}{4}\)

    (iv) The value of \(2 \int_{0}^{3}\left(1-\frac{x}{3}\right) d x\) is 

    (a) 0 (b) 1 (c) 2 (d) 3

    (v) Area of the smaller region bounded by the mirror and scratch is

    (a) \(3\left(\frac{\pi}{2}+1\right) \text { sq. units }\) (b) \(\left(\frac{\pi}{2}+1\right) \text { sq. units }\) (c) \(\left(\frac{\pi}{2}-1\right) \text { sq. units }\) (d) \(3\left(\frac{\pi}{2}-1\right) \text { sq. units }\)
  • 5)

    Consider the following equations of curves y = cos x, y = x + 1 and y = 0. On the basis of above information, answer the following questions.
    (i) The curves y = cos x and y = x + 1 meet at

    (a) (1, 0) (b) (0, 1) (c) (1, 1) (d) (0,0)

    (ii) y = cos x meet the x-axis at

    (a) \(\left(\frac{-\pi}{2}, 0\right)\) (b) \(\left(\frac{\pi}{2}, 0\right)\) (c) both (a) and (b) (d) None of these

    (iii) Value of the integral \(\int_{-1}^{0}(x+1) d x\) is

    (a) \(\frac{1}{2}\) (b) \(\frac{2}{3}\) (c) \(\frac{3}{4}\) (d) \(\frac{1}{3}\)

    (iv) Value of the integral \(\int_{0}^{\pi / 2} \cos x d x\) is

    (a) 0 (b) -1 (c) 2 (d) 1

    (v) Area bounded by the given curves is

    (a) \(\frac{1}{2} \mathrm{sq} . \text { unit }\) (b) \(\frac{3}{2} \text { sq. units }\) (c) \(\frac{3}{4} \text { sq. unit }\) (d) \(\frac{1}{4} \text { sq. unit }\)

CBSE 12th Standard Maths Subject Application of Derivatives Case Study Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Western music concert is organised every year in the stadium that can hold 36000 spectators. With ticket price of Rs. 10, the average attendance has been 24000. Some financial expert estimated that price of a ticket should be determined by the function \(p(x)=15-\frac{x}{3000}\), where x is the number of tickets sold.

    Based on the above information, answer the following questions.
    (i) The revenue, R as a function of xcan be represented as

    (a) \(15 x-\frac{x^{2}}{3000}\) (b) \(15-\frac{x^{2}}{3000}\) (c) \(15 x-\frac{1}{30000}\) (d) \(15 x-\frac{x}{3000}\)

    (ii) The range of x is

    (a) [24000, 36000] (b) [0, 24000] (c) [0, 36000] (d) none of these

    (iii) The value of xfor which revenue is maximum, is

    (a) 20000 (b) 21000 (c) 22500 (d) 25000

    (iv) When the revenue is maximum, the price of the ticket is

    (a) Rs. 5 (b) Rs. 5.5 (c) Rs. 7 (d) Rs. 7.5

    (v) How any spectators should be present to maximize the revenue?

    (a) 21500 (b) 21000 (c) 22000 (d) 22500
  • 2)

    A tin can manufacturer designs a cylindrical tin can for a company making sanitizer and disinfector. The tin can is made to hold 3 litres of sanitizer or disinfector.

    Based on the above in formation, answer the following questions. 
    (i) If r cm be the radius and h em be the height of the cylindrical tin can, then the surface area expressed as a function of r as

    (a)  \(2 \pi r^{2}\) (b) \(\sqrt{\frac{500}{\pi}} \mathrm{cm}\) (c) \(\sqrt[3]{\frac{1500}{\pi}} \mathrm{cm}\) (d) \(2 \pi r^{2}+\frac{6000}{r}\)

    (ii) The radius that will minimize the cost of the material to manufacture the tin can is

    (a) \(\sqrt[3]{\frac{600}{\pi}} \mathrm{cm}\) (b) \(\sqrt{\frac{500}{\pi}} \mathrm{cm}\) (c) \(\sqrt[3]{\frac{1500}{\pi}} \mathrm{cm}\) (d) \(\sqrt{\frac{1500}{\pi}} \mathrm{cm}\)

    (iii) The height thatt will minimize the cost of the material to manufacture the tin can is

    (a) \(\sqrt[3]{\frac{1500}{\pi}} \mathrm{cm}\) (b) \(2 \sqrt[3]{\frac{1500}{\pi}} \mathrm{cm}\) (c) \(\sqrt{\frac{1500}{\pi}}\) (d) \(2 \sqrt{\frac{1500}{\pi}}\)

    (iv) If the cost of material used to manufacture the tin can is Rs.100/m2 and \(\sqrt[3]{\frac{1500}{\pi}} \approx 7.8\) then minimum cost is approximately

    (a) Rs. 11.538 (b) Rs. 12 (c) Rs. 13 (d) Rs. 14

    (v) To minimize the cost of the material used to manufacture the tin can, we need to minimize the

    (a) volume (b) curved surface area (c) total surface area (d) surface area of the base
  • 3)

    Nitin wants to construct a rectangular plastic tank for his house that can hold 80 ft 3 of water. The top of the tank is open. The width of tank will be 5 ft but the length and heights are variables. Building the tank cost Rs.20 per sq. foot for the base and Rs. 10 per square foot for the side.

    Based on the above information, answer the following questions.

    (i) In order to make a least expensive water tank, Nitin need to minimize its

    (a) Volume (b) Base (c) Curved surface area (d) Cost

    (ii) Total cost of tank as a function of h can' be' represented as

    (a) c(h) = 100 h - 320 - 1600lh (b) (h) = 100 h - 320 h - 720 h2
    (c) c(h) = 100 + 220 h + 1600 h2 (d) \(c(h)=100 h+320+\frac{1600}{h}\)

    (iii) Range of h is

    (a) (3,5) (b) \((0, \infty)\) (c) (0,8) (d) (0,3)

    (iv) Value of h at which c(h) is minimum, is

    (a) 4 (b) 5 (c) 6 (d) 6.7

    (v) The cost ofleast expensive tank is

    (a) Rs. 1020 (b) Rs. 1100 (c) Rs. 1120 (d) Rs. 1220
  • 4)

    Shreya got a rectangular parallelopiped shaped box and spherical ball inside it as return gift. Sides of the box are x, 2x, and x/3, while radius of the ball is r.

    Based on the above information, answer the following questions.
    (i) If S represents the sum of volume of parallelopiped and sphere, then S can be written as

    (a) \(\frac{4 x^{3}}{3}+\frac{2}{2} \pi r^{2}\) \(\frac{2 x^{2}}{3}+\frac{4}{3} \pi r^{2}\) \(\frac{2 x^{3}}{3}+\frac{4}{3} \pi r^{3}\) \(\frac{2}{3} x+\frac{4}{3} \pi r\)

    (ii) If sum of the surface areas of box and ball are given to be constant k2 , then x is equal to

    (a) \(\sqrt{\frac{k^{2}-4 \pi r^{2}}{6}}\) (b) \(\sqrt{\frac{k^{2}-4 \pi r}{6}}\) (c) \(\sqrt{\frac{k^{2}-4 \pi}{6}}\) (d) none of these

    (iii) The radius of the ball, when S is minimum, is

    (a) \(\sqrt{\frac{k^{2}}{54+\pi}}\) (b) \(\sqrt{\frac{k^{2}}{54+4 \pi}}\) (c) \(\sqrt{\frac{k^{2}}{64+3 \pi}}\) (d) \(\sqrt{\frac{k^{2}}{4 \pi+3}}\)

    (iv) Relation between length of the box and radius of the ball can be represented as

    (a) x = 2r (b) \(x=\frac{r}{2}\) (c) \(x=\frac{r}{2}\) (d) \(\sqrt{\frac{k^{2}}{4 \pi+3}}\)

    (v) Minimum value of S is

    (a) \(\frac{k^{2}}{2(3 \pi+54)^{2 / 3}}\) (b) \(\frac{k}{(3 \pi+54)^{3 / 2}}\) (c) \(\frac{k^{3}}{3(4 \pi+54)^{1 / 2}}\) (d) none of these
  • 5)

    Kyra has a rectangular painting canvas a toatl area of 24ft2 which include a border of 0.5ft on the left,right and a border 0.75 ft on the bottom,top inside it.

    Based on the above information, answer the following questions.
    (i) If Kyra wants to paint in the maximum area: then she needs to maximize

    (a) Area of outer rectangle (b) Area of inner rectangle
    (c) Area of top border (d) None of these

    (ii) If x is the length of the outer rectangle, then area of inner rectangle in terms of x is

    (a) \((x+3)\left(\frac{24}{x}-2\right)\) (b) \((x-1)\left(\frac{24}{x}+1.5\right)\) (c) \((x-1)\left(\frac{24}{x}-1.5\right)\)  (d) \((x-1)\left(\frac{24}{x}\right)\)

    (iii) Find the range of x.

    (a) \((1, \infty)\) (b) (1, 16) (c) \((-\infty, 16)\) (d) (-1, 16)

    (iv) If area of inner rectangle is m~imum, then x is equal to

    (a) 2ft (b) 3ft (c) 4ft (d) 5 ft

    (v) If area of inner rectangle is maximum, then length and breadth of this rectangle are respectively

    (a) 3ft,4.Sft (b) 4.5ft,Sft (c) 1ft,2ft (d) 2ft,4ft

CBSE 12th Standard Maths Subject Application of Derivatives Case Study Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Megha wants to prepare a handmade gift box for her friend's birthday at home. For making lower part of box, she takes a square piece of cardboard of side 20 cm.

    Based on the above information, answer the following questions.
    (i) If x cm be the length of each side of the square cardboard which is to be cut off from corners of the square piece of side 20 cm, then possible value of x will be given by the interval

    (a) [0, 20] (b) (0, 10) (c) (0, 3) (d) None of these

    (ii) Volume of the open box formed by folding up the cutting corner can be expressed as

    (a) V = x(20 - 2x)(20 - 2x) (b) \(\begin{equation} V=\frac{x}{2}(20+x)(20-x) \end{equation}\)
    (c) \(\begin{equation} V=\frac{x}{3}(20-2 x)(20+2 x) \end{equation}\) (d) V = x(20 - 2x)(20 - x)

    (iii) The values of x for which \(\begin{equation} \frac{d V}{d x}=0 \end{equation}\) ,are

    (a) 3, 4 (b)  \(\begin{equation} 0, \frac{10}{3} \end{equation}\) (c) 0, 10 (d) \(\begin{equation} 10, \frac{10}{3} \end{equation}\)

    (iv) Megha is interested in maximising the volume of the box. So, what should be the side of the square to be cut off so that the volume of the box is maximum?

    (a) 12 cm (b) 8 cm (c) \(\begin{equation} \frac{10}{3} \mathrm{~cm} \end{equation}\) (d) 2 cm

    (v) The maximum value of the volume is

    (a) \(\begin{equation} \frac{17000}{27} \mathrm{~cm}^{3} \end{equation}\) (b) \(\begin{equation} \frac{11000}{27} \mathrm{~cm}^{3} \end{equation}\) (c) \(\begin{equation} \frac{8000}{27} \mathrm{~cm}^{3} \end{equation}\) (d) \(\begin{equation} \frac{16000}{27} \mathrm{~cm}^{3} \end{equation}\)
  • 2)

    Shobhit's father wants to construct a rectangular garden using a brick wall on one side of the garden and wire fencing for the other three sides as shown in' figure. He has 200 ft of wire fencing.
     
    Based on the above information, answer the following questions.
    (i) To construct a garden using 200 ft of fencing, we need to maximise its

    (a) volume (b) area (c) perimeter (d) length of the side

    (ii) If x denote the length of side of garden perpendicular to brick wall and y denote the length, of side parallel to brick wall, then find the relation representing total amount of fencing wire.

    (a) x + 2y = 150 (b) x+2y=50 (c) y+2x=200 (d) y+2x=100

    (iii) Area of the garden as a function of x, say A(x), can be represented as

    (a) 200 + 2x2 (b) x - 2x2 (c) 200x - 2x2 (d) 200-x2

    (iv) Maximum value of A(x) occurs at x equals

    (a) 50 ft (b) 30 ft (c) 26ft (d) 31 ft

    (v) Maximum area of garden will be

    (a) 2500 sq.ft (b) 4000 sq.ft (c) 5000 sq.ft (d) 6000 sq. ft
  • 3)

    The Government declare that farmers can get Rs.300 per quintal for their onions on 1st July and after that,the price will be dropped by Rs. 3 per quintal per extra day.
    Shyams father has 80 quintal of onions in the field on 1st July and he estimates that crop is increasing at the rate of 1 quintal per day.

