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?

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

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?

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? 

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

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?

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.

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?

Let A be the set of all students of class XII in a school and R be the relation, having the same sex in A, and then prove that R is an equivalence relation.
Do you think, Co-education may be helpful in child development and why?

A man is driving a car on the dangerous path given by:
\(f\left( x \right) =\begin{cases} \frac { 1-{ x }^{ n } }{ 1-x } ,\quad x\neq 1 \\ m-1,\quad \quad \quad x=1 \end{cases}m\epsilon N\)
Find the dangerous point on the path. Whether the driver should pass that point?

\(\frac { dy }{ dx } +\frac { y }{ x } =0,\) 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

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?

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.

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?

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?

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?

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.)

Find the value of \(\sin ^{ -1 }{ \left( \sin { \frac { 4\pi }{ 5 } } \right) } \). What value do you learn form it?

\(\tan ^{ -1 }{ 3x } +\tan ^{ -1 }{ 2x } =\frac { \pi }{ 4 } .\) What vlaue do you obseerve in real life scenario?

Let X be a non-empty set, Let * be a binary operation on the power set P(X) defined by A * B = A n B. What is the identify element for the operation * ? Given X is a set of people in a locality, A is a set of children and B is a set of citizens aged above 75 years in the same locality. Is * an invertible binary Feration for these sets as defined above?
What qualities would you suggest that elements of A should have towards elements of B?

The path of a moving bike is given by \(f(x)=\{2x-1, \ if\ x<0 \ 2x+1,\ if\ x\ge0\) 
Find the dangerous point on the path. Whether the rider should pass that point or not? Justify your answers.

Find the \(\lim _{ x\rightarrow 1 }{ \frac { { x }^{ 2 }-1 }{ x-1 } } \) can a person be successful without any failure in life?

The money to be spent for the welfare of the employees of a firm is proportional to the rate of change of its total revenue (marginal revenue). If the total revenue (in rupee) received from the sale of x units of a product is given by \(R(x)=3x^2+36x+5\)  find the marginal revenue when x=5 and write value does the question indicate.

Find the projection of the vector \(\overrightarrow { x } =\left( \hat { i } +2\hat { j } +\hat { k } \right) \) on \(\overrightarrow { y } =\left( 2\hat { i } +3\hat { j } +2\hat { k } \right) \). What value do you see from it?

The equation of tangent at (2, 3) on the curve y² = ax³ + b is y = 4x - 5. Find the values of a and b.