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REVISION TEST

12th Standard EM

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Maths

Time : 03:00:00 Hrs
Total Marks : 90
    PART-A
    20 x 1 = 20
  1. If A\(\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right] =\left[ \begin{matrix} 6 & 0 \\ 0 & 6 \end{matrix} \right] \), then A = 

    (a)

    \(\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right] \)

    (b)

    \(\left[ \begin{matrix} 1 & 2 \\ -1 & 4 \end{matrix} \right] \)

    (c)

    \(\left[ \begin{matrix} 4 & 2 \\ -1 & 1 \end{matrix} \right] \)

    (d)

    \(\left[ \begin{matrix} 4 & -1 \\ 2 & 1 \end{matrix} \right] \)

  2. If A is a non-singular matrix such that A-1 = \(\left[ \begin{matrix} 5 & 3 \\ -2 & -1 \end{matrix} \right] \), then (AT)−1 =

    (a)

    \(\left[ \begin{matrix} -5 & 3 \\ 2 & 1 \end{matrix} \right] \)

    (b)

    \(\left[ \begin{matrix} 5 & 3 \\ -2 & -1 \end{matrix} \right] \)

    (c)

    \(\left[ \begin{matrix} -1 & -3 \\ 2 & 5 \end{matrix} \right] \)

    (d)

    \(\left[ \begin{matrix} 5 & -2 \\ 3 & -1 \end{matrix} \right] \)

  3. If A is a square matrix that IAI = 2, than for any positive integer n, |An| =

    (a)

    0

    (b)

    2n

    (c)

    2n

    (d)

    n2

  4. 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)

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

  5. 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

  6. The principal value of the amplitude of (1+i) is

    (a)

    \(\frac { \pi }{ 4 } \)

    (b)

    \(\frac { \pi }{ 12 } \)

    (c)

    \(\frac { 3\pi }{ 4 } \)

    (d)

    \(\pi \)

  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 sin-1 x+sin-1 y+sin-1 z=\(\frac{3\pi}{2}\), the value of x2017+y2018+z2019\(-\frac { 9 }{ { x }^{ 101 }+{ y }^{ 101 }+{ z }^{ 101 } } \)is

    (a)

    0

    (b)

    1

    (c)

    2

    (d)

    3

  9. If cot-1 2 and cot-1 3 are two angles of a triangle, then the third angle is

    (a)

    \(\frac{\pi}{4}\)

    (b)

    \(\frac{3\pi}{4}\)

    (c)

    \(\frac{\pi}{6}\)

    (d)

    \(\frac{\pi}{3}\)

  10. The number of real solutions of the equation \(\sqrt { 1+cos2x } ={ 2sin }^{ -1 }\left( sinx \right) ,-\pi <x<\pi \) is

    (a)

    0

    (b)

    1

    (c)

    2

    (d)

    infinte

  11. The circle x2+y2=4x+8y+5intersects the line3x−4y=m at two distinct points if

    (a)

    15< m < 65

    (b)

    35< m <85

    (c)

    −85<m < −35

    (d)

    −35<m <15

  12. If \([\vec { a } ,\vec { b } ,\vec { c } ]=1\)\(\frac { \vec { a } .(\vec { b } \times \vec { c } ) }{ (\vec { c } \times \vec { a } ).\vec { b } ) } +\frac { \vec { b } .(\vec { c } \times \vec { a } ) }{ (\vec { a } \times \vec { b } ).\vec { c } } +\frac { \vec { c } .(\vec { a } \times \vec { b } ) }{ (\vec { c } \times \vec { b } ).\vec { a } } \) is

    (a)

    1

    (b)

    -1

    (c)

    2

    (d)

    3

  13. The position of a particle moving along a horizontal line of any time t is given by set) = 3t2 -2t- 8. The time at which the particle is at rest is

    (a)

    t= 0

    (b)

    \(\\ \\ \\ t=\cfrac { 1 }{ 3 } \)