    Based on the above information, answer the following questions.
    (i) If x is the number of days after 1st July, then price and quantity ofonion respectively can be expressed as

    (a) Rs. (300 - 3x), (80 + x) quintals (b) Rs. (300 - 3x), (80 - x) quintals
    (c) Rs. (300 + x), 80 quintals (d) None of these

    (ii) Revenue R as a function of x can be represented as

    (a) R(x) = 3x2 - 60x - 24000 (b) R(x) = -3x2 + 60x + 24000
    (c) R(x) = 3x2 + 40x - 16000 (d) R(x) = 3x2- 60x - 14000

    (iii) Find the number of days after 1stJuly, when Shyams father attain maximum revenue.

    (a) 10 (b) 20 (c) 12 (d) 22

    (iv) On which day should Shyam's father harvest the onions to maximise his revenue?

    (a) 11thuly (b) 20th July (c) 12th July (d) 22nd July

    (v) Maximum revenue is equal to

    (a) Rs. 20,000 (b) Rs. 24,000 (c) Rs. 24,300 (d) Rs. 24,700
  • 4)

    An owner of an electric bi~e rental company have determined that if they charge customers Rs. x per day to rent a bike, where 50 Rs. x Rs. 200, then number of bikes (n), they rent per day can be shown by linear function n(x) = 2000 - 10x. If they charge Rs. 50 per day or less, they will rent all their bikes. If they charge Rs. 200 or more per day, they will not rent any bike. Based on the above information, answer the following questions.


    Based on the above information, answer the following questions
    (i) Total revenue R as a function of x can be represented as

    (a) 2000x - 10x2 (b) 2000x + 10x2 (c) 2000 - 10x (d) 2000 - 5x2

    (ii) If R(x) denote the revenue, then maximum value of R(x) occur when x equals

    (a) 10 (b) 100 (c) 1000 (d) 50

    (iii) At x = 260, the revenue collected by the company is

    (a) Rs. 10 (b) Rs. 500 (c) Rs. 0 (d) Rs. 1000

    (iv) The number of bikes rented per day, if x = 105 is

    (a) 850 (b) 900 (c) 950 (d) 1000

    (v) Maximum revenue collected by company is

    (a) Rs. 40,000 (b) Rs. 50,000 (c) Rs. 75,000 (d) Rs. 1,00,000
  • 5)

    Mr. Sahil is the owner of a high rise residential society having 50 apartments. When he set rent at Rs. 10000/month, all apartments are rented. If he increases rent by Rs. 250/ month, one fewer apartment is rented. The maintenance cost for each occupied unit is Rs. 500/month. Based on the above information answer the following questions.

    Based on the above information answer the following questions.
    (i) If P is the rent price per apartment and N is the number of rented apartment, then profit is given by

    (a) NP (b) (N - 500)P (c) N(P - 500) (d) none of these

    (ii) If x represent the number of apartments which are not rented, then the profit expressed as a function of x is

    (a) (50 - x) (38 + x) (b) (50 + x) (38 - x) (c) 250(50 - x) (38 + x) (d) 250(50 + x) (38 - x)

    (iii) If P = 10500, then N =

    (a) 47 (b) 48 (c) 49 (d) 50

    (iv) If P = 11,000, then the profit is

    (a) Rs. 11000 (b) Rs. 11500 (c) Rs. 15800 (d) Rs.16500

CBSE 12th Standard Maths Subject Continuity and Differentiability Case Study Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Let f(x) be a real valued function, then its 
    Left Hand Derivative (L.H.D.) : \(\begin{equation} \mathrm{L} f^{\prime}(a)=\lim _{h \rightarrow 0} \frac{f(a-h)-f(a)}{-h} \end{equation}\) 
    Right Hand Derivative (R.H.D.) : \(\begin{equation} \mathrm{Rf}^{\prime}(a)=\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h} \end{equation}\) 
    Also, a function jfx) is said to be differentiable at x = a if its L.H.D. and R.H.D. at x = a exist and are equal 
    For the function \(\begin{equation} f(x)=\left\{\begin{array}{l} |x-3|, x \geq 1 \\ \frac{x^{2}}{4}-\frac{3 x}{2}+\frac{13}{4}, x<1 \end{array}\right. \end{equation}\) answer the following questions
    (i) R.H.D. of f(x) at x = 1is

    (a) 1 (b) -1 (c) 0 (d) 2

    (ii) L.H.D. of f(x) at x = 1 is

    (a) 1 (b) -1 (c) 0 (d) 2

    (iii) f(x) is non-differentiable at

    (a) x = 1 (b) x = 2 (c) x = 3 (d) x = 4

    (iv) Find the value of f'(2).

    (a) 1 (b) 2 (c) 3 (d) -1

    (v) The value of f'( -1) is

    (a) 2 (b) 1 (c) -2 (d) -1
  • 2)

    Let x = f(t) and y = get) be parametric forms with t as a parameter,
    then \(\begin{equation} \frac{d y}{d x}=\frac{d y}{d t} \times \frac{d t}{d x}=\frac{g^{\prime}(t)}{f^{\prime}(t)} \end{equation}\) ,where \(\begin{equation} f^{\prime}(t) \neq 0 \end{equation}\) \(\begin{equation} \frac{d y}{d x}=\frac{d y}{d t} \times \frac{d t}{d x}=\frac{g^{\prime}(t)}{f^{\prime}(t)} \end{equation}\) where \(\begin{equation} f^{\prime}(t) \neq 0 \end{equation}\).
    (i) The derivative off (tanx) w.r.t. \(\begin{equation} g(\sec x) \text { at } x=\frac{\pi}{4} \end{equation}\) ,where f'(1) and \(\begin{equation} g^{\prime}(\sqrt{2})=4 \end{equation}\) is

    (a) \(\begin{equation} \frac{1}{\sqrt{2}} \end{equation}\) (b) \(\begin{equation} \sqrt{2} \end{equation}\)  (c) 1 (d) 0

    (ii) The derivate of \(\begin{equation} \sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right) \end{equation}\) ,with respect to \(\begin{equation} \cos ^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right) \end{equation}\) is

    (a) -1 (b) 1 (c) 2 (d) 4

    (iii) The derivative of \(\begin{equation} e^{x^{3}} \end{equation}\) with respect to log x is

    (a) \(\begin{equation} e^{x^{3}} \end{equation}\) (b) \(\begin{equation} 3 x^{2} 2 e^{x^{3}} \end{equation}\) (c) \(\begin{equation} 3 x^{3} e^{x^{3}} \end{equation}\) (d) \(\begin{equation} 3 x^{2} e^{x^{3}}+3 x \end{equation}\)

    (iv) The derivative of \(\begin{equation} \cos ^{-1}\left(2 x^{2}-1\right) \end{equation}\) w.r.t. cos-1x is

    (a) 2 (b) \(\begin{equation} \frac{-1}{2 \sqrt{1-x^{2}}} \end{equation}\) (c) \(\begin{equation} \frac{2}{x} \end{equation}\) (d) 1 -x2

    (v) If \(\begin{equation} y=\frac{1}{4} u^{4} \end{equation}\)  and \(\begin{equation} u=\frac{2}{3} x^{3}+5 \end{equation}\) then \(\begin{equation} \frac{d y}{d x}= \end{equation}\)

    (a) \(\begin{equation} \frac{2}{27} x^{2}\left(2 x^{3}+15\right)^{3} \end{equation}\) (b) \(\begin{equation} \frac{2}{7} x^{2}\left(2 x^{3}+15\right)^{3} \end{equation}\) (c) \(\begin{equation} \frac{2}{27} x\left(2 x^{3}+5\right)^{3} \end{equation}\) (d) \(\begin{equation} \frac{2}{7}\left(2 x^{3}+15\right)^{3} \end{equation}\)
  • 3)

    Let \(\begin{equation} f: A \rightarrow B \end{equation}\) and \(\begin{equation} g: B \rightarrow C \end{equation}\) be two functions defined on non-empty sets A, B, C,
    then \(\begin{equation} \text { gof }: A \rightarrow C \end{equation}\)  be is called the composition off and g defined as, \(\begin{equation} g o f(x)=g\{f(x)\} \forall x \in A \end{equation}\) .
    Consider the functions \(\begin{equation} f(x)=\left\{\begin{array}{ll} \sin x, & x \geq 0 \\ 1-\cos x, & x \leq 0 \end{array}, g(x)=e^{x}\right. \end{equation}\) and
    then answer the following questions. 
    (i) The function gof(x) is defined as

    (a) \(\begin{equation} g o f(x)=\left\{\begin{array}{ll} e^{x} & , x \geq 0 \\ 1-e^{\cos x} & , x \leq 0 \end{array}\right. \end{equation}\) (b) \(\begin{equation} \operatorname{gof}(x)=\left\{\begin{array}{ll} e^{\sin x} & , x \leq 0 \\ e^{1-\cos x} & , x \geq 0 \end{array}\right. \end{equation}\) 
    (c) \(\begin{equation} g o f(x)=\left\{\begin{array}{ll} e^{\sin x} & , x \leq 0 \\ 1-e^{\cos x} & , x \geq 0 \end{array}\right. \end{equation}\) (d)  \(\begin{equation} g o f(x)=\left\{\begin{array}{ll} e^{\sin x} & , x \geq 0 \\ e^{1-\cos x} & , x \leq 0 \end{array}\right. \end{equation}\)

    (ii) \(\begin{equation} \frac{d}{d x}\{\operatorname{gof}(x)\}= \end{equation}\) 

    (a) \(\begin{equation} [g o f(x)]^{\prime}=\left\{\begin{array}{ll} \cos x \cdot e^{\sin x} & , x \geq 0 \\ e^{1-\cos x} \cdot \sin x & , x \leq 0 \end{array}\right. \end{equation}\) (b) \(\begin{equation} [g o f(x)]^{\prime}=\left\{\begin{array}{ll} \cos x \cdot e^{\sin x} & , x \geq 0 \\ -\sin x \cdot e^{1-\cos x} & , x \leq 0 \end{array}\right. \end{equation}\)
    (c) \(\begin{equation} [g o f(x)]^{\prime}=\left\{\begin{array}{ll} \cos x \cdot e^{\sin x} & , x \geq 0 \\ \sin x \cdot(1-\cos x) & , x \leq 0 \end{array}\right. \end{equation}\) (d) \(\begin{equation} [g o f(x)]^{\prime}=\left\{\begin{array}{ll} \cos x \cdot e^{\sin x} & , x \geq 0 \\ (1-\sin x) \cdot e^{1-\cos x} & , x \leq 0 \end{array}\right. \end{equation}\)

    (iii) R.H.D. of gof(x) at x = 0 is

    (a) 0 (b) 1 (c) -1 (d) 2

    (iv) L.H.D. of gof(x) at x = 0 is

    (a) 0 (b) 1 (c) -1 (d) 2

    (v) The value of \(\begin{equation} f^{\prime}(x) \text { at } x=\frac{\pi}{4} \end{equation}\) is 

    (a) 1/9 (b) \(\begin{equation} 1 / \sqrt{2} \end{equation}\) (c) 1/2 (d) not defined
  • 4)

    The function f(x) will be discontinuous at x = a if f(x) has 
    (a) Discontinuity of first kind \(\begin{equation} \lim _{h \rightarrow 0} f(a-h) \text { and } \lim _{h \rightarrow 0} f(a+h) \end{equation}\) both exist but are not equal. If is also known as irremovable discontinuity.
    (b)  Discontinuity of second kind: If none of the limits \(\begin{equation} \lim _{h \rightarrow 0} f(a-h) \text { and } \lim _{h \rightarrow 0} f(a+h) \end{equation}\) exist.
    (c) Removable discontinuity: \(\begin{equation} \lim _{h \rightarrow 0} f(a-h) \text { and } \lim _{h \rightarrow 0} f(a+h) \end{equation}\) both exist and equal but not equal to f(a). 
    Based on the above information, answer the following questions.
    (i) If \(\begin{equation} f(x)=\left\{\begin{array}{ll} \frac{x^{2}-9}{x-3}, & \text { for } x \neq 3 \\ 4, & \text { for } x=3 \end{array}\right. \end{equation}\) ,then at x= 3

    (a) f has removable discontinuity (b) f is continuous
    (c) f has irremovable discontinuity (d) none of these

    (ii) Let \(\begin{equation} f(x)=\left\{\begin{array}{ll} x+2, & \text { if } x \leq 4 \\ x+4, & \text { if } x>4 \end{array}\right. \end{equation}\) ,then at x = 4

