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12th Standard Maths English Medium Reduced Syllabus Model Question paper with answer key - 2021 Part - 2

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 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)

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

  2. Cramer's rule is applicable only when ______

    (a)

    Δ ≠ 0

    (b)

    Δ = 0

    (c)

    Δ =0, Δx =0

    (d)

    Δx = Δy = Δz =0

  3. In a homogeneous system if \(\rho\) (A) =\(\rho\)([A|0]) < the number of unknouns then the system has ________

    (a)

    trivial solution

    (b)

    only non - trivial solution

    (c)

    no solution

    (d)

    trivial solution and infinitely many non - trivial solutions

  4. The value of \(\sum _{ i=1 }^{ 13 }{ \left( { i }^{ n }+i^{ n-1 } \right) } \) is

    (a)

    1+ i

    (b)

    i

    (c)

    1

    (d)

    0

  5. If \(\omega \neq 1\) is a cubic root of unity and \(\left( 1+\omega \right) ^{ 7 }=A+B\omega \) ,then (A,B) equals

    (a)

    (1,0)

    (b)

    (−1,1)

    (c)

    (0,1)

    (d)

    (1,1)

  6. If z = \(\frac { 1 }{ (2+3i)^{ 2 } } \) then |z| =

    (a)

    \(\frac { 1 }{ 13 } \)

    (b)

    \(\frac { 1 }{ 5} \)

    (c)

    \(\frac { 1 }{ 12 } \)

    (d)

    none of these

  7. If z=1-cosθ + i sinθ, then |z| =

    (a)

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

    (b)

    2 cos\(\frac { \theta }{ 2 } \)

    (c)

    2|sin\(\frac { \theta }{ 2 } \)|

    (d)

    2|cos\(\frac { \theta }{ 2 } \)|

  8. If z = a + ib lies in quadrant then \(\frac { \bar { z } }{ z } \) also lies in the III quadrant if

    (a)

    a > b > 0

    (b)

    a < b < 0

    (c)

    b < a < 0

    (d)

    b > a > 0

  9. \(\frac { 1+e^{ -i\theta } }{ 1+{ e }^{ i\theta } } \) =

    (a)

    cosθ + i sinθ

    (b)

    cosθ - i sinθ

    (c)

    sinθ - i cosθ

    (d)

    sinθ + icosθ

  10. If a =cosα + i sinα, b= -cosβ + i sinβ then \(\left( ab-\frac { 1 }{ ab } \right) \) is _________

    (a)

    -2i sin(α - β)

    (b)

    2i sin(α - β)

    (c)

    2 cos(α - β)

    (d)

    -2 cos(α - β)

  11. The conjugate of \(\frac { 1+2i }{ 1-(1-i)^{ 2 } } \) is _______

    (a)

    \(\frac { 1+2i }{ 1-(1-i)^{ 2 } } \)

    (b)

    \(\frac { 5 }{ 1-(1-i)^{ 2 } } \)

    (c)

    \(\frac { 1-2i }{ 1+(1+i)^{ 2 } } \)

    (d)

    \(\frac { 1+2i }{ 1+(1-i)^{ 2 } } \)

  12. sin−1(cos x)\(=\frac{\pi}{2}-x \) is valid for

    (a)

    \(-\pi \le x\le 0\)

    (b)

    \(0\pi \le x\le 0\)

    (c)

    \(-\frac { \pi }{ 2 } \le x\le \frac { \pi }{ 2 } \)

    (d)

    \(-\frac { \pi }{ 4 } \le x\le \frac { 3\pi }{ 4 } \)

  13. ·If \(\alpha ={ tan }^{ -1 }\left( \cfrac { \sqrt { 3 } }{ 2y-x } \right) ,\beta ={ tan }^{ -1 }\left( \cfrac { 2x-y }{ \sqrt { 3y } } \right) \) then \(\alpha -\beta \)

    (a)

    \(\cfrac { \pi }{ 6 } \)

    (b)

    \(\cfrac { \pi }{ 3 } \)

    (c)

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

    (d)

    \(\cfrac { -\pi }{ 3 } \)