    (c)

    t =1

    (d)

    t = 3

  14. If u (x, y) = ex2+y2, then \(\frac { \partial u }{ \partial x } \) is equal to

    (a)

     ex2+y2

    (b)

    2xu

    (c)

    x2u

    (d)

    y2u

  15. The value of \(\int _{ 0 }^{ \infty }{ { e }^{ -3x }{ x }^{ 2 }dx } \\ \) is

    (a)

    \(\frac{7}{27}\)

    (b)

    \(\frac{5}{27}\)

    (c)

    \(\frac{4}{27}\)

    (d)

    \(\frac{2}{27}\)

  16. The order and degree of the differential equation \(\sqrt { sin\quad x } (dx+dy)=\sqrt { cos\quad x } (dx-dy)\)

    (a)

    1,2

    (b)

    2,2

    (c)

    1,1

    (d)

    2,1

  17. A random variable X has binomial distribution with n = 25 and p = 0.8 then standard deviation of X is

    (a)

    6

    (b)

    4

    (c)

    3

    (d)

    2

  18. Let X have a Bernoulli distribution with mean 0.4, then the variance of (2X - 3) is

    (a)

    0.24

    (b)

    0.48

    (c)

    0.6

    (d)

    0.96

  19. The operation * defined by a*b =\(\frac{ab}{7}\) is not a binary operation on

    (a)

    Q+

    (b)

    Z

    (c)

    R

    (d)

    C

  20. If a*b=\(\sqrt { { a }^{ 2 }+{ b }^{ 2 } } \) on the real numbers then * is

    (a)

    commutative but not associative

    (b)

    associative but not commutative

    (c)

    both commutative and associative

    (d)

    neither commutative nor associative

  21. PART-B

    WRITE ANY 7 QUESTIONS

    7 x 2 = 14
  22. Prove that \(\left[ \begin{matrix} \cos { \theta } & -\sin { \theta } \\ \sin { \theta } & \cos { \theta } \end{matrix} \right] \) is orthogonal

  23. If z1=3-2i and z2=6+4i, find \(\cfrac { { z }_{ 1 } }{ z_{ 2 } } \)

  24. Discuss the nature of the roots of the following polynomials:
    x2018+1947x1950+15x8+26x6+2019

  25. If \(\hat { a } =\hat { -3i } -\hat { j } +\hat { 5k } \)\(\hat{b}=\hat{i}-\hat{2j}+\hat{k} \)\(\hat{c}=\hat{4i}-\hat{4k} \)and \(\hat { a } .(\hat { b } \times \hat { c } )\)

  26. Find the slope of the tangent to the curves at the respective given points.
    y = x4 + 2x2 − x at x =1

  27. If w(x, y, z) = x2 y + y2z + z2x, x, y, z∈R, find the differential dw .

  28. Evaluate the following integrals using properties of integration:
    \(\int _{ -\frac { \pi }{ 4 } }^{ \frac { \pi }{ 4 } }{ { sin }^{ 2 }xdx } \)

  29. A differential equation, determine its order, degree (if exists)
    \(y\left( \frac { dy }{ dx } \right) =\frac { x }{ \left( \frac { dy }{ dx } \right) +{ \left( \frac { dy }{ dx } \right) }^{ 3 } } \)

  30. 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.

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

  32. PART-C

    WRITE ANY 7 QUESTIONS

    7 x 3 = 21
  33. Verify the property (AT)-1 = (A-1) with A = \(\left[ \begin{matrix} 2 & 9 \\ 1 & 7 \end{matrix} \right] \).

  34. If α, β and γ are the roots of the cubic equation x3+2x2+3x+4=0, form a cubic equation whose roots are, 2α, 2β, 2γ

  35. Find the domain of sin−1(2−3x2)

  36. Find the equation of the tangent and normal to the circle x2+y2−6x+6y−8=0 at (2,2) .

  37. Prove by vector method that the area of the quadrilateral ABCD having diagonals AC and BD is \(\frac { 1 }{ 2 } \left| \vec { AC } \times \vec { BC } \right| \).