    (a) f is continuous (b) f has removable discontinuity
    (c) f has irremovable discontinuity (d) none of these

    (iii) Consider the function f(x) defined \(\begin{equation} f(x)=\left\{\begin{array}{l} \frac{x^{2}-4}{x-2} \\ 5 \end{array}\right. \end{equation}\), for \(\begin{equation} x \neq 2 \end{equation}\) 

    (a) f has removable discontinuity (b) f has irremovable discontinuity
    (c) f is continuous (d) f is continuous if f(2) = 3

     (iv) If \(\begin{equation} f(x)=\left\{\begin{array}{cc} \frac{x-|x|}{x}, & if\ x \neq 0 \\ 2, & if\ x=0 \end{array}\right. \end{equation}\) ,then x = 0

    a) f is continuous (b) f has removable discontinuity
    (c) f has irremovable discontinuity (d) none of these

    (v) If \(\begin{equation} f^{\prime}(x)=\left\{\begin{array}{cl} \frac{e^{x}-1}{\log (1+2 x)}, & \text { if } x \neq 0 \\ 7, & \text { if } x=0 \end{array}\right. \end{equation}\), then at x = 0

    (a) f is continuous if f(0) = 2 (b) f is continuous
    (c) f has irremovable discontinuity (d) f has removable discontinuity
  • 5)

    If a real valued function f(x) is finitely derivable at any point of its domain, it is necessarily continuous at that point. But its converse need not be true.
    For example, every polynomial. constant function are both continuous as well as differentiable and inverse trigonometric functions are continuous and differentiable in its domains etc.
    Based on the above information, answer the following questions.
    (i) If \(\begin{equation} f(x)=\left\{\begin{array}{l} x, \text { for } x \leq 0 \\ 0, \text { for } x>0 \end{array}\right. \end{equation}\) , then at x = 0

    (a) f(x) is differentiable and continuous (b) j(x) is neither continuous nor differentiable
    (c) f(x) is continuous but not differentiable (d) none of these

    (ii) If \(\begin{equation} f(x)=|x-1|, x \in R \end{equation}\) ,then at x= 1 

    (a) f(x) is not continuous (b) f(x) is continuous but not differentiable
    (c) f(x) is continuous and differentiable (d) none of these

    (iii) f(x) = x3 is

    (a) continuous but not differentiable at x = 3 (b) continuous and differentiable at x = 3
    (c) neither continuous nor differentiable at x = 3 (d) none of these

    (iv) f(x) = [sin x], then which of the following is true? 

    (a) j(x) is continuous and differentiable at x = o. (b) j(x) is discontinuous at x = o.
    (c) j(x) is continuous at x = 0 but not differentiable (d) fix) is differentiable but not continuous at  \(\begin{equation} x=\pi / 2 \end{equation}\)

    (v) If f(x) = sin-1x, \(\begin{equation} -1 \leq x \leq 1 \end{equation}\), then

    (a) f(x) is both continuous and differentiable (b) f(x) is neither continuous nor differentiable.
    (c) f(x) is continuous but not differentiable (d) None of these

CBSE 12th Standard Maths Subject Continuity and Differentiability Case Study Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Let f(x) be a real valued function, then its 
    Left Hand Derivative (L.H.D.) : \(\begin{equation} \mathrm{L} f^{\prime}(a)=\lim _{h \rightarrow 0} \frac{f(a-h)-f(a)}{-h} \end{equation}\) 
    Right Hand Derivative (R.H.D.) : \(\begin{equation} \mathrm{Rf}^{\prime}(a)=\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h} \end{equation}\) 
    Also, a function jfx) is said to be differentiable at x = a if its L.H.D. and R.H.D. at x = a exist and are equal 
    For the function \(\begin{equation} f(x)=\left\{\begin{array}{l} |x-3|, x \geq 1 \\ \frac{x^{2}}{4}-\frac{3 x}{2}+\frac{13}{4}, x<1 \end{array}\right. \end{equation}\) answer the following questions
    (i) R.H.D. of f(x) at x = 1is

    (a) 1 (b) -1 (c) 0 (d) 2

    (ii) L.H.D. of f(x) at x = 1 is

    (a) 1 (b) -1 (c) 0 (d) 2

    (iii) f(x) is non-differentiable at

    (a) x = 1 (b) x = 2 (c) x = 3 (d) x = 4

    (iv) Find the value of f'(2).

    (a) 1 (b) 2 (c) 3 (d) -1

    (v) The value of f'( -1) is

    (a) 2 (b) 1 (c) -2 (d) -1
  • 2)

    The function f(x) will be discontinuous at x = a if f(x) has 
    (a) Discontinuity of first kind \(\begin{equation} \lim _{h \rightarrow 0} f(a-h) \text { and } \lim _{h \rightarrow 0} f(a+h) \end{equation}\) both exist but are not equal. If is also known as irremovable discontinuity.
    (b)  Discontinuity of second kind: If none of the limits \(\begin{equation} \lim _{h \rightarrow 0} f(a-h) \text { and } \lim _{h \rightarrow 0} f(a+h) \end{equation}\) exist.
    (c) Removable discontinuity: \(\begin{equation} \lim _{h \rightarrow 0} f(a-h) \text { and } \lim _{h \rightarrow 0} f(a+h) \end{equation}\) both exist and equal but not equal to f(a). 
    Based on the above information, answer the following questions.
    (i) If \(\begin{equation} f(x)=\left\{\begin{array}{ll} \frac{x^{2}-9}{x-3}, & \text { for } x \neq 3 \\ 4, & \text { for } x=3 \end{array}\right. \end{equation}\) ,then at x= 3

    (a) f has removable discontinuity (b) f is continuous
    (c) f has irremovable discontinuity (d) none of these

    (ii) Let \(\begin{equation} f(x)=\left\{\begin{array}{ll} x+2, & \text { if } x \leq 4 \\ x+4, & \text { if } x>4 \end{array}\right. \end{equation}\) ,then at x = 4

    (a) f is continuous (b) f has removable discontinuity
    (c) f has irremovable discontinuity (d) none of these

    (iii) Consider the function f(x) defined \(\begin{equation} f(x)=\left\{\begin{array}{l} \frac{x^{2}-4}{x-2} \\ 5 \end{array}\right. \end{equation}\), for \(\begin{equation} x \neq 2 \end{equation}\) 

    (a) f has removable discontinuity (b) f has irremovable discontinuity
    (c) f is continuous (d) f is continuous if f(2) = 3

     (iv) If \(\begin{equation} f(x)=\left\{\begin{array}{cc} \frac{x-|x|}{x}, & if\ x \neq 0 \\ 2, & if\ x=0 \end{array}\right. \end{equation}\) ,then x = 0

    a) f is continuous (b) f has removable discontinuity
    (c) f has irremovable discontinuity (d) none of these

    (v) If \(\begin{equation} f^{\prime}(x)=\left\{\begin{array}{cl} \frac{e^{x}-1}{\log (1+2 x)}, & \text { if } x \neq 0 \\ 7, & \text { if } x=0 \end{array}\right. \end{equation}\), then at x = 0

    (a) f is continuous if f(0) = 2 (b) f is continuous
    (c) f has irremovable discontinuity (d) f has removable discontinuity
  • 3)

    (a) A function f(x) is said to be continuous in an open interval (a, b), if it is continuous at every point in this interval.
    (b) A function f(x) is said to be continuous in the closed interval [a, b], if f(x) is continuous in (a, b) and \(\begin{equation} \lim _{h \rightarrow 0} f(a+h)=f(a) \text { and } \lim _{h \rightarrow 0} f(b-h)=f(b) \end{equation}\) 
    If function  \(\begin{equation} f(x)=\left\{\begin{array}{ll} \frac{\sin (a+1) x+\sin x}{x} & , x<0 \\ c & , x=0 \\ \frac{\sqrt{x+b x^{2}}-\sqrt{x}}{b x^{3 / 2}} & , x>0 \end{array}\right. \end{equation}\) is continuous at x = 0, then answer the following questions.
    (i) The value of a is

    (a)  -3/2 (b) 0  (c) 1/2 (d) -1/2

    (ii) The value of b is

    (a) 1 (b) -1 (c) 0 (d) any real number

    (iii) The value of c is

    (a) 1 (b) 1/2 (c) -1 (d) -1/2

    (iv) The value of a + c is

    (a) 1 (b) 0 (c) -1 (d) -2

    (v) The value oi c - a is

    (a) 1 (b) 0 (c) -1 (d) 2
  • 4)

    If y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x, then y = f[g(x)] is a differentiable function of x and \(\begin{equation} \frac{d y}{d x}=\frac{d y}{d u} \times \frac{d u}{d x} \end{equation}\). This rule is also known as CHAIN RULE.
    Based on the above information, find the derivative of functions w.r.t. x in the following questions
    (i) \(\begin{equation} \cos \sqrt{x} \end{equation}\) 

    (a) \(\begin{equation} \frac{-\sin \sqrt{x}}{2 \sqrt{x}} \end{equation}\) (b) \(\begin{equation} \frac{\sin \sqrt{x}}{2 \sqrt{x}} \end{equation}\) (c) \(\begin{equation} \sin \sqrt{x} \end{equation}\) (d) \(\begin{equation} -\sin \sqrt{x} \end{equation}\)

    (ii) \(\begin{equation} 7^{x+\frac{1}{x}} \end{equation}\) 

    (a) \(\begin{equation} \left(\frac{x^{2}-1}{x^{2}}\right) \cdot 7^{x+\frac{1}{x}} \cdot \log 7 \end{equation}\) (b) \(\begin{equation} \left(\frac{x^{2}+1}{x^{2}}\right) \cdot 7^{x+\frac{1}{x}} \cdot \log 7 \end{equation}\) (c) \(\begin{equation} \left(\frac{x^{2}-1}{x^{2}}\right) \cdot 7^{x-\frac{1}{x}} \cdot \log 7 \end{equation}\) (d) \(\begin{equation} \left(\frac{x^{2}+1}{x^{2}}\right) \cdot 7^{x-\frac{1}{x}} \cdot \log 7 \end{equation}\)

    (iii) \(\begin{equation} \sqrt{\frac{1-\cos x}{1+\cos x}} \end{equation}\) 

    (a) \(\begin{equation} \frac{-1}{x^{2}+b^{2}}+\frac{1}{x^{2}+a^{2}} \end{equation}\) (b) \(\begin{equation} \frac{1}{x^{2}+b^{2}}+\frac{1}{x^{2}+a^{2}} \end{equation}\) (c) \(\begin{equation} \frac{1}{x^{2}+b^{2}}-\frac{1}{x^{2}+a^{2}} \end{equation}\) (d) none of these

    (v) (d) :\(\begin{equation} \sec ^{-1} x+\operatorname{cosec}^{-1} \frac{x}{\sqrt{x^{2}-1}} \end{equation}\)  

    (a) \(\begin{equation} \frac{2}{\sqrt{x^{2}-1}} \end{equation}\) (b) \(\begin{equation} \frac{-2}{\sqrt{x^{2}-1}} \end{equation}\) (c) \(\begin{equation} \frac{1}{|x| \sqrt{x^{2}-1}} \end{equation}\) (d) \(\begin{equation} \frac{2}{|x| \sqrt{x^{2}-1}} \end{equation}\)
  • 5)

    If a relation between x and y is such that y cannot be expressed in terms of x, then y is called an implicit function of x.
    When a given relation expresses y as an implicit function of x and we want to find \(\begin{equation} \frac{d y}{d x} \end{equation}\).then
    we differentiate every term of the given relation w.r.t. x. remembering that a term in y is first differentiated w.r.t. y and then multiplied by \(\begin{equation} \frac{d y}{d x} \end{equation}\).
    Based on the above information, find the value of \(\begin{equation} \frac{d y}{d x} \end{equation}\) in each of the following questions 
    (i) x3+x2y+xy2+y3=81

    (a) \(\begin{equation} \frac{\left(3 x^{2}+2 x y+y^{2}\right)}{x^{2}+2 x y+3 y^{2}} \end{equation}\) (b) \(\begin{equation} \frac{-\left(3 x^{2}+2 x y+y^{2}\right)}{x^{2}+2 x y+3 y^{2}} \end{equation}\) (c) \(\begin{equation} \frac{\left(3 x^{2}+2 x y-y^{2}\right)}{x^{2}-2 x y+3 y^{2}} \end{equation}\) (d) \(\begin{equation} \frac{3 x^{2}+x y+y^{2}}{x^{2}+x y+3 y^{2}} \end{equation}\)

