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12th Standard English Medium Maths Reduced Syllabus Two Mark Important Questions with Answer key - 2021(Public Exam )

12th Standard

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Maths

Time : 02:45:00 Hrs
Total Marks : 100

    2 Marks

    50 x 2 = 100
  1. Find a matrix A if adj(A) = \(\left[ \begin{matrix} 7 & 7 & -7 \\ -1 & 11 & 7 \\ 11 & 5 & 7 \end{matrix} \right] \).

  2. Find the rank of the following matrices by minor method:
    \(\left[ \begin{matrix} 1 \\ 3 \end{matrix}\begin{matrix} -2 \\ -6 \end{matrix}\begin{matrix} -1 \\ -3 \end{matrix}\begin{matrix} 0 \\ 1 \end{matrix} \right] \)

  3. Find the rank of the following matrices by minor method:
    \(\left[ \begin{matrix} 1 & -2 & 3 \\ 2 & 4 & -6 \\ 5 & 1 & -1 \end{matrix} \right] \)

  4. Find the following \(\left| \cfrac { 2+i }{ -1+2i } \right| \)
     

  5. Which one of the points i,−2+i , and 3 is farthest from the origin?

  6. Represent the complex number −1−i

  7. Find the square roots of −6+8i

  8. Show that the following equations represent a circle, and, find its centre and radius
    \(\left| 2z+2-4i \right| =2\)

  9. Find the modulus and principal argument of the following complex numbers.
    \(\sqrt { 3 } \)-i

  10. Find a polynomial equation of minimum degree with rational coefficients, having 2i+3 as a root.

  11. Show that the polynomial 9x9+2x5-x4-7x2+2 has at least six imaginary roots.

  12. Find the period and amplitude of
    y=sin 7x

  13. Show that cot−1\(\left( \frac { 1 }{ \sqrt { { x }^{ 2 }-1 } } \right) ={ sec }^{ -1 }x,|x|>1\)

  14. Find the period and amplitude of
    y=4sin(−2x)

  15. Find all values of x such that
    -5\(\pi\le x \le 5\pi\) and cos x =1

  16. Find the principal value of
    \({ Sin }^{ -1 }\left( sin\left( -\frac { \pi }{ 3 } \right) \right) \)

  17. Determine whether x+y−1=0 is the equation of a diameter of the circle x2+y2−6x+4y+c = 0 for all possible values of c .

  18. If y=4x+c is a tangent to the circle x2+y2=9 , find c .

  19. 2x2−y2=7

  20. Find the length of the tangent from (2, -3) to the circle x2 + y2 - 8x - 9y + 12 = 0.

  21. Find the volume of the parallelepiped whose coterminus edges are given by the vectors \(\hat { 2i } -\hat { 3j } +\hat { 4k } \)\(\hat { i } -\hat { 2j } +\hat { 4k } \) and \(\hat {3 i } -\hat { j } +\hat { 2k } \)

  22. Find the volume of the parallelepiped whose coterminous edges are represented by the vectors \(-6\hat { i } +14\hat { j } +10\hat { k } ,14\hat { i } -10\hat { j } -6\hat { k } \) and \(2\hat { i } +4\hat { j } -2\hat { k } \)

  23. Find the intercepts cut off by the plane \(\vec { r } .(6\hat { i } +4\hat { j } -3\hat { k } )\)=12 on the coordinate axes.

  24. A person learnt 100 words for an English test. The number of words the person remembers in t days after learning is given by W(t) =100×(1− 0.1t)2, 0 ≤ t ≤ 10. What is the rate at which the person forgets the words 2 days after learning?

  25. Find the angle of intersection of the curve y = sin x with the positive x -axis.

  26. Find two positive numbers whose sum is 12 and their product is maximum.

  27.  

    Compute the limit  

     \(\underset{x\rightarrow 1}{lim}(\frac{x^{2}-3x+2}{x^{2}-4x+3})\).

