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12th Standard Maths English Medium Reduced Syllabus Important Questions with Answer key - 2021 Part - 1

12th Standard

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Maths

Time : 02:45:00 Hrs
Total Marks : 165

      Multiple Choice Questions

    15 x 1 = 15
  1. Let A = \(\left[ \begin{matrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{matrix} \right] \) and 4B = \(\left[ \begin{matrix} 3 & 1 & -1 \\ 1 & 3 & x \\ -1 & 1 & 3 \end{matrix} \right] \). If B is the inverse of A, then the value of x is

    (a)

    2

    (b)

    4

    (c)

    3

    (d)

    1

  2. Which of the following is not an elementary transformation?

    (a)

    Ri ↔️ Rj

    (b)

    Ri ⟶ 2Ri + Rj

    (c)

    Cj ⟶ Cj + Ci

    (d)

    Ri ⟶ Ri + Cj

  3. If (1, -3) is the centre of the circle x+ y+ ax + by + 9 = 0 its radius is

    (a)

    \(\sqrt{10}\)

    (b)

    1

    (c)

    5

    (d)

    \(\sqrt{19}\)

  4. The number of normals to the hyperbola \(\frac { { x }^{ 2 } }{ { a }^{ 2 } } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } \) = 1 from an external point is

    (a)

    2

    (b)

    4

    (c)

    6

    (d)

    5

  5. The two planes 3x + 3y - 3z - 1 = 0 and x + y - z + 5 = 0 are 

    (a)

    mutually perpendicular

    (b)

    parallel

    (c)

    inclined at 45o

    (d)

    inclined at 30

  6. The critical points of the function f(x) = \((x-2)^{ \frac { 2 }{ 3 } }(2x+1)\) are

    (a)

    -1, 2

    (b)

    1, \(\frac { 1 }{ 2 } \)

    (c)

    1, 2

    (d)

    none

  7. The function -3x+12 is ________ function on R.

    (a)

    decreasing

    (b)

    strictly decreasing

    (c)

    increasing

    (d)

    strictly increasing

  8. The function/(x) = x9 + 3x7 + 64 is increasing on ________

    (a)

    R

    (b)

    (-∞, 0)

    (c)

    (0, ∞)

    (d)

    None of these

  9. The approximate value of (627)\(\frac14\) is ................

    (a)

    5.002

    (b)

    5.003

    (c)

    5.005

    (d)

    5.004

  10. The area bounded by the parabola y = x2 and the line y = 2x is

    (a)

    \(\frac43\)

    (b)

    \(\frac23\)

    (c)

    \(\frac{51}{3}\)

    (d)

    \(\frac{30}{3}\)

  11. The solution of sec2x tan y dx+sec2y tan x dy=0 is

    (a)

    tan x+tan y =c

    (b)

    sec x+sec y=c

    (c)

    tan x tan y=c

    (d)

    sec x-sec y =c

  12. The differential equation of x2y = k is _________.

    (a)

    \({ x }^{ 2 }\frac { dy }{ dx } =0\)

    (b)

    \({ x }^{ 2 }\frac { dy }{ dx } +y=0\)

    (c)

    \({ x }\frac { dy }{ dx } +2y=0\)

    (d)

    \(y\frac { dy }{ dx } +2x=0\)

  13. Which one of the following is not a statement?

    (a)

    2 + 3 =5

    (b)

    How beautiful is this flower?

    (c)

    Delhi is the capital of Tamil Nadu

    (d)

    A triangle has found angles.

  14. Which of the following is a contradiction?

    (a)

    p v q

    (b)

    p ∧ q

    (c)

    q v ~ q

    (d)

    q ∧ ~ q

  15. If p is true and q is unknown, then _________

    (a)

    ~ p is true

    (b)

    p v (~p) is false

    (c)

    p ∧ (~p) is true

    (d)

    p v q is true

    1. 2 Marks

    15 x 2 = 30
  16. Reduce the matrix \(\left[ \begin{matrix} 3 & -1 & 2 \\ -6 & 2 & 4 \\ -3 & 1 & 2 \end{matrix} \right] \) to a row-echelon form.