    (ii) xy = c- y

    (a) \(\begin{equation} \frac{x-y}{(1+\log x)} \end{equation}\) (b) \(\begin{equation} \frac{x+y}{(1+\log x)} \end{equation}\) (c) \(\begin{equation} \frac{x-y}{x(1+\log x)} \end{equation}\) (d) \(\begin{equation} \frac{x+y}{x(1+\log x)} \end{equation}\)

    (iii) esiny = xy

    (a) \(\begin{equation} \frac{-y}{x(y \cos y-1)} \end{equation}\) (b) \(\begin{equation} \frac{y}{y \cos y-1} \end{equation}\) (c) \(\begin{equation} \frac{y}{y \cos y+1} \end{equation}\) (d) \(\begin{equation} \frac{y}{x(y \cos y-1)} \end{equation}\)

    (iv) sin2 x + cos2y = 1

    (a) \(\begin{equation} \frac{\sin 2 y}{\sin 2 x} \end{equation}\) (b) \(\begin{equation} -\frac{\sin 2 x}{\sin 2 y} \end{equation}\) (c) \(\begin{equation} -\frac{\sin 2 y}{\sin 2 x} \end{equation}\) (d) \(\begin{equation} \frac{\sin 2 x}{\sin 2 y} \end{equation}\)

    (v) \(\begin{equation} y=(\sqrt{x})^{\sqrt{x}} \end{equation}\) 

    (a) \(\begin{equation} \frac{-y^{2}}{x(2-y \log x)} \end{equation}\) (b) \(\begin{equation} \frac{y^{2}}{2+y \log x} \end{equation}\) (c) \(\begin{equation} \frac{y^{2}}{x(2+y \log x)} \end{equation}\) (d) \(\begin{equation} \frac{y^{2}}{x(2-y \log x)} \end{equation}\)

CBSE 12th Standard Maths Subject Matrices Case Study Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    In a city there are two factories A and B. Each factory produces sports clothes for boys and girls. There are three types of clothes produced in both the factories, type I, II and III. For boys the number of units of types I, II and III respectively are 80, 70'and 65 in factory A and 85, 65 and 72 are in factory B. For girls the number of units of types I, II and III respectively are 80, 75, 90 in factory A and 50, 55, 80 are in factory B.
     
    Based on the above information, answer the following questions.
    (i) If P represents the matrix of number of units of each type produced by factory A for both boys and girls, then P is given by

    (ii) If Q represents the matrix of number of units of each type produced by factory B for both boys and girls, then Q is given by
     
    (iii) The total- production of sports clothes of each type for boys is given by the matrix
     
    (iv) The total production of sports clothes of each type for girls is given by the matrix

    (v) Let R be a 3 x 2 matrix that represent the total production of sports clothes of each type for boys and girls, then transpose of R is(iv) The total production of sports clothes of each type for girls is given by the matrix

  • 2)

    To promote the making of toilets for women, an organisation tried to generate awareness through (i) house calls (ii) emails and (iii) announcements. The cost for each mode per attempt is given below:

    (i) Rs.50 (ii) Rs.20 (iii) Rs.40
    The number of attempts made in the villages X, Y and Z are given below:
    \(\begin{array}{llll} & (\mathrm{i}) & (\mathrm{ii}) & (\mathrm{iii}) \\ X & 400 & 300 & 100 \\ Y & 300 & 250 & 75 \\ Z & 500 & 400 & 150 \end{array}\) 
    Also, the chance of making of toilets corresponding to one attempt of given modes is
    (i) 2% (ii) 4% (iii) 20%
    Based on the above information, answer the following questions.
    (i) The cost incurred by the organisation on village X is

     (a) 10000   (b)  Rs.15000   (c) 30000  (d) Rs.20000

    (ii) The cost incurred by the organisation on village Y is

      (a) Rs.25000  (b) Rs.18000 (c) Rs.23000  (d) Rs.28000

    (iii) The cost incurred by the organisation on village Z is

     (a)  Rs.19000  (b)  Rs.39000  (c)  Rs.4500  (d)  Rs.5000

    (iv) The total number of toilets that can be expected after the promotion in village X, is

    (a)  20 (b)  30  (c)  40 (d)  50

    (v) The total number of toilets that can be expected after the promotion in village Z, is

    (a) 26  (b) 36  (c)  46  (d)  56
  • 3)

    Two farmers Shyam and Balwan Singh cultivate only three varieties of pulses namely Urad, Masoor and Mung. The sale (in Rs.) of these varieties of pulses by both the farmers in the month of September and October are given by the following matrices A and B.
     
     
    Using algebra of matrices, answer the following questions.
    (i) The combined sales of Masoor in September and October, for farmer Balwan Singh, is

    (a) Rs. 80000 (b) Rs. 90000 (c) Rs. 40000 (d) Rs. 135000

    (ii) The combined sales of Urad in September and October, for farmer Shyam is

    (a) Rs. 20000 (b) Rs. 30000 (c) Rs. 36000 (d) Rs. 15000

    (iii) Find the decrease in sales of Mung from September to October, for the farmer Shyam.

    (a) Rs. 24000 (b) Rs. 10000 (c) Rs. 30000 (d) No change

    (iv) If both farmers receive 2% profit on gross sales, compute the profit for each farmer and for each variety sold in October. 
     
    (v) Which variety of pulse has the highest selling value in the month of September for the farmer Balwan Singh?

    (a) Urad (b) Masoor (c) Mung (d) All of these have the same price
  • 4)

    If \(A=\left[a_{i j}\right]_{m \times n} \text { and } B=\left[b_{i j}\right]_{m \times n}\) are two matrices, then \(A \pm B\) is of order m x n and is defined as \((A \pm B)_{i j}=a_{i j} \pm b_{i j}\), where i = 1,2, , m and  j = 1,2, ..., n
    If \(A=\left[a_{i j}\right]_{m \times n} \text { and } B=\left[b_{j k}\right]_{n \times p}\) are two matrices, then AB is of order m x p and is defined as \((A B)_{i k}=\sum_{r=1}^{n} a_{i r} b_{r k}=a_{i 1} b_{1 k}+a_{i 2} b_{2 k}+\ldots . .+a_{i n} b_{n k}\) 
    Consider \(A=\left[\begin{array}{cc} 2 & -1 \\ 3 & 4 \end{array}\right], B=\left[\begin{array}{ll} 5 & 2 \\ 7 & 4 \end{array}\right], C=\left[\begin{array}{ll} 2 & 5 \\ 3 & 8 \end{array}\right] \text { and } D=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]\) 
    Using the concept of matrices answer the following questions. 
    (i) Find the product AB. 

    (a) \(\left[\begin{array}{cc} 3 & 0 \\ 43 & 22 \end{array}\right]\)  (b) \(\left[\begin{array}{cc} 0 & 3 \\ 22 & 43 \end{array}\right]\) (c) \(\left[\begin{array}{cc} 43 & 22 \\ 0 & 3 \end{array}\right]\) (d) \(\left[\begin{array}{cc} 22 & 43 \\ 3 & 0 \end{array}\right]\) 

    (ii) If A and B are any other two matrices such that AB exists, then

    (a) BA does not exist  (b) BA will be equal to AB  (c) BA mayor may not exist  (d) None of these

    (iii) Find the values of a and c in the matrix D such than CD - AB = 0.

    (a)  a = 77, c=-191 (b) a = -191, c=77 (c) a =  191, c=77 (d) a = 91, c = 70

    (iv) Find the values of band d in the matrix D such that CD - AB = 0.

    (a) b = 44, d = -110  (b) b = 110, d = 44  (c) b = -110, d = 44  (d) b = -44, d = 110

    (v) Find B + D.

    (a) \(\left[\begin{array}{cc} 80 & 200 \\ 115 & 105 \end{array}\right]\) (b) \(\left[\begin{array}{cc} 84 & 48 \\ 180 & 181 \end{array}\right]\)  (c) \(\left[\begin{array}{ll} 186 & 108 \\ -84 & -48 \end{array}\right]\) (d) \(\left[\begin{array}{cc} -186 & -108 \\ 84 & 48 \end{array}\right]\)
  • 5)

    A trust fund has Rs. 35000 that must be invested in two different types of bonds, say X and Y. The first bond pays 10% interest p.a. which will be given to an old age home and second one pays 8% interest p.a. which will be given to WWA (Women Welfare Association). Let A be a 1 x 2 matrix and B be a 2 x 1 matrix, representing the investment and interest rate on each bond respectively.

    Based on the above information, answer the following questions.
    (i) . If Rs.15000 is invested in bond X, then
     
    (ii) If Rs.15000 is invested in bond X, then total amount.of interest received on both bonds is

    (a) Rs.2000 (b) Rs.2100 (c) Rs. 3100 (d) Rs.4000

    (iii) If the trust fund obtains an annual total interest of Rs.3200, then the investment in two bonds is

    (a) Rs. 15000 in X,  Rs. 20000 in Y (b) Rs. 17000 in X,  Rs. 18000 in Y (c) Rs. 20000 in X,  Rs. 15000 in Y (d) Rs. 18000 in X,  Rs. 17000 in Y

    (iv) The total amount of interest received on both bonds is given by

    (a) AB (b) A'B (c) B'A (d) none of these

    (v) If the amount of interest given to old age home is Rs.500, then the amount of investment in bond Y is

    (a) Rs. 20000 (b) Rs. 30000 (c) Rs. 15000 (d) Rs. 25000

CBSE 12th Standard Maths Subject Matrices Case Study Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    In a city there are two factories A and B. Each factory produces sports clothes for boys and girls. There are three types of clothes produced in both the factories, type I, II and III. For boys the number of units of types I, II and III respectively are 80, 70'and 65 in factory A and 85, 65 and 72 are in factory B. For girls the number of units of types I, II and III respectively are 80, 75, 90 in factory A and 50, 55, 80 are in factory B.
     
    Based on the above information, answer the following questions.
    (i) If P represents the matrix of number of units of each type produced by factory A for both boys and girls, then P is given by

    (ii) If Q represents the matrix of number of units of each type produced by factory B for both boys and girls, then Q is given by
     
    (iii) The total- production of sports clothes of each type for boys is given by the matrix
     
    (iv) The total production of sports clothes of each type for girls is given by the matrix

    (v) Let R be a 3 x 2 matrix that represent the total production of sports clothes of each type for boys and girls, then transpose of R is(iv) The total production of sports clothes of each type for girls is given by the matrix

  • 2)

    Three car dealers, say A, Band C, deals in three types of cars, namely Hatchback cars, Sedan cars, SUV cars. The sales figure of 2019 and 2020 showed that dealer A sold 120 Hatchback, 50 Sedan, 10 SUV cars in 2019 and 300 Hatchback, 150 Sedan, 20 SUV cars in 2020; dealer B sold 100 Hatchback, 30 Sedan,S SUV cars in 2019 and 200 Hatchback, 50 Sedan, 6 SUV cars in 2020; dealer C sold 90 Hatchback, 40 Sedan, 2 SUV cars in 2019 and 100 Hatchback, 60 Sedan,S SUV cars in 2020.
     
    Based on the above information, answer the following questions.
    (i) The matrix summarizing sales data of 2019 is
     
    (ii) The matrix summarizing sales data of 2020 is
     
     (iii) The total number of cars sold in two given years, by each dealer, is given by the matrix
     
    (iv) The increase in sales from 2019 to 2020 is given by the matrix
     
    (v) If each dealer receive profit of Rs. 50000 on sale of a Hatchback, Rs. 100000 on sale of a Sedan and Rs. 200000 on sale of a SUV (v) then amount of profit received in the year 2020 by each dealer is given by the matrix.

     

  • 3)

    Three schools A, Band C organized a mela for collecting funds for helping the rehabilitation of flood victims. They sold hand made fans, mats and plates from recycled material at a cost of Rs. 25, Rs.100 and Rs.50 each. The number of articles sold by school A, B, C are given below.

    Artilcle\School A B C
    Fans 40 25 35
    Mats 50 40 50
    Plates 20 30 40

    Based on above information, answer the following questions.
    (i) If P be a 3 x 3 matrix represent the sale of handmade fans, mats and plates by three schools A, Band C, then
     
    (ii) If Q be a 3 x 1 matrix represent the sale prices (in Rs) of given products per unit, then

    (iii) The funds collected by school A by selling the given articles is

    (a) Rs. 7000 (b) Rs. 6125 (c) Rs. 7875 (d) Rs. 8000

    (iv) The funds collected by school B by selling the given articles is

    (a) Rs. 5125 (b) Rs. 6125 (c) Rs. 7125 (d) Rs. 8125

    (v) The total funds collected for the required purpose is

    (a) Rs. 20000 (b) Rs. 21000 (c) Rs. 30000 (d) Rs. 35000
  • 4)

    A manufacturer produces three types of bolts, x, y and z which he sells in two markets. Annual sales (in Rs) are indicated below:

    Markets Products
    x y z
    I 10000 2000 18000
    II 6000 20000 8000

    If unit sales prices of x, y and z are Rs.2.50, Rs.1.50 and Rs.1.00 respectively, then answer the following questions using the concept of matrices.
    (i) Find the total revenue collected from the Market-I.

