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12th Standard Maths English Medium Reduced Syllabus Important Questions with Answer key - 2021 Part - 2

12th Standard

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Maths

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

      Multiple Choice Questions

    15 x 1 = 15
  1. If A = \(\left[ \begin{matrix} 2 & 0 \\ 1 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & 4 \\ 2 & 0 \end{matrix} \right] \) then |adj (AB)| = 

    (a)

    -40

    (b)

    -80

    (c)

    -60

    (d)

    -20

  2. If ATA−1 is symmetric, then A2 =

    (a)

    A-1

    (b)

    (AT)2

    (c)

    AT

    (d)

    (A-1)2

  3. Which of the following is/are correct?
    (i) Adjoint of a symmetric matrix is also a symmetric matrix.
    (ii) Adjoint of a diagonal matrix is also a diagonal matrix.
    (iii) If A is a square matrix of order n and λ is a scalar, then adj(λA) = λn adj(A).
    (iv) A(adjA) = (adjA)A = |A| I

    (a)

    Only (i)

    (b)

    (ii) and (iii)

    (c)

    (iii) and (iv)

    (d)

    (i), (ii) and (iv)

  4. The area of the triangle formed by the complex numbers z,iz, and z+iz in the Argand’s diagram is

    (a)

    \(\cfrac { 1 }{ 2 } \left| z \right| ^{ 2 }\)

    (b)

    |z|2

    (c)

    \(\cfrac { 3 }{ 2 } \left| z \right| ^{ 2 }\)

    (d)

    2|z|2

  5. The conjugate of a complex number is \(\cfrac { 1 }{ i-2 } \)/Then the complex number is

    (a)

    \(\cfrac { 1 }{ i+2 } \)

    (b)

    \(\cfrac { -1 }{ i+2 } \)

    (c)

    \(\cfrac { -1 }{ i-2 } \)

    (d)

    \(\cfrac { 1 }{ i-2 } \)

  6. If z is a non zero complex number, such that 2iz2=\(\bar { z } \) then |z| is

    (a)

    \(\cfrac { 1 }{ 2 } \)

    (b)

    1

    (c)

    2

    (d)

    3

  7. According to the rational root theorem, which number is not possible rational root of 4x7+2x4-10x3-5?

    (a)

    -1

    (b)

    \(\frac { 5 }{ 4 } \)

    (c)

    \(\frac { 4 }{ 5 } \)

    (d)

    5

  8. If a, b, c ∈ Q and p +√q (p,q ∈ Q) is an irrational root of ax2+bx+c=0 then the other root is

    (a)

    -p+√q

    (b)

    p-iq

    (c)

    p-√q

    (d)

    -p-√q

  9. Let a > 0, b > 0, c >0. h n both th root of th quatlon ax2+b+C= 0 are

    (a)

    real and negative

    (b)

    real and positive

    (c)

    rational numb rs

    (d)

    none

  10. A stone is thrown up vertically. The height it reaches at time t seconds is given by x = 80t -16t2. The stone reaches the maximum height in time t seconds is given by

    (a)

    2

    (b)

    2.5

    (c)

    3

    (d)

    3.5

  11. The maximum value of the function x2 e-2x,

    (a)

    \(\cfrac { 1 }{ e } \)

    (b)

    \(\cfrac { 1 }{ 2e } \)

    (c)

    \(\cfrac { 1 }{ { e }^{ 2 } } \)

    (d)

    \(\cfrac { 4 }{ { e }^{ 4 } } \)

  12. If f (x, y) = exy then \(\frac { { \partial }^{ 2 }f }{ \partial x\partial y } \) is equal to

    (a)

    xyexy

    (b)

    (1 +xy)exy

    (c)

    (1 +y)exy

    (d)

    (1 + x)exy

  13. If we measure the side of a cube to be 4 cm with an error of 0.1 cm, then the error in our calculation of the volume is

    (a)

    0.4 cu.cm

    (b)

    0.45 cu.cm

    (c)

    2 cu.cm

    (d)

    4.8 cu.cm

  14. If p and q are the order and degree of the differential equation \(y=\frac { dy }{ dx } +{ x }^{ 3 }\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) +xy=cosx,\)When

    (a)

    p < q

    (b)

    p = q

    (c)

    p>q

    (d)

    p exists and q does not exist

  15. A binary operation * is defined on the set of positive rational numbers Q+ by a*b = \(\frac { ab }{ 4 } \). Then 3 * \(\left( \frac { 1 }{ 5 } *\frac { 1 }{ 2 } \right) \) is

    (a)

    \(\frac { 3 }{ 160 } \)

    (b)

    \(\frac { 5 }{ 160 } \)

    (c)

    \(\frac { 3 }{ 10 } \)

    (d)

    \(\frac { 3 }{ 40 } \)

    1. 2 Marks

    15 x 2 = 30
  16. Find the modulus of the following complex numbers
    \(\cfrac { 2i }{ 3+4i } \)

  17. Find the modulus of the following complex numbers
    (1-i)10

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

  19. Find the modulus of the complex number i25.

  20. If α, β, γ  and \(\delta\) are the roots of the polynomial equation 2x4+5x3−7x2+8=0 , find a quadratic equation with integer coefficients whose roots are α + β + γ + \(\delta\) and αβ૪\(\delta\).