  17. Find the rank of the following matrices which are in row-echelon form :
    \(\left[ \begin{matrix} 2 & 0 & -7 \\ 0 & 3 & 1 \\ 0 & 0 & 1 \end{matrix} \right] \)

  18. Find the rank of the following matrices which are in row-echelon form :
    \(\left[ \begin{matrix} -2 & 2 & -1 \\ 0 & 5 & 1 \\ 0 & 0 & 0 \end{matrix} \right] \)

  19. Find the rank of the following matrices which are in row-echelon form :
    \(\left[ \begin{matrix} 6 \\ \begin{matrix} 0 \\ \begin{matrix} 0 \\ 0 \end{matrix} \end{matrix} \end{matrix}\begin{matrix} 0 \\ \begin{matrix} 2 \\ \begin{matrix} 0 \\ 0 \end{matrix} \end{matrix} \end{matrix}\begin{matrix} -9 \\ \begin{matrix} 0 \\ \begin{matrix} 0 \\ 0 \end{matrix} \end{matrix} \end{matrix} \right] \)

  20. Find the square root of 6−8i .

  21. Write the following in the rectangular form:
     \(\overline { 3i } +\cfrac { 1 }{ 2-i } \).

  22. Solve the equations:
    6x4-35x3+62x2-35x+6=0

  23. Find the principal value of sin-1(2), if it exists.

  24. Find the principal value of
     \({ Sin }^{ -1 }\left( \frac { 1 }{ \sqrt { 2 } } \right) \)

  25. Find the value of sec−1\(\left( -\frac { 2\sqrt { 3 } }{ 3 } \right) \)

  26. Find the value, if it exists. If not, give the reason for non-existence.
    sin-1 [sin5]

  27. Find the distance between the parallel planes x+2y-2z=0 and 2x+4y-4z+5=0

  28. 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.

  29. 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.

  30. Write the converse, inverse, and contrapositive of each of the following implication.
    (i) If x and y are numbers such that x = y, then x2 = y2
    (ii) If a quadrilateral is a square then it is a rectangle.

    1. 3 Marks

    15 x 3 = 45
  31. Find the inverse of each of the following by Gauss-Jordan method:
    \(\left[ \begin{matrix} 2 & -1 \\ 5 & -2 \end{matrix} \right] \)
     

  32. Find the inverse (if it exists) of the following:
    \(\left[ \begin{matrix} 5 & 1 & 1 \\ 1 & 5 & 1 \\ 1 & 1 & 5 \end{matrix} \right] \)

  33. Find the approximate value of f (3.02) where f(x) = 3x2 + 5x +3.

  34. Find the linear approximation to \(g(z)=\sqrt [ 4 ]{ zat } z=2\)

  35. Evaluate : \(\underset { \left( x,y \right) \rightarrow \left( 2,0 \right) }{ lim } \cfrac { \sqrt { 2x-y-2 } }{ 2x-y-4 } \)

  36. Evaluate : \(\underset { \left( x,y \right) \rightarrow \left( 0,0 \right) }{ lim } \cfrac { { x }^{ 2 }-xy }{ \sqrt { x } -\sqrt { y } } \)

  37. z is a homogeneous function in x and y of degree n then prove that \(x\cfrac { \partial z }{ \partial x } +y\cfrac { \partial z }{ \partial y } =(ax+by+n)\) ,where v=zeax+by

  38. Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { sin }^{ 0 }xdx } \)

  39. Form the D.E of family of curves represented by y=c(x-c)2.where c is the parameter.

  40. Show that the function y=Acos2x-Bsin2x is a solution of the D.E y2+4y=0

  41. Solve:\(\cfrac { dy }{ dx } =\cfrac { 1-cosx }{ 1+cosx } \)

  42. A six sided die is marked '2' on one face, '3' on two ofits faces, and '4' on remaining three faces. The die is thrown twice. If X denotes the total score in two throws, find the values of the random variable and number of points in its inverse images.