    (a) Rs. 44000 (b) Rs. 48000 (c) Rs. 46000 (d) Rs. 53000

    (ii) Find the total revenue collected from the Market-II.

    (a) Rs. 5100  (b) Rs. 5300  (c ) Rs. 46000  (d) Rs. 49000

    (iii) If the unit costs of the above three commodities are Rs.2.00, Rs.1.00 and 50 paise respectively, then find the gross profit from both the markets.

    (a) Rs. 53000  (b) Rs. 46000  (c) Rs. 34000  (d) Rs. 32000

    (iv) If matrix \(4=\left[a_{i j}\right]_{2 \times 2}\) , where \(a_{i j}=1, \text { if } i \neq j\) , and \(a_{i j}=0 \text { if } i=j\) , then A2 is equal to

    (a) I  (b) A   (c) 0  (d) none of these

    (v) If A and B are matrices of same order, then (AB' - BA') is a

    (a) skew-symmetric matrix  (b) null matrix  (c) symmetric matrix  (d) unit matrix
  • 5)

    If \(A=\left[a_{i j}\right]_{m \times n} \text { and } B=\left[b_{i j}\right]_{m \times n}\) are two matrices, then \(A \pm B\) is of order m x n and is defined as \((A \pm B)_{i j}=a_{i j} \pm b_{i j}\), where i = 1,2, , m and  j = 1,2, ..., n
    If \(A=\left[a_{i j}\right]_{m \times n} \text { and } B=\left[b_{j k}\right]_{n \times p}\) are two matrices, then AB is of order m x p and is defined as \((A B)_{i k}=\sum_{r=1}^{n} a_{i r} b_{r k}=a_{i 1} b_{1 k}+a_{i 2} b_{2 k}+\ldots . .+a_{i n} b_{n k}\) 
    Consider \(A=\left[\begin{array}{cc} 2 & -1 \\ 3 & 4 \end{array}\right], B=\left[\begin{array}{ll} 5 & 2 \\ 7 & 4 \end{array}\right], C=\left[\begin{array}{ll} 2 & 5 \\ 3 & 8 \end{array}\right] \text { and } D=\left[\begin{array}{ll} a & b \\ c & d \end{array}\right]\) 
    Using the concept of matrices answer the following questions. 
    (i) Find the product AB. 

    (a) \(\left[\begin{array}{cc} 3 & 0 \\ 43 & 22 \end{array}\right]\)  (b) \(\left[\begin{array}{cc} 0 & 3 \\ 22 & 43 \end{array}\right]\) (c) \(\left[\begin{array}{cc} 43 & 22 \\ 0 & 3 \end{array}\right]\) (d) \(\left[\begin{array}{cc} 22 & 43 \\ 3 & 0 \end{array}\right]\) 

    (ii) If A and B are any other two matrices such that AB exists, then

    (a) BA does not exist  (b) BA will be equal to AB  (c) BA mayor may not exist  (d) None of these

    (iii) Find the values of a and c in the matrix D such than CD - AB = 0.

    (a)  a = 77, c=-191 (b) a = -191, c=77 (c) a =  191, c=77 (d) a = 91, c = 70

    (iv) Find the values of band d in the matrix D such that CD - AB = 0.

    (a) b = 44, d = -110  (b) b = 110, d = 44  (c) b = -110, d = 44  (d) b = -44, d = 110

    (v) Find B + D.

    (a) \(\left[\begin{array}{cc} 80 & 200 \\ 115 & 105 \end{array}\right]\) (b) \(\left[\begin{array}{cc} 84 & 48 \\ 180 & 181 \end{array}\right]\)  (c) \(\left[\begin{array}{ll} 186 & 108 \\ -84 & -48 \end{array}\right]\) (d) \(\left[\begin{array}{cc} -186 & -108 \\ 84 & 48 \end{array}\right]\)

CBSE 12th Standard Maths Subject Determinants Case Study Questions With Solutions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Let \(\begin{equation} A=\left[\begin{array}{ll} 1 & 0 \\ 2 & 1 \end{array}\right] \end{equation}\) , and U1 U2 are first and second columns respectively of a 2 x 2 matrix U.
    Also, let the column matrices UI and U2 satisfying \(\begin{equation} A U_{1}=\left[\begin{array}{l} 1 \\ 0 \end{array}\right] \end{equation}\) and \(\begin{equation} A U_{2}=\left[\begin{array}{l} 2 \\ 3 \end{array}\right] \end{equation}\) 
    Based on the above information, answer the following questions
    (i) The matrix U1 + U2 is equal to

    (a) \(\begin{equation} \left[\begin{array}{c} 1 \\ -1 \end{array}\right] \end{equation}\) (b) \(\begin{equation} \left[\begin{array}{c} 2 \\ -2 \end{array}\right] \end{equation}\) (c) \(\begin{equation} \left[\begin{array}{c} 3 \\ -3 \end{array}\right] \end{equation}\) (d) \(\begin{equation} \left[\begin{array}{c} 4 \\ -4 \end{array}\right] \end{equation}\)

    (ii) The value of IUI is

    (a) 2 (b) -2 (c) 3 (d) -3

    (iii) If \(\begin{equation} X=\left[\begin{array}{ll} 3 & 2 \end{array}\right] U\left[\begin{array}{l} 3 \\ 2 \end{array}\right] \end{equation}\),then the value of IXI =

    (a) 3 (b) -3 (c) -5 (d) 5

    (iv) The minor of element at the position a22 in U is

    (a) 1 (b) 2 (c) -2 (d) -1

    (v) If \(\begin{equation} U=\left[a_{i j}\right]_{2 \times 2} \end{equation}\) , then the value of a11A11 + a12Al2 where Aij denotes the cofactor of aij is

    (a) 1 (b) 2 (c) -3 (d) 3
  • 2)

    Two schools A and B want to award their selected students on the values of Honesty, Hard work and Punctuality. The school A wants to award Rs. x each, Rs. y each and Rs.z each for the three respective values to its 3, 2 and 1 students respectivefy with a total award money of Rs. 2200. School B wants to spend Rs. 3100 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as school A). The total amount of award for one prize on each value is Rs. 1200.

    Using the concept of matrices and determinants, answer the following questions.
    (i) What is the award money for Honesty?

    (a) Rs.350 (b) Rs.300 (c) Rs.500 (d)Rs.400

    (ii) What is the award money for Punctuality?

    (a) Rs.300 (b) Rs.280 (c) Rs.450 (d) Rs.500

    (iii) What is the award money for Hard work?

    (a) Rs 500 (b) Rs.400 (c) 0 (d) none of these

    (iv) If a matrix P is both symmetric and skew-symmetric, then IPI is equal to

    (a) 1 (b) -1 (c) 0 (d) none of these

    (v) If P and Q are two matrices such that PQ = Q and QP = P, then IQ21 is equal to

    (a) IQI (b) IPI (c) 1 (d) 0
  • 3)

    Three shopkeepers Salim, Vijay and Venket are using polythene bags, handmade bags (prepared by prisoners) and newspaper's envelope as carry bags. It is found that the shopkeepers Salim, Vijay and Venket are using (20, 30, 40), (30, 40, 20) and (40, 20, 30) polythene bags, handmade bags and newspaper's envelopes respectively. The shopkeepers Salim, Vijay and Venket spent Rs. 250, Rs. 270 and Rs. 200 on these carry bags respectively.

    Using the concept of matrices and determinants, answer the following questions.
    (i) What is the cost of one polythene bag?

    (a) Rs. 1 (b) Rs. 2 (c) Rs. 3 (d) Rs. 5

    (ii) What is the cost of one handmade bag?

    (a) Rs. 1 (b) Rs. 2 (c) Rs. 3 (d) Rs. 5

    (iii) What is the cost of one newspaper envelope

    (a) Rs. 1 (b) Rs. 2 (c) Rs. 3 (d) Rs. 5

    (iv) Keeping in mind the social conditions, which shopkeeper is better?

    (a) Salim (b) Vijay (c) Venket (d) None of these

    (v) Keeping in mind the environmental conditions, which shopkeeper is better?

    (a) Salim (b) Vijay (c) Venket (d) None of these
  • 4)

    Minor of an element aij of a determinant is the determinant obtained by deleting its ith row and /h column in aij lies and is denoted by Mij.
    Cofactor of an element aij'denoted by Aij,is defined by \(\begin{equation} A_{i j}=(-1)^{i+j} M_{i j} \end{equation}\), where Mij is minor of aij.
    Also, the determinant of a square matrix A is the sum of the products of the elements of any row (or column with their corresponding cofactors.
    For example if \(\begin{equation} A=\left[a_{i j}\right]_{3 \times 3}, \text { then }|A|=a_{11} A_{11}+a_{12} A_{12}+a_{13} A_{13} \end{equation}\) .
    Based on the above information, answer the following questions
    (i) Find the sum of the cofactors of all the elements of \(\begin{equation} \left|\begin{array}{cc} 1 & -2 \\ 4 & 3 \end{array}\right| \end{equation}\) 

    (a) 1 (b) -2 (c) 4 (d) 1

    (ii) Find the minor of a21 of \(\begin{equation} \left|\begin{array}{ccc} 5 & 6 & -3 \\ -4 & 3 & 2 \\ -4 & -7 & 3 \end{array}\right| \end{equation}\) 

    (a) 3 (b) -3 (c) 39 (d) -39

    (iii) In the determinant \(\begin{equation} \left|\begin{array}{ccc} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{array}\right| \end{equation}\) find the value of a32·A32

    (a) 27 (b) -110 (c) 110 (d) -27

    (iv) If \(\begin{equation} \Delta=\left|\begin{array}{lll} 5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3 \end{array}\right| \end{equation}\), find the value of a32·A32 .

    (a) -10 (b) -7 (c) 10 (d) 7

    (v) If \(\begin{equation} \Delta=\left|\begin{array}{ccc} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{array}\right| \end{equation}\), then find the value of \(\begin{equation} |\Delta| \end{equation}\).

    (a) 26 (b)28 (c) 72 (d) 46
  • 5)

    Gaurav purchased 5 pens, 3 bags and 1 instrument box and pays Rs. 16. From the same shop, Dheeraj purchased 2 pens, 1 bag and 3 instrument boxes and pays Rs. 19, while Ankur purchased 1 pen, 2 bags and 4 instrument boxes and pays Rs. 25.
     
    Using the concept of matrices and determinants, answer the following questions.
    (i) The cost of one pen is

    (a) Rs. 2 (b) Rs. 5 (c) Rs. 1 (d) Rs. 3

    (ii) What is the cost of one pen and one bag?

    (a) Rs. 3 (b) Rs. 5 (c) Rs. 7 (d) Rs. 8

    (iii) What is the cost of one pen and one instrument box?

    (a) Rs. 7 (b) Rs. 6 (c) Rs. 8 (d) Rs. 9

    (iv) Which of the following is correct?

    (a) Determinant is a square matrix. (b) Determinant is a number associated to a matrix
    (c) Determinant is a number associated to a square matrix (d) All of the above

    (v) From the matrix equation AB = AC, it can be concluded that B = C provided

    (a) A is singular (b) A is non-singular (c) A is symmetric (d) A is square

CBSE 12th Standard Maths Subject Determinants Case Study Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    A company produces three products every day. Their production on certain day is 45 tons. It is found that the production of third product exceeds the production of first product by 8 tons while the total production of first and third product is twice the production of second product.

    Using the concepts of matrices and determinants, answer the following questions.
    (i) If x, y and z respectively denotes the quantity (in tons) of first, second and third product produced, then which of the following is true?

