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Public Exam Model Question Paper III 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 0 ≤ θ  ≤ π and the system of equations x + (sinθ)y - (cosθ)z = 0, (cosθ)x - y +z = 0, (sinθ)x + y - z = 0 has a non-trivial solution then θ is

    (a)

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

    (b)

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

    (c)

    \(\frac { 5\pi }{ 6 } \)

    (d)

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

  2. If A = [2 0 1] then the rank of AAT is ______

    (a)

    1

    (b)

    2

    (c)

    3

    (d)

    0

  3. If a=cosθ + i sinθ, then \(\frac { 1+a }{ 1-a } \) =

    (a)

    cot \(\frac { \theta }{ 2 } \)

    (b)

    cot θ

    (c)

    i cot \(\frac { \theta }{ 2 } \)

    (d)

    i tan\(\frac { \theta }{ 2 } \)

  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 ax2 + bx + c = 0, a, b, c E R has no real zeros, and if a + b + c < 0, then __________

    (a)

    c>0

    (b)

    c<0

    (c)

    c=0

    (d)

    c≥0

  6. The equation tan-1 x-cot-1 x=tan-1\(\left( \frac { 1 }{ \sqrt { 3 } } \right) \)has

    (a)

    no solution

    (b)

    unique solution

    (c)

    two solutions

    (d)

    infinite number of solutions

  7. If \({ sin }^{ -1 }x-cos^{ -1 }x=\cfrac { \pi }{ 6 } \) then

    (a)

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

    (b)

    \(\cfrac { \sqrt { 3 } }{ 2 } \)

    (c)

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

    (d)

    none of these

  8. The eccentricity of the ellipse (x−3)2 +(y−4)2 =\(\frac { { y }^{ 2 } }{ 9 } \) is

    (a)

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

    (b)

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

    (c)

    \(\frac { 1 }{ 3\sqrt { 2 } } \)

    (d)

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

  9. The angle between the lines \(\frac { x-2 }{ 3 } =\frac { y+1 }{ -2 } \), z=2 and \(\frac { x-1 }{ 1 } =\frac { 2y+3 }{ 3 } =\frac { z+5 }{ 2 } \)

    (a)

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

    (b)

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

    (c)

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

    (d)

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

  10. The point of inflection of the curve y = (x - 1)3 is

    (a)

    (0,0)

    (b)

    (0,1)

    (c)

    (1,0)

    (d)

    (1,1)

  11. \(\underset { x\rightarrow 0 }{ lim } \frac { x }{ tanx } \) is _________

    (a)

    1

    (b)

    -1

    (c)

    0

    (d)

  12. Linear approximation for g(x) = cos x at x=\(\frac{-\pi}{2}\) is

    (a)

    x + \(\frac{-\pi}{2}\)

    (b)

    - x + \(\frac{\pi}{2}\)

    (c)

    x - \(\frac{\pi}{2}\)

    (d)

    - x + \(\frac{\pi}{2}\)

  13. If the radius of the sphere is measured as 9 em with an error of 0.03 cm, the approximate error in calculating its volume is

    (a)

    9.72 cm3

    (b)

    0.972 cm3

    (c)

    0.972π cm3

    (d)

    9.72π cm3

  14. The value of \(\int _{ -4 }^{ 4 }{ \left[ { tan }^{ -1 }\left( \frac { { x }^{ 2 } }{ { x }^{ 4 }+1 } \right) +{ tan }^{ -1 }\left( \frac { { x }^{ 4 }+1 }{ { x }^{ 2 } } \right) \right] dx } \) is

    (a)

    \(\pi\)

    (b)

    \(2\pi\)

    (c)

    \(3\pi\)

    (d)

    \(4\pi\)

  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 integrating factor of the differential equation \(\frac{dy}{dx}\)+P(x)y=Q(x)is x, then P(x)

    (a)

    x

    (b)

    \(\frac { { x }^{ 2 } }{ 2 } \)

    (c)

    \(\frac{1}{x}\)

    (d)

    \(\frac{1}{x^2}\)

  17. The differential equation associated with the family of concentric circles having their centres at the origin is _________.

    (a)

    \(\frac { dy }{ dx } =\frac { -x }{ y } \)

    (b)

    \(\frac { dy }{ dx } =\frac { -y }{ x } \)

    (c)

    \(\frac { dy }{ dx } =\frac { x }{ y } \)

    (d)

    \(\frac { dy }{ dx } =\frac { y }{ x } \)

  18. If the function  \(f(x)=\cfrac { 1 }{ 12 } \) for. a < x < b, represents a probability density function of a continuous random variable X, then which of the followingcannot be the value of a and b?