  43. Verify the
    (i) closure property,
    (ii) commutative property,
    (iii) associative property
    (iv) existence of identity and
    (v) existence of inverse for the arithmetic operation - on Z.

  44. The pair (S,*) identify and inverse aximos, prove that * is commutative if and only if \(\left( a\ast b \right) ^{ 2 }={ a }^{ a }\ast { b }^{ 2 }\).

  45. In in (S,*) satisfying closure, associative, identity and inverse axioms and\((a\ast b)^{ -1 }={ a }^{ -1 }\ast { b }^{ -1 }\) ∀a,b∈S, then prove that * is commutative. 

    1. 5 Marks

    15 x 5 = 75
  46. Show that the matrix \(\left[ \begin{matrix} 3 & 1 & 4 \\ 2 & 0 & -1 \\ 5 & 2 & 1 \end{matrix} \right] \) is non-singular and reduce it to the identity matrix by elementary row transformations.

  47. Find the inverse of A = \(\left[ \begin{matrix} 2 & 1 & 1 \\ 3 & 2 & 1 \\ 2 & 1 & 2 \end{matrix} \right] \) by Gauss-Jordan method.

  48. From a lot of 10 items containing 3 defective items, 4 items are drawn at random. Find the mean and variance of the number of defective items drawn.

  49. A box contains 4 red and 5 black marbles. Find the probability distribution of the red marbles in a random draw of three marbles. Also find the mean and standard deviation of the distribution

  50. The difference between the mean and variance of a binomial distribution is 1 and the difference of their squares is 11. Find the distribution.

  51. In a business venture a man can make a profit of Rs.2,000 with a probability of 0.4 or have a loss of Rs.1,000 with a probability of 0.6. What is his expectation, variance and S.D of profit?

  52. The p.d.f of a continuous random variable X is \(f(x)=\begin{cases} a+b{ x }^{ 2 },0\le x\le 1 \\ 0,otherwise \end{cases}\) where a and b are some constants. Find
    (a) a and b if \(E(x)=\cfrac { 3 }{ 5 } \) 
    (b) Var (X)

  53. The probability distribution of a random variable X is given by

    X 0 1 2 3
    P(X) 0.1 0.3 0.5 0.1

    If Y=X2+3X, find the mean and the variance of Y.

  54. Verify
    (i) closure property,
    (ii) commutative property,
    (iii) associative property,
    (iv) existence of identity, and
    (v) existence of inverse for the operation ×11 on a subset A = {1,3,4,5,9}
    of the set of remainders {0,1,2,3,4,5,6,7,8,9,10}

  55. Define an operation∗ on Q as follows:  a*b=\(\left( \frac { a+b }{ 2 } \right) \); a,b ∈Q. Examine the existence of identity and the existence of inverse for the operation * on Q.

  56. Let S be a non-empty set and 0 be a binary operation on s defined by x 0 y = x; x, Y \(\in \) s. Determine whether 0 is commutative and association.

  57. \(M=\left\{ \left( \begin{matrix} a & 0 \\ 0 & a \end{matrix} \right) /\alpha \neq 0\ and\ \alpha \epsilon R \right\} \) Show that M satisfies the closure, associative, inverse, identity and commutative axioms under multiplication.

  58. Show that Z7-{[0]} satisfies the closure, associative, identity, inverse and commutative axioms under multiplication modulo 5.

  59. Examine whether there is a nonempty subset S of the set of real numbers such that it satisfies closure, associative, identity and inverse properties under a binary operation * defined as a*b=k where k is a fixed real number.

  60. Prove that (2019)10+(2020)10≡1025(mod 2018)

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