    (a)  x + y + z = 45 (b)  x + 8 = z (c)  -2y+z=0 (d) all of these

    (ii) If \(\left(\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 0 & -2 \\ 1 & -1 & 1 \end{array}\right)^{-1}=\frac{1}{6}\left(\begin{array}{ccc} 2 & 2 & 2 \\ 3 & 0 & -3 \\ 1 & -2 & 1 \end{array}\right)\) , then the inverse of \(\left(\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 0 & -1 \\ 1 & -2 & 1 \end{array}\right)\) is

    (a) \(\left(\begin{array}{lll} \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ \frac{1}{2} & 0 & \frac{-1}{2} \\ \frac{1}{6} & \frac{-1}{3} & \frac{1}{6} \end{array}\right)\) (b) \(\left(\begin{array}{ccc} \frac{1}{2} & 0 & -\frac{1}{2} \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ \frac{1}{6} & \frac{-1}{3} & \frac{1}{6} \end{array}\right)\) (c) \(\left(\begin{array}{ccc} \frac{1}{3} & \frac{1}{2} & \frac{1}{6} \\ \frac{1}{3} & 0 & \frac{-1}{3} \\ \frac{1}{3} & \frac{-1}{2} & \frac{1}{6} \end{array}\right)\) (d) none of these

    (iii) x :y : z is equal to

    (a) 12: 13: 20  (b)  11:15:19  (c) 15: 19: 11  (d)  13: 12: 20

    (iv) Which of the following is not true?

    (a) IAI = IA'I  (b) (A'rl = (A-I),  (c) A is skew symmetric-matrix of odd order, then IAI = 0  (d) IABI = IAI + IBI
  • 2)

    Each triangular face of the Pyramid of Peace in Kazakhstan is made up of 25 smaller equilateral triangles as shown in the figure.

    Using the above information and concept of determinants, answer the following questions
    (i) If the vertices of one of the smaller equilateral triangle are (0, 0), \(\begin{equation} (3, \sqrt{3}) \end{equation}\) and \(\begin{equation} (3,-\sqrt{3}) \end{equation}\) then the area of such triangle is

    (a) \(\begin{equation} \sqrt{3} \text { sq. units } \end{equation}\) (b) \(\begin{equation} 2 \sqrt{3} \text { sq. units } \end{equation}\)  (c) \(\begin{equation} 3 \sqrt{3} \text { sq. units } \end{equation}\) (d) none of these

    (ii) The area of a face of the Pyramid is

    (a)  \(\begin{equation} 25 \sqrt{3} \text { sq. units } \end{equation}\) (b) \(\begin{equation} 50 \sqrt{3} \text { sq. units } \end{equation}\) (c) \(\begin{equation} 75 \sqrt{3} \text { sq. units } \end{equation}\) (d)  \(\begin{equation} 35 \sqrt{3} \text { sq. units } \end{equation}\)

    (iii) The length of a altitude of a smaller equilateral triangle is

    (a) 2 units (b) 3 units (c) \(\begin{equation} \sqrt{3} \text { units } \end{equation}\) (d) 4 units

    (iv) If (2, 4), (2, 6) are two vertices of a smaller equilateral triangle, then the third vertex will lie on the line represented by

    (a) x +y = 5 (b) \(\begin{equation} x=1+\sqrt{3} \end{equation}\) (c) \(\begin{equation} x=2+\sqrt{3} \end{equation}\) (d) 2x + y = 5

    (v) Let A(a, 0), B(O, b) and C(1, 1) be three points. If \(\begin{equation} \frac{1}{a}+\frac{1}{b}=1 \end{equation}\) ,then the three points are

    (a) vertices of an equilateral triangle (b) vertices of a right angled triangle (c)collinear (d) vertices of an isosceles triangle
  • 3)

    Two schools A and B want to award their selected students on the values of Honesty, Hard work and Punctuality. The school A wants to award Rs. x each, Rs. y each and Rs.z each for the three respective values to its 3, 2 and 1 students respectivefy with a total award money of Rs. 2200. School B wants to spend Rs. 3100 to award its 4, 1 and 3 students on the respective values (by giving the same award money to the three values as school A). The total amount of award for one prize on each value is Rs. 1200.

    Using the concept of matrices and determinants, answer the following questions.
    (i) What is the award money for Honesty?

    (a) Rs.350 (b) Rs.300 (c) Rs.500 (d)Rs.400

    (ii) What is the award money for Punctuality?

    (a) Rs.300 (b) Rs.280 (c) Rs.450 (d) Rs.500

    (iii) What is the award money for Hard work?

    (a) Rs 500 (b) Rs.400 (c) 0 (d) none of these

    (iv) If a matrix P is both symmetric and skew-symmetric, then IPI is equal to

    (a) 1 (b) -1 (c) 0 (d) none of these

    (v) If P and Q are two matrices such that PQ = Q and QP = P, then IQ21 is equal to

    (a) IQI (b) IPI (c) 1 (d) 0
  • 4)

    Area of a triangle whose vertices are (x1, y1), (x2' y2) and (x3, y3) is given by thedeterminant
    \(\begin{equation} \Delta=\frac{1}{2}\left|\begin{array}{lll} x_{1} & y_{1} & 1 \\ x_{2} & y_{2} & 1 \\ x_{3} & y_{3} & 1 \end{array}\right| \end{equation}\) 
    Since, area is a positive quantity, so we always take the absolute value of the determinant Δ. Also, the area of the triangle formed by three collinear points is zero.
    Based on the above information, answer the following questions
    (i) Find the area of the triangle whose vertices are (-2, 6), (3, -6) and (1, 5).

    (a) 30 sq. units (b) 35 sq. units (c) 40 sq. units (d) 15.5 sq. units

    (ii) If the points (2, -3), (k, -1) and (0, 4) are collinear, then find the value of 4k

    (a) 4                      (b) \(\begin{equation} \frac{7}{140} \end{equation}\)                  (c) 4                       (d) \(\begin{equation} \frac{40}{7} \end{equation}\)

    (iii) If the area of a triangle ABC, with vertices A(1, 3), B(O, 0) and C(k, 0) is 3 sq. units, then a value of k is

    (a) 2 (b) 3  (c) 4         (d)   5

    (iv) Using determinants, find the equation of the line joining the points A(1, 2) and B(3, 6).

    (a) y = 2x                (b) x = 3y               (c) y = x                   (d) 4x-y = 5

    (v) If A = (11, 7), B = (5, 5) and C = (-1, 3), then

    (a) \(\begin{equation} \Delta A B C \end{equation}\) is scalene triangle (b) \(\begin{equation} \Delta A B C \end{equation}\) is equilateral triangle
    (c) A, B and C are collinear (d) None of these
  • 5)

    Gaurav purchased 5 pens, 3 bags and 1 instrument box and pays Rs. 16. From the same shop, Dheeraj purchased 2 pens, 1 bag and 3 instrument boxes and pays Rs. 19, while Ankur purchased 1 pen, 2 bags and 4 instrument boxes and pays Rs. 25.
     
    Using the concept of matrices and determinants, answer the following questions.
    (i) The cost of one pen is

    (a) Rs. 2 (b) Rs. 5 (c) Rs. 1 (d) Rs. 3

    (ii) What is the cost of one pen and one bag?

    (a) Rs. 3 (b) Rs. 5 (c) Rs. 7 (d) Rs. 8

    (iii) What is the cost of one pen and one instrument box?

    (a) Rs. 7 (b) Rs. 6 (c) Rs. 8 (d) Rs. 9

    (iv) Which of the following is correct?

    (a) Determinant is a square matrix. (b) Determinant is a number associated to a matrix
    (c) Determinant is a number associated to a square matrix (d) All of the above

    (v) From the matrix equation AB = AC, it can be concluded that B = C provided

    (a) A is singular (b) A is non-singular (c) A is symmetric (d) A is square

CBSE 12th Standard Maths Subject Relations and Functions Case Study Questions With Solution 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    A relation R on a set A is said to be an equivalence relation on A iff it is
    (a) Reflexive i.e.., \((a, a) \in R \ \forall \ a \in A\)
    (b) Symmetric i.e., \((a, b) \in R \Rightarrow(b, a) \in R \ \forall \ a, b \in A\) 
    (c) Transitive i.e., \((a, b) \in R\) and \((b, c) \in R \Rightarrow(a, c) \in R\ \forall\ a, b, c \in A\) 
    Based on the above information, answer the following questions.
    (i) If the relation R = {(1, 1), (1, 2), (1, 3), (2,2), (2, 3), (3,1), (3, 2), (3, 3)} defined on the set A = {1, 2, 3}, then R is

    (a) reflexive  (b) symmetric  (c) transitive  (d) equivalence

    (ii) If the relation R = {(1, 2), (2,1), (1, 3), (3, I)} defined on the setA = {1, 2, 3}, then R is

    (a) reflexive  (b) symmetric  (c) transitive  (d) equivalence

     (iii) If the relation R on the set N of all natural numbers defined as R = {(x, y) : y = x + 5 and x < 4}, then R is

    (a) reflexive  (b) symmetric  (c) transitive  (d) equivalence

    (iv) If the relation R on the set A = {1, 2, 3, , 13, 14}defined as R = {(x, y) : 3x - y = 0}, then R is

    (a) reflexive  (b) symmetric  (c) transitive  (d) equivalence
  • 2)

    Consider the mapping \(f: A \rightarrow B\) is defined by \(f(x)=\frac{x-1}{x-2}\) such that f is a bijection. 
    Based on the above information, answer the following questions.
    (i) Domain of f is

    (a) R - {2}  (b) R (C) R-{1,2}  (d) R-{0}

    (ii) Range of f is

    (a) R (b) R -{1} (C) R-{0}  (d) R-{1,2}

    (iii) If g: \(R-\{2\} \rightarrow R-\{1\}\) is defined by g(x) = 2f(x) - I, then g(x) in terms of x is

    (a) \(\frac{x+2}{x}\) (b) \(\frac{x+1}{x-2}\) (c) \(\frac{x-2}{x}\) (d) \(\frac{x}{x-2}\)

    (iv) The function g defined above, is

    (a) One-one (b) Many-one (c) into (d) None of these

    (v) A function J(x) is said to be one-one iff

    (a) \(f\left(x_{1}\right)=f\left(x_{2}\right) \Rightarrow-x_{1}=x_{2}\)   (b) \(f\left(-x_{1}\right)=f\left(-x_{2}\right) \Rightarrow-x_{1}=x_{2}\)   (c) \(f\left(x_{1}\right)=f\left(x_{2}\right) \Rightarrow x_{1}=x_{2}\)   (d) None of these

CBSE 12th Standard Maths Subject Relations and Functions Case Study Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    A relation R on a set A is said to be an equivalence relation on A iff it is
    (a) Reflexive i.e.., \((a, a) \in R \ \forall \ a \in A\)
    (b) Symmetric i.e., \((a, b) \in R \Rightarrow(b, a) \in R \ \forall \ a, b \in A\) 
    (c) Transitive i.e., \((a, b) \in R\) and \((b, c) \in R \Rightarrow(a, c) \in R\ \forall\ a, b, c \in A\) 
    Based on the above information, answer the following questions.
    (i) If the relation R = {(1, 1), (1, 2), (1, 3), (2,2), (2, 3), (3,1), (3, 2), (3, 3)} defined on the set A = {1, 2, 3}, then R is

    (a) reflexive  (b) symmetric  (c) transitive  (d) equivalence

    (ii) If the relation R = {(1, 2), (2,1), (1, 3), (3, I)} defined on the setA = {1, 2, 3}, then R is

    (a) reflexive  (b) symmetric  (c) transitive  (d) equivalence

     (iii) If the relation R on the set N of all natural numbers defined as R = {(x, y) : y = x + 5 and x < 4}, then R is

    (a) reflexive  (b) symmetric  (c) transitive  (d) equivalence

    (iv) If the relation R on the set A = {1, 2, 3, , 13, 14}defined as R = {(x, y) : 3x - y = 0}, then R is

    (a) reflexive  (b) symmetric  (c) transitive  (d) equivalence
  • 2)

    Consider the mapping \(f: A \rightarrow B\) is defined by \(f(x)=\frac{x-1}{x-2}\) such that f is a bijection. 
    Based on the above information, answer the following questions.
    (i) Domain of f is

    (a) R - {2}  (b) R (C) R-{1,2}  (d) R-{0}

    (ii) Range of f is

    (a) R (b) R -{1} (C) R-{0}  (d) R-{1,2}

    (iii) If g: \(R-\{2\} \rightarrow R-\{1\}\) is defined by g(x) = 2f(x) - I, then g(x) in terms of x is

    (a) \(\frac{x+2}{x}\) (b) \(\frac{x+1}{x-2}\) (c) \(\frac{x-2}{x}\) (d) \(\frac{x}{x-2}\)

