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12th Standard Maths English Medium Reduced Syllabus Model Question paper - 2021 Part - 1

12th Standard

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Maths

Time : 01:00:00 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 P = \(\left[ \begin{matrix} 1 & x & 0 \\ 1 & 3 & 0 \\ 2 & 4 & -2 \end{matrix} \right] \) is the adjoint of 3 × 3 matrix A and |A| = 4, then x is

    (a)

    15

    (b)

    12

    (c)

    14

    (d)

    11

  2. If ATA−1 is symmetric, then A2 =

    (a)

    A-1

    (b)

    (AT)2

    (c)

    AT

    (d)

    (A-1)2

  3. If \(4{ cos }^{ -1 }x+{ sin }^{ -1 }x=\pi \) then x is

    (a)

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

    (b)

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

    (c)

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

    (d)

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

  4. The value of tan \(\left( { cos }^{ -1 }\cfrac { 3 }{ 5 } +{ tan }^{ -1 }\cfrac { 1 }{ 4 } \right) \) is ______

    (a)

    \(\cfrac { 19 }{ 8 } \)

    (b)

    \(\cfrac { 8 }{ 19 } \)

    (c)

    \(\cfrac { 19 }{ 12 } \)

    (d)

    \(\cfrac { 3 }{ 4 } \)

  5. Equation of tangent at (-4, -4) on x2 = -4y is

    (a)

    2x - y + 4 = 0

    (b)

    2x + y - 4 = 0

    (c)

    2x - y - 12 = 0

    (d)

    2x + y + 4 = 0

  6. The distance between the foci of a hyperbola is 16 and e = \(\sqrt { 2 } \) Its equation is

    (a)

    x2 - y2 = 32

    (b)

    y2 - x2 = 32

    (c)

    x2 - y2 = 16

    (d)

    y2 - x2 = 16

  7. If the foci of the ellipse \(\frac { { x }^{ 2 } }{ 16 } +\frac { { y }^{ 2 } }{ { b }^{ 2 } } \) = 1 and the hyperbola \(\frac { { x }^{ 2 } }{ 144 } -\frac { { y }^{ 2 } }{ 81 } =\frac { 1 }{ 25 } \) coincide then b2 is

    (a)

    1

    (b)

    5

    (c)

    7

    (d)

    9

  8. The length of major and minor axes of 4x2 + 3y2 = 12 are ____________

    (a)

    4, 2\(\sqrt3\)

    (b)

    2, \(\sqrt3\)

    (c)

    2\(\sqrt3\), 4

    (d)

    \(\sqrt3\), 2

  9. The tangent at any point P on the ellipse \(\frac { { x }^{ 2 } }{ 6 } +\frac { { y }^{ 2 } }{ 3 } \) = 1 whose centre C meets the major axis at T and PN is the perpendicular to the major axis; The CN CT = ______________

    (a)

    \(\sqrt6\)

    (b)

    3

    (c)

    \(\sqrt3\)

    (d)

    6

  10. If \(\lambda \overset { \wedge }{ i } +2\lambda \overset { \wedge }{ j } +2\lambda \overset { \wedge }{ k } \) is a unit vector, then the value of λ is 

    (a)

    土 \(\frac { 1 }{ 3 } \)

    (b)

    土 \(\frac { 1 }{ 4 } \)

    (c)

    土 \(\frac { 1 }{ 9 } \)

    (d)

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

  11. If the vectors \(a\overset { \wedge }{ i } +\overset { \wedge }{ j } +\overset { \wedge }{ k } \)\(\overset { \wedge }{ i } +b\overset { \wedge }{ j } +\overset { \wedge }{ k } \) and \(\overset { \wedge }{ i } +\overset { \wedge }{ j } +c\overset { \wedge }{ k } \) (a ≠ b ≠ c ≠ 1) are coplaner, then \(\frac { 1 }{ 1-a } +\frac { 1 }{ 1-b } +\frac { 1 }{ 1-c } =\)

    (a)

    0

    (b)

    1

    (c)

    2

    (d)

    \(\frac { abc }{ (1-a)(1-b)(1-c) } \)

  12. The equation of the tangent to the curve y=x2-4x+2 at (4,2) is

    (a)

    x + 4y + 12 = 0

    (b)

    4x + y + 12 = 0

    (c)

    4x - y - 14 = 0

    (d)

    x + 4y - 12 = 0

  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 \(\int _{ -\pi }^{ \pi }{ { sin }^{ 3 }x \ { cos }^{ 3 }x \ } dx\) is

    (a)

    0

    (b)

    \(\pi \)

    (c)

    2\(\pi \)

    (d)

    4\(\pi \)

  15. The I.F of y log y\(\frac{dx}{dy}+x-log\ y=0\) is

    (a)

    log(log y)

    (b)

    log y

    (c)

    \(\frac{1}{log\ y}\)

    (d)

    \(\frac{1}{log(log\ y)}\)

  16. If a random variable X has the p.d.f.\(f(x)=\cfrac { k }{ { x }^{ 2 }+1 } ,0<x<\infty \) then k is

    (a)

    \(\pi \)

    (b)

    \(\cfrac { 1 }{ \pi } \)

    (c)

    1

    (d)

    \(\cfrac { 2 }{ \pi } \)

  17. The sum of the mean and variance of a binomial distribution for 6 total is 2.16. Then the probability of success p=__________

    (a)

    0.4

    (b)

    0.6

    (c)

    0.8

    (d)

    0.2

  18. The Identity element of \(\left\{ \left( \begin{matrix} x & x \\ x & x \end{matrix} \right) \right\} \) |x\(\in \)R, x≠0} under matrix multiplication is