  38. Find the points on the curve y2 - 4xy = x2 + 5 for which the tangent is horizontal.

  39. Let F(x, y) = x3 y + y2x + 7 for all (x, y)∈ R2. Calculate \(\frac { \partial F }{ \partial x } \)(-1,3) and \(\frac { \partial F }{ \partial y } \)(-2,1).

  40. Solve \(\frac { dy }{ dx } +2y={ e }^{ -x }\)

  41. If X- B(n, p) such that 4P(X = 4) = P(X = 2) and n = 6 • Find the distribution, mean and standard deviation of X.

  42. Construct the truth table for \((p\overset { \_ \_ }{ \vee } q)\wedge (p\overset { \_ \_ }{ \vee } \neg q)\)

  43. PART-D 

    WRITE ANY 7 QUESTIONS

    7 x 5 = 35
  44. If F(\(\alpha\)) = \(\left[ \begin{matrix} \cos { \alpha } & 0 & \sin { \alpha } \\ 0 & 1 & 0 \\ -\sin { \alpha } & 0 & \cos { \alpha } \end{matrix} \right] \), show that [F(\(\alpha\))]-1 = F(-\(\alpha\)).

  45. If \(2cosa=x+\cfrac { 1 }{ x } \) and \(2cos\beta =y+\cfrac { 1 }{ y } \), show that 
    i) \(\cfrac { x }{ y } +\cfrac { y }{ x } =2cos\left( \alpha -\beta \right) \).
    ii) \(xy-\cfrac { 1 }{ xy } =2isin\left( \alpha +\beta \right) \)
    iii)
    \(\cfrac { { x }^{ m } }{ { y }^{ n } } -\cfrac { { y }^{ n } }{ { x }^{ m } } =2isin\left( m\alpha -n\beta \right) \)
    iv)
    \({ x }^{ m }{ y }^{ n }+\cfrac { 1 }{ { x }^{ m }{ y }^{ n } } =2cos(m\alpha +n\beta )\)

  46. Solve the equation (x-2)(x-7)(x-3)(x+2)+19=0

  47. If \(\vec { a } =-2\hat { i } +3\hat { j } -2\hat { k } ,\vec { b } =3\hat { i } -\hat { j } +3\hat { k } ,\vec { c } =2\hat { i } -5\hat { j } +\hat { k } \) find \((\vec { a } \times \vec { b } )\times \vec { c } \) and \((\vec { a } \times \vec { b } )\times \vec { c } \). State whether they are equal.

  48. Find the equation of tangent and normal to the curve given by x = 7 cos t and y = 2sin t, t ∈ R at any point on the curve.

  49. For each of the following functions find the fx, fy, and show that fxy =fyx
    f(x,y) = \(\frac { 3x }{ y+sinx \ } \) 

  50. Evaluate: \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ (\sqrt { tan\quad x } +\sqrt { cot\quad x } )dx } \)

  51. Solve the Linear differential equation:
    \(\frac { dy }{ dx } +\frac { 3y }{ x } =\frac { 1 }{ { x }^{ 2 } } \), given that y=2 when x=1 

  52. Suppose the amount of milk sold daily at a milk booth is distributed with a minimum of 200 Iitres and a maximum of 600 litres with probability density function
    \(\begin{cases} \begin{matrix} k & 200\le x\le 600 \end{matrix} \\ \begin{matrix} 0 & otherwise \end{matrix} \end{cases}\) 
    Find
    (i) the value of k
    (ii) the distribution function
    (iii) the probability that daily sales will fall between 300 litres and 500 litres?

  53. Let M=\(\left\{ \left( \begin{matrix} x & x \\ x & x \end{matrix} \right) :x\in R-\{ 0\} \right\} \) and let * be the matrix multiplication. Determine whether M is closed under ∗. If so, examine the commutative and associative properties satisfied by ∗ on M.

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