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Public Exam Model Question Paper II 2019 - 2020

12th Standard EM

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Maths

Time : 02:45:00 Hrs
Total Marks : 90

    Part I

    Answer all the questions.

    Choose the most suitable answer from the given four alternatives and write the option code with the corresponding answer.

    20 x 1 = 20
  1. If A = \(\left[ \begin{matrix} \frac { 3 }{ 5 } & \frac { 4 }{ 5 } \\ x & \frac { 3 }{ 5 } \end{matrix} \right] \) and AT = A−1 , then the value of x is

    (a)

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

    (b)

    \(\frac { -3 }{ 5 } \)

    (c)

    \(\frac { 3 }{ 5 } \)

    (d)

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

  2. The system of linear equations x + y + z  = 6, x + 2y + 3z =14 and 2x + 5y + λz =μ (λ, μ \(\in \) R) is consistent with unique solution if

    (a)

    λ = 8

    (b)

    λ = 8, μ ≠ 36

    (c)

    λ ≠ 8

    (d)

    none

  3. \(\frac { (cos\theta +isin\theta )^{ 6 } }{ (cos\theta -isin\theta )^{ 5 } } \) = ________

    (a)

    cos 11θ - isin 11θ

    (b)

    cos 11θ + isin 11θ

    (c)

    cosθ + i sinθ

    (d)

    \(cos\frac { 6\theta }{ 5 } +isin\frac { 6\theta }{ 5 } \)

  4. If f and g are polynomials of degrees m and n respectively, and if h(x) =(f 0 g)(x), then the degree of h is

    (a)

    mn

    (b)

    m+n

    (c)

    mn

    (d)

    nm

  5. If x2 - hx - 21 = 0 and x2 - 3hx + 35 = 0 (h > 0) have a common root, then h = ___________

    (a)

    0

    (b)

    1

    (c)

    4

    (d)

    3

  6. \({ sin }^{ -1 }\frac { 3 }{ 5 } -{ cos }^{ -1 }\frac { 12 }{ 13 } +{ sec }^{ -1 }\frac { 5 }{ 3 } { -cosec }^{ 1- }\frac { 13 }{ 2 } \)is equal to

    (a)

    2\(\pi\)

    (b)

    \(\pi\)

    (c)

    0

    (d)

    tan-1\(\frac{12}{65}\)

  7. If \({ tan }^{ -1 }\left\{ \cfrac { \sqrt { 1+{ x }^{ 2 } } -\sqrt { 1-{ x }^{ 2 } } }{ \sqrt { 1+{ x }^{ 2 } } +\sqrt { 1-{ x }^{ 2 } } } \right\} =\alpha \) then x2 =

    (a)

    \(sin2\alpha \)

    (b)

    \(sin\alpha \)

    (c)

    \(cos2\alpha \)

    (d)

    \(cos\alpha \)

  8. The equation of the circle passing through(1,5) and (4,1) and touching y -axis is x2+y2−5x−6y+9+(4x+3y−19)=0 whereλ is equal to

    (a)

    \(0,-\frac { 40 }{ 9 } \)

    (b)

    0

    (c)

    \(\frac { 40 }{ 9 } \)

    (d)

    \(\frac { -40 }{ 9 } \)

  9. The volume of the parallelepiped with its edges represented by the vectors \(\hat { i } +\hat { j } ,\hat { i } +2\hat { j } ,\hat { i } +\hat { j } +\pi \hat { k } \) is

    (a)

    \(\frac { \pi }{ 2 } \)

    (b)

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

    (c)

    \( { \pi }\)

    (d)

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

  10. The maximum value of the product of two positive numbers, when their sum' of the squares is 200, is

    (a)

    100

    (b)

    \(25\sqrt { 7 } \)

    (c)

    28

    (d)

    \(24\sqrt { 14 } \)

  11. The angle made by any tangent to the curve y = x5 + 8x + 1 with the X-axis is a

    (a)

    obtuse

    (b)

    right angle

    (c)

    acute angle

    (d)

    no angle

  12. If u(x, y) = x2+ 3xy + y - 2019, then \(\frac { \partial u }{ \partial x } \)(4, -5) is equal to

    (a)

    -4

    (b)

    -3

    (c)

    -7

    (d)

    13

  13. If f(x,y) = 2x2 - 3xy + 5y + 7 then f(0, 0) and f(1, 1) is

    (a)

    7,11

    (b)

    11,7

    (c)

    0,7

    (d)