    (a)

    \(\left( \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right) \)

    (b)

    \(\left( \begin{matrix} \frac { 1 }{ 4x } & \frac { 1 }{ 4x } \\ \frac { 1 }{ 4x } & \frac { 1 }{ 4x } \end{matrix} \right) \)

    (c)

    \(\left( \begin{matrix} \frac { 1 }{ 2 } & \frac { 1 }{ 2 } \\ \frac { 1 }{ 2 } & \frac { 1 }{ 2 } \end{matrix} \right) \)

    (d)

    \(\left( \begin{matrix} \frac { 1 }{ 2x } & \frac { 1 }{ 2x } \\ \frac { 1 }{ 2x } & \frac { 1 }{ 2x } \end{matrix} \right) \)

  19. 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

  20. Which of the following is a statement?

    (a)

    7+2<10

    (b)

    Wish you all success

    (c)

    All the best

    (d)

    How old are you?

  21. Part II

    Answer any 7 questions. Question no. 27 is compulsory.

    7 x 2 = 14
  22. If \(\overset { \rightarrow }{ a } =\overset { \wedge }{ i } +2\overset { \wedge }{ j } +3\overset { \wedge }{ k } \)\(\overset { \rightarrow }{ b } =-\overset { \wedge }{ i } +2\overset { \wedge }{ j } +\overset { \wedge }{ k } \) and \(\overset { \rightarrow }{ c } =3\overset { \wedge }{ i } +\overset { \wedge }{ j } \) find \(\frac { \lambda }{ c } \) such that \(\overset { \rightarrow }{ a } +\lambda \overset { \rightarrow }{ b } \) is perpendicular to \(\overset { \rightarrow }{ c } \)

    ()

    -a

  23.  

    Compute the limit  

     \(\underset{x\rightarrow 1}{lim}(\frac{x^{2}-3x+2}{x^{2}-4x+3})\).

  24. Find df for f(x) = x2 + x 3 and evaluate it for
    x = 3 and dx = 0.02

  25. If w=log(x2+y2),x=cosθ,y=sinθ, find \(\cfrac { dw }{ d\theta } \)

  26. Evaluate: \(\int _{ 0 }^{ 3 }{ (3{ x }^{ 2 }-4x+5) } dx\)

  27. A differential equation, determine its order, degree (if exists)
    \({ \left( \frac { dy }{ dx } \right) }^{ 3 }=\sqrt { 1+\left( \frac { dy }{ dx } \right) } \)

  28. 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. Part III

      Answer any 7 questions. Question no. 34 is compulsory.

    7 x 3 = 21
  29. Find,the rank of the matrix math \(\left[ \begin{matrix} 4 \\ -2 \\ 1 \end{matrix}\begin{matrix} 4 \\ 3 \\ 4 \end{matrix}\begin{matrix} 0 \\ -1 \\ 8 \end{matrix}\begin{matrix} 3 \\ 5 \\ 7 \end{matrix} \right] \).

  30. Prove that \({ tan }^{ -1 }\left( \cfrac { m }{ n } \right) -{ tan }^{ -1 }\left( \cfrac { m-n }{ m+n } \right) =\cfrac { \pi }{ 4 } \)
     

  31. Show that the lines \(\frac { x-1 }{ 3 } =\frac { y+1 }{ 2 } =\frac { z-1 }{ 5 } \) and \(\frac { x+2 }{ 4 } =\frac { y-1 }{ 3 } =\frac { z+1 }{ -2 } \) do not intersect

    ()

    a, b, c

  32. A ball is thrown vertically upwards, moves according to the law s = 13.8 t - 4.9 t2 where s
    is in metres and t is in seconds.
    (i) Find the acceleration at t = 1
    (ii) Find velocity at t = 1
    (iii) Find the maximum height reached by the ball?

  33. If v=log(tanx+tany+tanz), then prove that \(\Sigma sin2x\cfrac { \partial v }{ \partial x } =2\)

  34. Solve : ydx+(x-y2)dy=0

  35. A random variable X has the following probability distribution

    x 0 1 2 3 4 5 6 7
    P(X) 0 k 2k 2k 3k k2 2k2 7k2+k


    Evaluate (i) k
    (ii) \(\\ \\ P(X\ge 6)\)
    (iii) P(0<X<3)

    1. Part IV

      Answer all the questions.

    7 x 5 = 35
  36. Solve: (2x2 - 3x + 1) (2x2 + 5x + 1) = 9x2.

  37. The guides of a railway bridge is a parabola with its vertex at the highest point 15 m above the ends. If the span is 120 m, find the height of the bridge at 24 m from the middle point.

  38. Gas is escaping from a spherical balloon at the rate of 900 cm3/sec. How fast is the surface area and radius of the balloon shrinking when the radius of the balloon is 30 cm?

  39. Find the intervals for which the function f(x)=2x2-9x2-12x+1 is increasing or decfreasing and find the local extermems.

  40. Find the local maximum and local minimum values for f(x)=12x2-2x2-x4.

  41. Let X be a random variable denoting the life time of an electrical equipment having probability density function
    \(f(x)=\begin{cases} \begin{matrix} { ke }^{ -2x } & forx>0 \end{matrix} \\ \begin{matrix} 0 & forx\le 0 \end{matrix} \end{cases}\) 
    Find
    (i) the value of k
    (ii) Distribution function 
    (iii) P(X < 2)
    (iv) calculate the probability that X is at least for four unit of time 
    (v) P(X = 3)

  42. Prove that p➝(¬q V r) ≡ ¬pV(¬qVr) using truth table.

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