    (iv) The function g defined above, is

    (a) One-one (b) Many-one (c) into (d) None of these

    (v) A function J(x) is said to be one-one iff

    (a) \(f\left(x_{1}\right)=f\left(x_{2}\right) \Rightarrow-x_{1}=x_{2}\)   (b) \(f\left(-x_{1}\right)=f\left(-x_{2}\right) \Rightarrow-x_{1}=x_{2}\)   (c) \(f\left(x_{1}\right)=f\left(x_{2}\right) \Rightarrow x_{1}=x_{2}\)   (d) None of these

CBSE 12th Standard Maths Subject Case Study Questions 2021 - by Shalini Sharma - Udaipur View & Read

  • 1)

    Deepa rides her car at 25 km/hr, She has to spend Rs. 2 per km on diesel and if she rides it at a faster speed of 40 km/hr, the diesel cost increases to Rs. 5 per km. She has Rs. 100 to spend on diesel. Let she travels x kms with speed 25 km/hr and y kms with speed 40 km/hr. The feasible region for the LPP is shown below:
    Based on the above information, answer the following questions

    Based on the above information, answer the following questions.
    (i) What is the point of intersection of line l1 and l2,

    \(\text { (a) }\left(\frac{40}{3}, \frac{50}{3}\right)\) \(\text { (b) }\left(\frac{50}{3}, \frac{40}{3}\right)\) \(\text { (c) }\left(\frac{-50}{3}, \frac{40}{3}\right)\) \(\text { (d) }\left(\frac{-50}{3}, \frac{-40}{3}\right)\)

    (ii) The corner points of the feasible region shown in above graph are

    \(\text { (a) }(0,25),(20,0),\left(\frac{40}{3}, \frac{50}{3}\right)\) \(\text { (b) }(0,0),(25,0),(0,20)\) \(\text { (c) }(0,0),\left(\frac{40}{3}, \frac{50}{3}\right),(0,20)\) \(\text { (d) }(0,0),(25,0),\left(\frac{50}{3}, \frac{40}{3}\right),(0,20)\)

    (iii) If Z = x + y be the objective function and max Z = 30. The maximum value occurs at point

    \(\text { (a) }\left(\frac{50}{3}, \frac{40}{3}\right)\) (b) (0, 0) (c) (25, 0) (d) (0, 20)

    (iv) If Z = 6x - 9y be the objective function, then maximum value of Z is

    (a) -20 (b) 150 (c) 180 (d) 20

    (v) If Z = 6x + 3y be the objective function, then what is the minimum value of Z?

    (a) 120 (b) 130 (c) 0 (d) 150
  • 2)

    Suppose a dealer in rural area wishes to purpose a number of sewing machines. He has only Rs. 5760 to invest and has space for at most 20 items for storage. An electronic sewing machine costs him Rs. 360 and a manually operated sewing machine Rs. 240. He can sell an electronic sewing machine at a profit of Rs. 22 and a manually operated sewing machine at a profit of Rs. 18.
    Based on the above information, answer the following questions.

    (i) Let x and y denotes the number of electronic sewing machines and manually operated sewing machines purchased by the dealer. If it is assume that the dealer purchased atleast one of the given machines, then

    (a) x + y ≥  0 (b) x + y < 0 (c) x + y > 0 (d) x + y ≤ 0

    (ii) Let the constraints in the given problem is represented by the following inequalities
    x + y ≤ 20
    360x + 240y ≤ 5760
    x, y ≥ 0
    Then which of the following point lie in its feasible region.

    (a) (0, 24) (b) (8, 12) (c) (20, 2) (d) None of these

    (iii) If the objective function of the given problem is maximise z = 22x + 18y, then its optimal value occur at

    (a) (0, 0) (b) (16, 0) (c) (8, 12) (d) (0, 20)

    (iv) Suppose the following shaded region APDO, represent the feasible region corresponding to mathematical formulation. of given problem. Then which of the following represent the coordinates of one of its corner points.

    (a) (0, 24) (b) (12, 8) (c) (8, 12) (d) (6, 14)

    (v) If an LPP admits optimal solution at two consecutive vertices of a feasible region, then

    (a) the required optimal solution is at the midpoint of the line joining two points. (b) the optimal solution occurs at every point on the line joining these two points.
    (c) the LPP under consideration is not solvable. (d) the LPP under consideration must be reconstructed.
  • 3)

    A student Arun is running on a playground along the curve given by y = x2 + 7. Another student Manita standing at point (3, 7) on playground wants to hit Arun by paper ball when Arun is nearest to Manita .
     
    Based on above information, answer the following questions.
    (i) Arun's position at any value of x will be

    (a) x2,y-7 (b) (x2,y+7) (c) (x, x2 + 7) (d) (x2,x-7)

    (ii) Distance (say D) between Arun and Manita will be

    (a) (x - 1)(2x2 + 2x + 3) (b) (x-3)2+x4 (c) \(\sqrt{(x-3)^{2}+x^{4}}\) (d) \(\sqrt{(x-1)\left(2 x^{2}+2 x+3\right)}\)

    (iii) For which real value(s) of x, first derivative of D2 w.r.t. 'x' will  Vanish?

    (a) 1 (b) 2 (c) 3 (d) 4

    (iv) Find the position of Arun when Manita will hit the paper hall.

    (a) (5, 32) (b) (1, 8) (c) (3, 7) (d) (3, 16)

    (v) The minimum value of D is

    (a) 3 (b) \(\sqrt{3}\) (c) 5 (d) \(\sqrt{5}\)
  • 4)

    Rohan, a student of class XII, visited his uncle's flat with his father. He observe that the window of the house is in the form of a rectangle surmounted by a semicircular opening having perimeter 10m as shown in the figure.

    (i) If x and y represents the length and breadth of the rectangular region, then relation between x and y can be represented as

    (a) \(x+y+\frac{\pi}{2}=10\) \(x+2 y+\frac{\pi x}{2}=10\) (c) 2x + 2y = 10 (d) \(x+2 y+\frac{\pi}{2}=10\)

    (ii) The area (A) of the window can be given by

    (a) \(A=x-\frac{x^{3}}{8}-\frac{x^{2}}{2}\) (b) \(A=5 x-\frac{x^{2}}{2}-\frac{\pi x^{2}}{8}\) (c) \(A=x+\frac{\pi x^{3}}{8}-\frac{3 x^{2}}{8}\) (d) \(A=5 x+\frac{x^{2}}{2}+\frac{\pi x^{2}}{8}\)

    (iii) Rohan is interested in maximizing the area of the whole window, for this to happen, the value of x should be

    (a) \(\frac{10}{2-\pi}\) (b) \(\frac{20}{4-\pi}\) (c) \(\frac{20}{4+\pi}\) (d) \(\frac{10}{2+\pi}\)

    (iv) Maximum area of the window is

    (a) \(\frac{30}{4-\pi}\) (b) \(\frac{30}{4+\pi}\) (c) \(\frac{50}{4-\pi}\) (d) \(\frac{50}{4+\pi}\)

    (v) For maximum value of A, the breadth of rectangular part of the window is

    (a) \(\frac{10}{4+\pi}\) (b) \(\frac{10}{4-\pi}\) (c) \(\frac{20}{4+\pi}\) (d) \(\frac{20}{4-\pi}\)
  • 5)

    A student is preparing for the competitive examinations LIC AAO, SSC CGL and Bank P.O. The probabilities that the student is selected independently in competitive examination of LIC AAO, SSC CGL and Bank P.O. are a, b and c respectively. Of these examinations, students has 50% chance of selection in at least one, 40% chance of selection in at least two and 30% chance of selection in exactly two examinations.

    Based on the above information, answer the following questions.
    (i) The value of a + b + c - ab - bc - ca + abc is

    (a) 0.3 (b) 0.5 (c) 0.7 (d) 0.6

    (ii) The value of ab + bc + ac - 2abc is

    (a) 0.5 (b) 0.3 (c) 0.4 (d) 0.6

    (iii) The value of abc is

    (a) 0.1 (b) 0.5 (c) 0.7 (d) 0.3

    (iv) The value of ab + bc + dc is

    (a) 0.1 (b) 0.6 (c) 0.5 (d) 0.3

    (v) The value of a + b + c is

    (a) 1 (b) 1.5 (c) 1.6 (d) 1.4

CBSE 12th Standard Maths Subject Case Study Questions With Solution 2021 Part - II - by Shalini Sharma - Udaipur View & Read

  • 1)

    Three car dealers, say A, Band C, deals in three types of cars, namely Hatchback cars, Sedan cars, SUV cars. The sales figure of 2019 and 2020 showed that dealer A sold 120 Hatchback, 50 Sedan, 10 SUV cars in 2019 and 300 Hatchback, 150 Sedan, 20 SUV cars in 2020; dealer B sold 100 Hatchback, 30 Sedan,S SUV cars in 2019 and 200 Hatchback, 50 Sedan, 6 SUV cars in 2020; dealer C sold 90 Hatchback, 40 Sedan, 2 SUV cars in 2019 and 100 Hatchback, 60 Sedan,S SUV cars in 2020.
     
    Based on the above information, answer the following questions.
    (i) The matrix summarizing sales data of 2019 is
     
    (ii) The matrix summarizing sales data of 2020 is
     
     (iii) The total number of cars sold in two given years, by each dealer, is given by the matrix
     
    (iv) The increase in sales from 2019 to 2020 is given by the matrix
     
    (v) If each dealer receive profit of Rs. 50000 on sale of a Hatchback, Rs. 100000 on sale of a Sedan and Rs. 200000 on sale of a SUV (v) then amount of profit received in the year 2020 by each dealer is given by the matrix.

     

  • 2)

    Three schools A, Band C organized a mela for collecting funds for helping the rehabilitation of flood victims. They sold hand made fans, mats and plates from recycled material at a cost of Rs. 25, Rs.100 and Rs.50 each. The number of articles sold by school A, B, C are given below.

    Artilcle\School A B C
    Fans 40 25 35
    Mats 50 40 50
    Plates 20 30 40

    Based on above information, answer the following questions.
    (i) If P be a 3 x 3 matrix represent the sale of handmade fans, mats and plates by three schools A, Band C, then
     
    (ii) If Q be a 3 x 1 matrix represent the sale prices (in Rs) of given products per unit, then

    (iii) The funds collected by school A by selling the given articles is

    (a) Rs. 7000 (b) Rs. 6125 (c) Rs. 7875 (d) Rs. 8000

    (iv) The funds collected by school B by selling the given articles is

    (a) Rs. 5125 (b) Rs. 6125 (c) Rs. 7125 (d) Rs. 8125

    (v) The total funds collected for the required purpose is

    (a) Rs. 20000 (b) Rs. 21000 (c) Rs. 30000 (d) Rs. 35000
  • 3)

    The upward speed v(t) of a rocket at time t is approximated by \(\begin{equation} v(t)=a t^{2}+b t+c, 0 \leq t \leq 100 \end{equation}\) ,where a, band c are constants. It has been found that the speed at times t = 3, t = 6 and t = 9 seconds are respectively 64, 133 and 208 miles per second.

    If \(\begin{equation} \left(\begin{array}{ccc} 9 & 3 & 1 \\ 36 & 6 & 1 \\ 81 & 9 & 1 \end{array}\right)^{-1}=\frac{1}{18}\left(\begin{array}{ccc} 1 & -2 & 1 \\ -15 & 24 & -9 \\ 54 & -54 & 18 \end{array}\right) \end{equation}\) ,then answer the following questions
    (i) The value of b + c is

    (a) 20 (b) 21 (c) 3/4 (d) 4/3

    (ii) The value of a + c is

    (a) 1 (b) 20 (c) 4/3 (d) none of these

    (iii) v(t) is given by

    (a) t2+20t+l (b) \(\begin{equation} \frac{1}{3} t^{2}+20 t+1 \end{equation}\) (c) \(\begin{equation} t^{2}+\frac{1}{3} t+20 \end{equation}\) (d) P + t + 1

    (iv) The speed at time t = 15 seconds is

    (a) 346 miles/see (b) 356 miles/see (c) 366 miles/see (d) 376 miles/see

    (v) The time at which the speed of rocket is 784 miles/see is

    (a) 20 seconds (b) 30 seconds (c) 25 seconds (d) 27 seconds

     

  • 4)

    Minor of an element aij of a determinant is the determinant obtained by deleting its ith row and /h column in aij lies and is denoted by Mij.
    Cofactor of an element aij'denoted by Aij,is defined by \(\begin{equation} A_{i j}=(-1)^{i+j} M_{i j} \end{equation}\), where Mij is minor of aij.
    Also, the determinant of a square matrix A is the sum of the products of the elements of any row (or column with their corresponding cofactors.
    For example if \(\begin{equation} A=\left[a_{i j}\right]_{3 \times 3}, \text { then }|A|=a_{11} A_{11}+a_{12} A_{12}+a_{13} A_{13} \end{equation}\) .
    Based on the above information, answer the following questions
    (i) Find the sum of the cofactors of all the elements of \(\begin{equation} \left|\begin{array}{cc} 1 & -2 \\ 4 & 3 \end{array}\right| \end{equation}\) 

    (a) 1 (b) -2 (c) 4 (d) 1

    (ii) Find the minor of a21 of \(\begin{equation} \left|\begin{array}{ccc} 5 & 6 & -3 \\ -4 & 3 & 2 \\ -4 & -7 & 3 \end{array}\right| \end{equation}\) 

    (a) 3 (b) -3 (c) 39 (d) -39

    (iii) In the determinant \(\begin{equation} \left|\begin{array}{ccc} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{array}\right| \end{equation}\) find the value of a32·A32

    (a) 27 (b) -110 (c) 110 (d) -27

    (iv) If \(\begin{equation} \Delta=\left|\begin{array}{lll} 5 & 3 & 8 \\ 2 & 0 & 1 \\ 1 & 2 & 3 \end{array}\right| \end{equation}\), find the value of a32·A32 .