    1,0

  14. For any value of n∈Z, \(\int _{ 0 }^{ \pi }{ e{ cos }^{ 2x }{ cos }^{ 3 } } [(2n+1)x]\) is

    (a)

    \(\frac{\pi}{2}\)

    (b)

    \(\pi\)

    (c)

    0

    (d)

    2

  15. \(\int _{ 0 }^{ \frac { \pi }{ 4 } }{ { cos }^{ 3 }2x \ dx= } \)

    (a)

    \(\frac { 2 }{ 3 } \)

    (b)

    \(\frac { 1 }{ 3 } \)

    (c)

    0

    (d)

    \(\frac { 2\pi }{ 3 } \)

  16. The degree of the differential equation y \(y(x)=1+\frac { dy }{ dx } +\frac { 1 }{ 1.2 } { \left( \frac { dy }{ dx } \right) }^{ 2 }+\frac { 1 }{ 1.2.3 } { \left( \frac { dy }{ dx } \right) }^{ 3 }+....\) is

    (a)

    2

    (b)

    3

    (c)

    1

    (d)

    4

  17. Using y = vx, the differential equation \(\frac { dy }{ dx } =\frac { y }{ x+\sqrt { xy } } \) is reduced to ________.

    (a)

    x(1+\(\sqrt{v}\))dv=v\(\sqrt{v}\)dx

    (b)

    x(1-\(\sqrt{v}\))dv=v\(\sqrt{v}\)dx

    (c)

    x(1+\(\sqrt{v}\))dv=-v\(\sqrt{v}\)dx

    (d)

    v(1+\(\sqrt{v}\))dx-v\(\sqrt{v}\)dv=0

  18. A computer salesperson knows from his past experience that he seUs computers to one in every twenty customers who enter the showroom. What is the probability that he will seU a computer to exactly two of the next three customers?

    (a)

    \(\cfrac { 57 }{ { 20 }^{ 3 } } \)

    (b)

    \(\cfrac { 57 }{ { 20 }^{ 2 } } \)

    (c)

    \(\cfrac { { 19 }^{ 3 } }{ { 20 }^{ 3 } } \)

    (d)

    \(\cfrac { 57 }{ 20 } \)

  19. The proposition p ∧ (¬p ∨ q) is

    (a)

    a tautology

    (b)

    a contradiction

    (c)

    logically equivalent to p ∧ q

    (d)

    logically equivalent to p ∨ q

  20. Define * on Z by a*b = a+b+1 ∀ a,b \(\in \) Z. Then the identity element of z is

    (a)

    1

    (b)

    0

    (c)

    1

    (d)

    -1

  21. Part II

    Answer any 7 questions. Question no. 30 is compulsory.

    10 x 2 = 20
  22. Find the rank of the matrix A =\(\left[ \begin{matrix} 4 \\ 7 \end{matrix}\begin{matrix} 5 \\ -3 \end{matrix}\begin{matrix} -6 \\ 0 \end{matrix}\begin{matrix} 1 \\ 8 \end{matrix} \right] \).

  23. Find the number of positive and negative roots of the equation x7 - 6x6 + 7x5 + 5x2+2x+2

  24. Find the equation of the parabola with vertex at the origin, passing through (2, -3) and symmetric about x-axis

  25. The volume of the parallelepiped whose coterminus edges are \(7\hat { i } +\lambda \hat { j } -3\hat { k } ,\hat { i } +2\hat { j } -\hat { k } \)\(-3\hat { i } +7\hat { j } +5\hat { k } \) is 90 cubic units. Find the value of λ.

  26. A particle moves in a line so that x=\(\sqrt { t } \). Show that the acceleration is negative and proportional to the cube of the velocity.

  27. Evaluate \(\begin{matrix} lim \\ (x,y)\rightarrow (1,2) \end{matrix}\), if the limit exists, where \((x,y)=\frac { { 3x }^{ 2 }-xy }{ { x }^{ 2 }+{ y }^{ 2 }+3 } \)

  28. Evaluate \(\int { \sum _{ r=0 }^{ \infty }{ \cfrac { { x }^{ r }{ 2 }^{ r } }{ r! } } dx } \)

  29. Determine the order and degree (if exists) of the following differential equations: 
    \(\frac { { d }^{ 2 }y }{ { dx }^{ 2 } } +3{ \left( \frac { dy }{ dx } \right) }^{ 2 }={ x }^{ 2 }log\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) \)

  30. The time to failure in thousands of hours of an electronic equipment used in a manufactured computer has the density function \(f(x)=\begin{cases} \begin{matrix} { 3e }^{ -3x } & x>0 \end{matrix} \\ \begin{matrix} 0 & elsewhere \end{matrix} \end{cases}\) 
    Find the expected life of this electronic equipment.