  14. 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

  15. y2 - 2x - 2y + 5 = 0 is a

    (a)

    circle

    (b)

    parabola

    (c)

    ellipse

    (d)

    hyperbola

  16. 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

  17. The line y = mx +1 is a tangent to the parabola y2 = 4x if m = ______________

    (a)

    1

    (b)

    2

    (c)

    3

    (d)

    4

  18. 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)

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

  19. A pair of dice numbered 1, 2, 3, 4, 5, 6 of a six-sided die and 1, 2, 3, 4 of a four-sided die is rolled and the sum is determined. Let the random variable X denote this sum. Then the number of elements in the inverse image of 7 is

    (a)

    1

    (b)

    2

    (c)

    3

    (d)

    4

  20. Which one of the following is a binary operation on N?

    (a)

    Subtraction

    (b)

    Multiplication

    (c)

    Division

    (d)

    All the above

  21. If the function f(x)sin-1(x2-3), then x belongs to

    (a)

    [-1,1]

    (b)

    [\(\sqrt2\),2]

    (c)

    \(\\ \\ \\ \left[ -2,-\sqrt { 2 } \right] \cup \left[ \sqrt { 2 } ,2 \right] \)

    (d)

    \(\left[ -2,-\sqrt { 2 } \right] \cap \left[ \sqrt { 2 } ,2 \right] \)

  22. If sin-1 x+sin-1 y=\(\frac{2\pi}{3};\)then cos-1x+cos-1 y is equal to

    (a)

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

    (b)

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

    (c)

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

    (d)

    \(\pi\)

  23. In a \(\Delta ABC\)  if C is a right angle, then  \({ tan }^{ -1 }\left( \cfrac { a }{ b+c } \right) +{ tan }^{ -1 }\left( \cfrac { b }{ c+a } \right) =\) 

    (a)

    \(\cfrac { \pi }{ 3 } \)

    (b)

    \(\cfrac { \pi }{ 4 } \)

    (c)

    \(\cfrac { 5\pi }{ 2 } \)

    (d)

    \(\cfrac { \pi }{ 6 } \)

  24. If \(\theta ={ sin }^{ -1 }\left( sin(-{ 60 }^{ 0 }) \right) \) then one of the possible values of \(\theta\) is _________

    (a)

    \(\cfrac { \pi }{ 3 } \)

    (b)

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

    (c)

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

    (d)

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

  25. \({ tan }^{ -1 }\left( tan\cfrac { 9\pi }{ 8 } \right) \)

    (a)

    \(\cfrac { 9\pi }{ 8 } \)

    (b)

    \(\cfrac { 9\pi }{ 8 } \)

    (c)

    \(\cfrac { \pi }{ 8 } \)

    (d)

    \(\cfrac { -\pi }{ 8 } \)

    1. Part II

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


    7 x 2 = 14
  26. For any 2 x 2 matrix, if A (adj A) =\(\left[ \begin{matrix} 10 & 0 \\ 0 & 10 \end{matrix} \right] \) then find |A|.

  27. The time T, taken for a complete oscillation of a single pendulum with length l, is given by the equation T = 2ㅠ\(\sqrt { \frac { 1 }{ g } } \), where g is a constant. Find the approximate percentage error in the calculated value of T corresponding to an error of 2 percent in the value of 1

  28. A differential equation, determine its order, degree (if exists)
    \(x={ e }^{ xy\left( \frac { dy }{ dx } \right) }\)

  29. Show that y = e−x + mx + n is a solution of the differential equation ex \(\left( \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } \right) \) -1 = 0

  30. Suppose X is the number of tails occurred when three fair coins are tossed once simultaneously. Find the values ofthe random variable X and number of points in its inverse images.

  31. Let p: Jupiter is a planet and q: India is an island be any two simple statements. Give
    verbal sentence describing each of the following statements.
    (i) ¬p
    (ii) p ∧ ¬q
    (iii) ¬p ∨ q
    (iv) p➝ ¬q
    (v) p↔q

  32. Fill in the following table so that the binary operation ∗ on A = {a,b,c} is commutative.

    * a b c
    a b    
    b c b a
    c a   c
  33. Find the square root of 6−8i .