  21. Find a polynomial equation of minimum degree with rational coefficients, having 2-\(\sqrt{3}\)i as a root.

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

  23. Examine for the rational roots of x8-3x+1=0

  24. Find the eccentricity of the hyperbola. with foci on the x-axis if the length of its conjugate axis is \({ \left( \frac { 3 }{ 4 } \right) }^{ th }\) of the length of its tranverse axis.

  25. Solve :\(\cfrac { dy }{ dx } =\cfrac { 2x }{ { x }^{ 2 }+1 } \)

  26. Define Distribution function.

  27. Define Probability Density function

  28. 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.

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

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

  30. Check whether dot product is defined on the set of vectors. Explain?

    1. 3 Marks

    15 x 3 = 45
  31. Given the complex number z=2+3i, represent the complex numbers in Argand diagram
    z, iz , and z+iz

  32. Given the complex number z=2+3i, represent the complex numbers in Argand diagram
    z, −iz , and z−iz

  33. For the hyperbola 3x2 - 6y2 = -18, find the length of transverse and conjugate axes and eccentricity.

  34. Find the equation of the plane passing through the line of intersection of the planes \(\vec { r } .(2\hat { i } -7\hat { j } +4\hat { k } )=3\) and 3x - 5y + 11 = 0, and the point (-2, 1, 3)

  35. Find the equation of the plane which passes through the point (3, 4, -1) and is parallel to the plane 2x - 3y + 5z = 0. Also, find the distance between the two planes.

  36. Find the equation of the plane passing through the intersection of the planes \(\vec { r } .(\hat { i } +\hat { j } +\hat { k } )+1=0\) and \(\vec { r } .(2\hat { i } -3\hat { j } +5\hat { k } )=2\) and the point (-1, 2, 1)

  37. Find the absolute extrema of the following function on the given closed interval
    f(x) = x2 -12x + 10; [1,2]

  38. A right circular cylinder has radius r =10 cm. and height h = 20 cm. Suppose that the radius of the cylinder is increased from 10 cm to 10. 1 cm and the height does not change. Estimate the change in the volume of the cylinder. Also, calculate the relative error and percentage error.

  39. Assuming log10e = 0.4343, find an approximate value of log10 1003

  40. Find the partial derivatives of the following functions at the indicated point
    h (x, y, z) = x sin (xy) + z2x, \(\left( 2,\frac { \pi }{ 4 }, 1\right) \) 

  41. A firm produces two types of calculators each week, x number of type A and y number of type B. The weekly revenue and cost functions (in rupees) are R(x, y) = 80x + 90y + 0.04xy − 0.05x2 − 0.05y2 and C(x, y) = 8x + 6y + 2000 respectively
    Find the profit function P(x, y) 

  42. Let U(x, y, z) = xyz, x = e-t, y = e-t cos t, z = sin t, t ∈ R. Find \(\frac{dU}{dt}\)

  43. Evaluate the following definite integrals:
    \(\int _{ 0 }^{ 1 }{ \sqrt { \frac { 1-x }{ 1+x } } } dx\)

  44. On the set Q of rational numbers, an operation * is defined as a*b=k(a+b) where k is a given non zero number. Is it associative

  45. If on the set Q of rational numbers, a binary operation * is defined as a*b=λ(a+b) were λ is a nonzero fixed number and its given that * is associative, then the value of λ and what can we say about the operation*?

    1. 5 Marks

    15 x 5 = 75
  46. For what value of λ, the system of equations x+y+z=1, x+2y+4z=λ, x+4y+10z=λ2 is consistent.

  47. Find a polynomial equation of minimum degree with rational coefficients, having \(\sqrt{5}\)\(\sqrt{3}\) as a root.

  48. Find the domain of the following functions
    (i) f(x) = sin-1(2x - 3)
    (ii) f(x) = sin-1x + cos x

  49. Show that the equation of the normal to the curve x=cos3θ,y=asin3θ at 'θ' is xcosθ-ysinθ=acos2θ.

  50. Verify Euler’s theorem for the function \(f(x,y)=\cfrac { 1 }{ \sqrt { { x }^{ 2 }+{ y }^{ 2 } } } \)

  51. Find the ratio of the area between the curves y=cosx and y=cos2x and x- axis from x=0 to \(x=\cfrac { \pi }{ 3 } \)

  52. Find the area bounded by the curve y2(2a-x)=x2 and the line x=2a.

  53. Solve : \({ e }^{ \frac { dy }{ dx } }=x+1,y(0)=5\)

  54. Solve :x2dy+y(x+y)dx=0 given that y=1 when x=1.

  55. Solve :\(x\cfrac { dy }{ dx } sin\left( \cfrac { y }{ x } \right) +x-ysin\left( y\cfrac { y }{ x } \right) =,y(1)=\cfrac { \pi }{ 2 } \)

  56. Four bad oranges are accidentally mixed with sixteen good ones. Find the probability distribution of bad oranges in a draw of two oranges. Also find the mean, variance and standard deviation of the distribution.

  57. Two cards are drawn simultaneously from a well shuffled pack of 52 cards. Find the mean and variance of the number of red cards.

  58. Let Q, be the set of all nonzero rational numbers and k is a nonzero fixed rational number and * be a binary operation defined as a*b=kab. Show that (Q,*) satisfies closure, associative, inverse and commutative properties.

  59. Show that (2018)2017+(2020)2017≡0(mod2019).

  60. Prove by using truth table \(\sim (pV(qVr)\equiv \left( \sim p \right) \wedge \left( \sim q\wedge \sim r \right) \) 

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