  31. An operation * is defined as mXn=mn-nm.Is it binary on N?

  32. Part III

    Answer any 7 questions. Question no. 40 is compulsory.

    10 x 3 = 30
  33. For what value of t will the system tx +3y - z = 1, x + 2y + z = 2, -tx + y + 2z = -1 fail to have unique solution?

  34. If \(\cfrac { z+3 }{ z-5i } =\cfrac { 1+4i }{ 2 } \), find the complex number z

  35. If α and β are the roots of the quadratic equation 2x2−7x+13 = 0 , construct a quadratic equation whose roots are α2 and β2.

  36. Solve: 2x+2x-1+2x-2=7x+7x-1+7x-2

  37. If \(sin\left( { sin }^{ -1 }\cfrac { 1 }{ 5 } +{ cos }^{ -1 }x \right) =1\) then find the value ofx.

  38. Find the equations of the tangent and normal to hyperbola 12x2−9y2=108 at \(\theta =\frac { \pi }{ 3 } \) (Hint: use parametric form)

  39. Prove that \(\left[ \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ c } \right] \)=\(\left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] \)

    ()

    0

  40. Find the local extremum of the function f (x) = x4 + x 32x

  41. Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ \cfrac { dx }{ 1+{ tan }^{ 3 }x } } \)

  42. Construct the truth table for (-p) v (q ∧ r)

  43. Part IV

    Answer all the questions.

    14 x 5 = 70
  44. The sum of three numbers is 20. If we multiply the third number by 2 and add the first number to the result we get 23. By adding second and third numbers to 3 times the first number we get 46. Find the numbers using Cramer's rule.

  45. Show that \(\left( \frac { i+\sqrt { 3 } }{ -i+\sqrt { 3 } } \right) ^{ 2\omega }+\left( \frac { i-\sqrt { 3 } }{ i+\sqrt { 3 } } \right) ^{ 2\omega }\)=-1

  46. Determine k and solve the equation 2x3-6x2+3x+k=0 if one of its roots is twice the sum of the other two roots.

  47. Find the principal value of
    sec−1(−2).

  48. Provethat \({ tan }^{ -1 }\left( \cfrac { 1-x }{ 1+x } \right) -{ tan }^{ -1 }\left( \cfrac { 1-y }{ 1+y } \right) ={ sin }^{ -1 }\left( \cfrac { y-x }{ \sqrt { 1+{ x }^{ 2 } } .\sqrt { 1+{ y }^{ 2 } } } \right) \\ \)

  49. The foci of a hyperbola coincides with the foci of the ellipse \(\frac { { x }^{ 2 } }{ 25 } +\frac { y^{ 2 } }{ 9 } =1\). Find the equation of the hyperbola if its eccentricity is 2.

  50. Find the vector and Cartesian equation of the plane passing through the point (1,1, -1) and perpendicular to the planes x + 2y + 3z - 7 = 0 and 2x - 3y + 4z = 0

  51. Evaluate: \(\underset{x \rightarrow1}{lim} \ x^{\frac{1}{1-x}}\)

  52. Let f(x, y) = sin(xy2) + ex3+5y for all ∈ R2. Calculate \(\frac { \partial f }{ \partial x } ,\frac { \partial f }{ \partial y } ,\frac { { \partial }^{ 2 }f }{ { \partial y\partial x } } \)and \(\frac { { \partial }^{ 2 }f }{ { \partial x\partial y } } \)

  53. Show that the area under the curve y = sin x and y = sin 2x between x = 0 and x = \(\frac { \pi }{ 3 } \) and x axis are as 2:3

    1. If the probability that a fluorescent light has a useful life of at least 600 hours is 0.9, find the probabilities that among 12 such lights
      (i) exactly 10 will have a useful life of at least 600 hours;
      (ii) at least 11 will have a useful life of at I least 600 hours; I
      (iii) at least 2 will not have a useful life of at : least 600 hours.

    2. Construct the truth table for (p ∧ q) v r.

    1. Find the differential equations of the family of all the ellipses having foci on the y-axis and centre at the origin.

    2. Sovle : (x+y+1)2dy=dx,y(-1)=0

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