    (a) -10 (b) -7 (c) 10 (d) 7

    (v) If \(\begin{equation} \Delta=\left|\begin{array}{ccc} 2 & -3 & 5 \\ 6 & 0 & 4 \\ 1 & 5 & -7 \end{array}\right| \end{equation}\), then find the value of \(\begin{equation} |\Delta| \end{equation}\).

    (a) 26 (b)28 (c) 72 (d) 46
  • 5)

    If a real valued function f(x) is finitely derivable at any point of its domain, it is necessarily continuous at that point. But its converse need not be true.
    For example, every polynomial. constant function are both continuous as well as differentiable and inverse trigonometric functions are continuous and differentiable in its domains etc.
    Based on the above information, answer the following questions.
    (i) If \(\begin{equation} f(x)=\left\{\begin{array}{l} x, \text { for } x \leq 0 \\ 0, \text { for } x>0 \end{array}\right. \end{equation}\) , then at x = 0

    (a) f(x) is differentiable and continuous (b) j(x) is neither continuous nor differentiable
    (c) f(x) is continuous but not differentiable (d) none of these

    (ii) If \(\begin{equation} f(x)=|x-1|, x \in R \end{equation}\) ,then at x= 1 

    (a) f(x) is not continuous (b) f(x) is continuous but not differentiable
    (c) f(x) is continuous and differentiable (d) none of these

    (iii) f(x) = x3 is

    (a) continuous but not differentiable at x = 3 (b) continuous and differentiable at x = 3
    (c) neither continuous nor differentiable at x = 3 (d) none of these

    (iv) f(x) = [sin x], then which of the following is true? 

    (a) j(x) is continuous and differentiable at x = o. (b) j(x) is discontinuous at x = o.
    (c) j(x) is continuous at x = 0 but not differentiable (d) fix) is differentiable but not continuous at  \(\begin{equation} x=\pi / 2 \end{equation}\)

    (v) If f(x) = sin-1x, \(\begin{equation} -1 \leq x \leq 1 \end{equation}\), then

    (a) f(x) is both continuous and differentiable (b) f(x) is neither continuous nor differentiable.
    (c) f(x) is continuous but not differentiable (d) None of these

CBSE 12th Standard Maths Subject Case Study Questions With Solution 2021 Part - I - by Shalini Sharma - Udaipur View & Read

  • 1)

    Consider the mapping \(f: A \rightarrow B\) is defined by \(f(x)=\frac{x-1}{x-2}\) such that f is a bijection. 
    Based on the above information, answer the following questions.
    (i) Domain of f is

    (a) R - {2}  (b) R (C) R-{1,2}  (d) R-{0}

    (ii) Range of f is

    (a) R (b) R -{1} (C) R-{0}  (d) R-{1,2}

    (iii) If g: \(R-\{2\} \rightarrow R-\{1\}\) is defined by g(x) = 2f(x) - I, then g(x) in terms of x is

    (a) \(\frac{x+2}{x}\) (b) \(\frac{x+1}{x-2}\) (c) \(\frac{x-2}{x}\) (d) \(\frac{x}{x-2}\)

    (iv) The function g defined above, is

    (a) One-one (b) Many-one (c) into (d) None of these

    (v) A function J(x) is said to be one-one iff

    (a) \(f\left(x_{1}\right)=f\left(x_{2}\right) \Rightarrow-x_{1}=x_{2}\)   (b) \(f\left(-x_{1}\right)=f\left(-x_{2}\right) \Rightarrow-x_{1}=x_{2}\)   (c) \(f\left(x_{1}\right)=f\left(x_{2}\right) \Rightarrow x_{1}=x_{2}\)   (d) None of these
  • 2)

    To promote the making of toilets for women, an organisation tried to generate awareness through (i) house calls (ii) emails and (iii) announcements. The cost for each mode per attempt is given below:

    (i) Rs.50 (ii) Rs.20 (iii) Rs.40
    The number of attempts made in the villages X, Y and Z are given below:
    \(\begin{array}{llll} & (\mathrm{i}) & (\mathrm{ii}) & (\mathrm{iii}) \\ X & 400 & 300 & 100 \\ Y & 300 & 250 & 75 \\ Z & 500 & 400 & 150 \end{array}\) 
    Also, the chance of making of toilets corresponding to one attempt of given modes is
    (i) 2% (ii) 4% (iii) 20%
    Based on the above information, answer the following questions.
    (i) The cost incurred by the organisation on village X is

     (a) 10000   (b)  Rs.15000   (c) 30000  (d) Rs.20000

    (ii) The cost incurred by the organisation on village Y is

      (a) Rs.25000  (b) Rs.18000 (c) Rs.23000  (d) Rs.28000

    (iii) The cost incurred by the organisation on village Z is

     (a)  Rs.19000  (b)  Rs.39000  (c)  Rs.4500  (d)  Rs.5000

    (iv) The total number of toilets that can be expected after the promotion in village X, is

    (a)  20 (b)  30  (c)  40 (d)  50

    (v) The total number of toilets that can be expected after the promotion in village Z, is

    (a) 26  (b) 36  (c)  46  (d)  56
  • 3)

    Three car dealers, say A, Band C, deals in three types of cars, namely Hatchback cars, Sedan cars, SUV cars. The sales figure of 2019 and 2020 showed that dealer A sold 120 Hatchback, 50 Sedan, 10 SUV cars in 2019 and 300 Hatchback, 150 Sedan, 20 SUV cars in 2020; dealer B sold 100 Hatchback, 30 Sedan,S SUV cars in 2019 and 200 Hatchback, 50 Sedan, 6 SUV cars in 2020; dealer C sold 90 Hatchback, 40 Sedan, 2 SUV cars in 2019 and 100 Hatchback, 60 Sedan,S SUV cars in 2020.
     
    Based on the above information, answer the following questions.
    (i) The matrix summarizing sales data of 2019 is
     
    (ii) The matrix summarizing sales data of 2020 is
     
     (iii) The total number of cars sold in two given years, by each dealer, is given by the matrix
     
    (iv) The increase in sales from 2019 to 2020 is given by the matrix
     
    (v) If each dealer receive profit of Rs. 50000 on sale of a Hatchback, Rs. 100000 on sale of a Sedan and Rs. 200000 on sale of a SUV (v) then amount of profit received in the year 2020 by each dealer is given by the matrix.

     

  • 4)

    Three schools A, Band C organized a mela for collecting funds for helping the rehabilitation of flood victims. They sold hand made fans, mats and plates from recycled material at a cost of Rs. 25, Rs.100 and Rs.50 each. The number of articles sold by school A, B, C are given below.

    Artilcle\School A B C
    Fans 40 25 35
    Mats 50 40 50
    Plates 20 30 40

    Based on above information, answer the following questions.
    (i) If P be a 3 x 3 matrix represent the sale of handmade fans, mats and plates by three schools A, Band C, then
     
    (ii) If Q be a 3 x 1 matrix represent the sale prices (in Rs) of given products per unit, then

    (iii) The funds collected by school A by selling the given articles is

    (a) Rs. 7000 (b) Rs. 6125 (c) Rs. 7875 (d) Rs. 8000

    (iv) The funds collected by school B by selling the given articles is

    (a) Rs. 5125 (b) Rs. 6125 (c) Rs. 7125 (d) Rs. 8125

    (v) The total funds collected for the required purpose is

    (a) Rs. 20000 (b) Rs. 21000 (c) Rs. 30000 (d) Rs. 35000
  • 5)

    A manufacturer produces three types of bolts, x, y and z which he sells in two markets. Annual sales (in Rs) are indicated below:

    Markets Products
    x y z
    I 10000 2000 18000
    II 6000 20000 8000

    If unit sales prices of x, y and z are Rs.2.50, Rs.1.50 and Rs.1.00 respectively, then answer the following questions using the concept of matrices.
    (i) Find the total revenue collected from the Market-I.

    (a) Rs. 44000 (b) Rs. 48000 (c) Rs. 46000 (d) Rs. 53000

    (ii) Find the total revenue collected from the Market-II.

    (a) Rs. 5100  (b) Rs. 5300  (c ) Rs. 46000  (d) Rs. 49000

    (iii) If the unit costs of the above three commodities are Rs.2.00, Rs.1.00 and 50 paise respectively, then find the gross profit from both the markets.

    (a) Rs. 53000  (b) Rs. 46000  (c) Rs. 34000  (d) Rs. 32000

    (iv) If matrix \(4=\left[a_{i j}\right]_{2 \times 2}\) , where \(a_{i j}=1, \text { if } i \neq j\) , and \(a_{i j}=0 \text { if } i=j\) , then A2 is equal to

    (a) I  (b) A   (c) 0  (d) none of these

    (v) If A and B are matrices of same order, then (AB' - BA') is a

    (a) skew-symmetric matrix  (b) null matrix  (c) symmetric matrix  (d) unit matrix

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CBSE 12th Standard Maths Subject 2015 Main Exam Panchkula Set 3 Question Paper With Answer Key - by users_admin View & Read

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CBSE 12th Standard Maths Subject 2015 Main Exam Patna Set 3 Question Paper With Answer Key - by users_admin View & Read

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CBSE 12th Standard Maths Subject 2015 Main Exam Guwahati Set 3 Question Paper With Answer Key - by users_admin View & Read

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CBSE 12th Standard Maths Subject 2015 Main Exam Chennai Set 3 Question Paper With Answer Key - by users_admin View & Read

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CBSE 12th Standard Maths Subject 2015 Main Exam Allahabad Set 3 Question Paper With Answer Key - by users_admin View & Read

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CBSE 12th Standard Maths Subject 2015 Compartment Exam Delhi Set 3 Question Paper With Answer Key - by users_admin View & Read

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CBSE 12th Standard Maths Subject 2014 Board Paper Set 3 Question Paper With Answer Key - by users_admin View & Read

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CBSE 12th Standard Maths Subject 2013 Delhi Board Paper Set 1 Question Paper With Answer Key - by users_admin View & Read

CBSE 12th Standard Maths Subject 2013 Main Exam Delhi Set 1 Question Paper With Answer Key - by users_admin View & Read

CBSE 12th Standard Maths Subject 2012 Main Exam Delhi Set 1 Question Paper With Answer Key - by users_admin View & Read

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CBSE 12th Standard Maths Subject 2009 Delhi Board Paper Set 2 Question Paper With Answer Key - by users_admin View & Read

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CBSE 12th Standard Mathematics Vectors and 3D Key Points - by users_admin View & Read

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12th Standard CBSE Maths Application Of Derivatives Important Quetions - by Shalini Sharma - Udaipur View & Read

  • 1)

    If P(A) = \(\frac12\), P(B) = 0, then P(A|B) is ______.

  • 2)

    Let f : R ➝ R be defined as f (x) = 3x. Choose the correct answer

  • 3)

    Consider a binary operation * on N defined as a * b = a3 + b3. Choose the correct answer

  • 4)

    If A = {1,3,5,7} and define a relation, such that R = { (a,b) a,b ∈ A : |a+b| = 8}. Then how many elements are there in the relation R

  • 5)

     If A = diag(3, -1), then matrix A is

12th Standard Mathematics Delhi Main Exam Annual Question paper with Answer key Set 1 - 2020 - by users_admin View & Read

12th Standard Maths Syllabus - 2021 - by users_admin View & Read

CBSE 12th Standard Mathematics Reduced Syllabus 2020- 21 - by users_admin View & Read