    (a)

    0 and 12

    (b)

    5 and 17

    (c)

    7 and 19

    (d)

    16 and 24

  19. Determine the truth value of each of the following statements:
    (a) 4+2=5 and 6+3=9
    (b) 3+2=5 and 6+1=7
    (c) 4+5=9 and1+2= 4
    (d) 3+2=5 and 4+7=11

    (a)
    (a) (b) (c) (d)
    F T T T
    (b)
    (a) (b) (c) (d)
    T F T F
    (c)
    (a) (b) (c) (d)
    T T F F
    (d)
    (a) (b) (c) (d)
    F F T T
  20. The identity element in the group {R - {1},x} where a * b = a + b - ab is

    (a)

    0

    (b)

    1

    (c)

    \(\frac { 1 }{ a-1 } \)

    (d)

    \(\frac { a }{ a-1 } \)

  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 th Int rval for a for which 3x2+2(a2+1) x+(a2-3n+2) possesses roots of opposite sign.

  24. Find the locus of a point which divides so that the sum of its.distances from (-4, 0) and (4, 0) is 10 units.

  25. Find the angle between the following lines.
    2x = 3y =  −z and 6x = − y = −4z.

  26. Determine the domain of convexity of the function y=ex

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

  28. Find the area of the region enclosed by the curve y = \(\sqrt x\) + 1, the axis of x and the lines x=0, x=4.

  29. Solve the Linear differential equation:
    \(\frac { dy }{ dx } =\frac { { sin }^{ 2 }x }{ 1+{ x }^{ 3 } } -\frac { { 3x }^{ 2 } }{ 1+{ x }^{ 3 } } y\)

  30. 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 .

  31. A and B are Boolean matrices of order 2X2.If AVB=A, is it necessary that \(B=\left( \begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix} \right) \)

  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. Simplify the following:
    i 1729

  35. 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

  36. Solve:(x-1)4+(x-5)4=82

  37. Solve: \({ tan }^{ -1 }\left( \cfrac { x-1 }{ x-2 } \right) +{ tan }^{ -1 }\left( \cfrac { x+1 }{ x+2 } \right) =\cfrac { \pi }{ 4 } \)

  38. Identify the type of the conic for the following equations:
    (1) 16y2=−4x2+64
    (2) x2+y2=−4x−y+4
    (3) x2−2y=x+3
    (4) 4x2−9y2−16x+18y−29 = 0

  39. If \(\overset { \rightarrow }{ a } =\overset { \wedge }{ i } -\overset { \wedge }{ j } ,\overset { \rightarrow }{ b } =\overset { \wedge }{ j } -\overset { \wedge }{ k } ,\overset { \rightarrow }{ c } =\overset { \wedge }{ k } -\overset { \wedge }{ i } \) then find \(\left[ \overset { \rightarrow }{ a } -\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ b } -\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ c } -\overset { \rightarrow }{ a } \right] \)

    ()

    parametric form od vector equation

  40. Compute the value of 'c' satisfied by Rolle’s theorem for the function \(f(x)=log(\frac{x^{2}+6}{5x})\) in the interval [2,3]

  41. Find the area bounded by the curve y=x|x|,x-axis and the ordinates x=-1,x=1

  42. Give an example (S,*), where a2=e for all a∈S.Given that e is identity element.

  43. Part IV

    Answer all the questions.

    14 x 5 = 70
  44. Write thefunction\(f(x)={ tan }^{ -1 }\sqrt { \cfrac { a-x }{ a+x } } -a<x<a\) in the simplest form
     

  45. 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.

  46. If \(\left| \overset { \rightarrow }{ A } \right| =\overset { \wedge }{ i } +\overset { \wedge }{ j } +\overset { \wedge }{ k } \) and \(\overset { \wedge }{ i } =\overset { \wedge }{ j } -\overset { \wedge }{ k } \) are two given vector, then find a vector B satisfying the equations \(\overset { \rightarrow }{ A } \times \overset { \rightarrow }{ B } \)\(\overset { \rightarrow }{ C } \) and \(\overset { \rightarrow }{ A } \).\(\overset { \rightarrow }{ B } \)=3

  47. Using the l’Hôpital Rule prove that, \(\underset{x\rightarrow 0^{+}}{lim}(1+x)^{\frac{1}{x}}=e\)

  48. Prove that f(x, y) = x3 - 2x2y + 3xy2 + y3 is taomogeneous; what is the degree? Verify fuler's Theorem for f.

  49. Find the area of the region common to the circle x2+y2=16 and the parabola y2=6x.

  50. The equation of electromotive force for an electric circuit containing resistance and self inductance is E=Ri L\(\frac{di}{dt},\) Where E is the electromotive force is given to the circuit, R the resistance and L, the coefficient of induction. Find the current i at time t when E = 0.

  51. Sovle \(\left( x+2 \right) \cfrac { dy }{ dx } =x2+4x-9\) .Also find the domain of the function.

    1. If X is the random variable with distribution function F(x) given by,
      \(F(x)=\begin{cases} \begin{matrix} 0 & -\infty
      then find (i) the probability density function f{x) 
      (ii) P(0.3 ≤ X ≤ 0.6)

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

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

    2. 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

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

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

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