  28. Find the asymptotes of the curve \(f(x)=\frac { { 2x }^{ 2 }-8 }{ { x }^{ 2 }-16 } \)

  29. Find the equation of the tangent to the curve y2=4x+5 and which is parallel to y=2x+7

  30. Evaluate the following limits, if necessary using L’Hopitalrule
    (i) \(\underset { x\rightarrow 2 }{ lim } \cfrac { sin\pi x }{ 2-x } \) 
    (ii) \(\cfrac { lim }{ x\rightarrow 2 } \cfrac { { x }^{ n }-{ a }^{ n } }{ x-2 } \) 
    (iii) \(\underset { x\rightarrow \infty }{ lim } \cfrac { sin\frac { 2 }{ x } }{ \frac { 1 }{ x } } \)
    (iv) \(\underset { x\rightarrow \infty }{ lim } \cfrac { { x }^{ 2 } }{ { e }^{ x } } \)

  31. If the radius of a sphere, with radius 10 cm, has to decrease by 0 1. cm, approximately how much will its volume decrease?

  32. Find differential dy for each of the following function
    y = (3 + sin(2x)) 2/3 

  33. If u=x2+3xy2+y2, then prove that \(\cfrac { { \partial }^{ 2 }u }{ \partial x\partial y } =\cfrac { { \partial }^{ 2 }u }{ \partial y\partial x } \)

  34. Evaluate \(\int _{ 0 }^{ 1 }{ \left( \cfrac { { e }^{ 5logx }-{ e }^{ 4logx } }{ { e }^{ 3logx }-{ e }^{ 2logx } } \right) } \)

  35. Find the slope of the tangent to the curve \(y=\int _{ 0 }^{ x }{ \cfrac { dt }{ 1+{ t }^{ 3 } } stx=1 } \) 

  36. Find the area enclosed between the parabola y2=4ax and the line x=a, x=9a.

  37. Express the physical statement in the form of the differential equation.
    For a certain substance, the rate of change of vapor pressure P with respect to temperature T is proportional to the vapor pressure and inversely proportional to the square of the temperature.

  38. Find the differential equation of the family of all non-vertical lines in a plane.

  39. Determine the order and degree of \(\cfrac { \left[ 1+\left( \frac { dy }{ dx } \right) ^{ 2 } \right] ^{ \frac { 3 }{ 2 } } }{ \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } } =k\)

  40. Find the order and degree of \(\left( \cfrac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) ^{ 2 }+cos\left( \cfrac { dy }{ dx } \right) =0\)

  41. An urn contains 5 mangoes and 4 apples Three fruits are taken at randaom If the number of apples taken is a random variable, then find the values of the random variable and number of points in its inverse images.

  42. A six sided die is marked '1' on one face, '3' on two of its faces, and '5' on remaining three faces. The die is thrown twice. If X denotes the total score in two throws, find
    (i) the probability mass function
    (ii) the cumulative distribution function
    (iii) P(4 ≤ X < 10)
    (iv) P(X ≥ 6)

  43. A commuter train arrives punctually at a station every half hour. Each morning, a student leaves his house to the train station.Let X denote- the amount of time, in minutes that the student waits for the train from the time he reaches the train station. It is known  that the pdf of X is 
    \(f(x)=\begin{cases} \begin{matrix} \frac { 1 }{ 3 } & 0<x<30 \end{matrix} \\ \begin{matrix} 0 & elsewhere \end{matrix} \end{cases}\) 
    Obtain and interpret the expected value of the random variable X .

  44. A coin is tossed until a head appears or the tail appears 4 times in succession. Find the probability distribution of the number of tosses.

  45. The probability distribution of a random variable X is given under :

    Find (i) k
    (ii) E(X)

  46. Is it possible that the mean of a binomial distribution is 15 and its standard deviation is 5?

  47. Let A =\(\begin{bmatrix} 0 & 1 \\ 1 & 1 \end{bmatrix},B=\begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\)be any two boolean matrices of the same type. Find AvB and A^B.

  48. Determine whether ∗ is a binary operation on the sets given below.
    a*b=min (a,b) on A={1,2,3,4,5)

  49. Determine the truth value of each of the following statements
    (i) If 6 + 2 = 5 , then the milk is white.
    (ii) China is in Europe or \(\sqrt3\) is an integer
    (iii) It is not true that 5 + 5 = 9 or Earth is a planet
    (iv) 11 is a prime number and all the sides of a rectangle are equal

  50. Give the truth value of (~pvq)v(~q)

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