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

  35. Simplify the following
    \(\sum _{ n=1 }^{ 12 }{ { i }^{ n } } \)

    1. Part III

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


    7 x 3 = 21
  36. 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

  37. Evaluate the following limits, if necessary use L’Hopitals rule
    (i) \(\underset { x\rightarrow { 0 }^{ + } }{ lim } { x }^{ sinx }\)
    (ii) \(\underset { x\rightarrow 0 }{ lim } \cfrac { cotx }{ cot2x } \) 
    (iii) \(\underset { x\rightarrow \frac { { \pi }^{ - } }{ 2 } }{ lim } \left( tanx \right) ^{ cosx }\)

  38. If w=xy+z and x=cot, y=sint, z=t then find \(\cfrac { dw }{ dt } \)

  39. Find the area bounded by x=0,x=6+5y-y2

  40. In a game, a man wins Rs.5 for getting a number greater than 4 and loses Re.1 otherwise when a fair dice is thrown. The man decided to throw a die thrice but to quit as and when he gets a number greater than 4. Find the expected value of the amount he wins/loses

  41. A random variable X has the following probability distribution.

    xi -2 -1 0 1 2 3
    Pi 0.1 k 0.2 2k 0.3 k

    i) find k
    ii) find the mean of the distribution

  42. Let G = {1, i,-1, -i} under the binary operation multiplication. Find the inverse of all the elements.

  43. A search light has a parabolic reflector (has a cross-section that forms a ‘bowl’). The parabolic bowl is 40cm wide from rim to rim and 30cm deep. The bulb is located at the focus.
    (1) What is the equation of the parabola used for reflector?
    (2) How far from the vertex is the bulb to be placed so that the maximum distance covered?

  44. An equation of the elliptical part of an optical lens system is \(\frac { { x }^{ 2 } }{ 16 } +\frac { { y }^{ 2 } }{ 9 } \) =1. The parabolic part of the system has a focus in common with the right focus of the ellipse .The vertex of the parabola is at the origin and the parabola opens to the right. Determine the equation of the parabola.

  45. Identify the type of conic and find centre, foci, vertices, and directrices of each of the following:
    \(\frac { { x }^{ 2 } }{ 25 } +\frac { { y }^{ 2 } }{ 9 } =1\)

    1. Part IV

      Answer all the questions.


    7 x 5 = 35
    1. Sovle \(\left( x+2 \right) \cfrac { dy }{ dx } =x2+4x-9\) .Also find the domain of the function.

    2. Solve : \(\cfrac { dy }{ dx } =\left( { sin }^{ 2 }x{ cos }^{ 2 }x+{ xe }^{ x } \right) dx\)

    1. Solve : \(\cfrac { dy }{ dx } =-\cfrac { x+ycos }{ 1+sinx } \) .Also find the domain of the function.

    2. It is given that the rate at which some bacteria multiply is proportional to the instantaneous number present. If the original number of bacteria doubles in two hours, in how many hours will it be five times.

    1. Ten coins are tossed simultaneously. What is the probability of getting (a) exactly 6 heads (b) at least 6 heads (c) at most 6 heads?

    2. Show that (Z7-[0],X7) satisfies closure, identity, inverse and commutative properties.

  46. Simplify: (1+i)18

  47. Show that \(\left( \cfrac { \sqrt { 3 } }{ 2 } +\cfrac { i }{ 2 } \right) ^{ 5 }+\left( \cfrac { \sqrt { 3 } }{ 2 } -\cfrac { i }{ 2 } \right) ^{ 5 }=-\sqrt { 3 } \)

  48. 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)
    \({ x }^{ m }{ y }^{ n }+\cfrac { 1 }{ { x }^{ m }{ y }^{ n } } =2cos(m\alpha +n\beta )\)

  49. If 1, ω, ω2 are the cube roots of unity then show that (1+5ω24) (1+5ω+ω2) (5+ω+ω5) =64

  50. Find all the roots \((2-2i)^{ \frac { 1 }{ 3 } }\) and also find the product of its roots.

  51. Find the radius and centre of the circle \(z\bar { z } \)-(2+3i)z-(2-3i)\(\bar { z } \)+9 =0 where z is a complex number.

    1. Prove that (2019)10+(2020)10≡1025(mod 2018)

    2. Let z1,z2, and z3 be complex numbers such that \(\left| { z }_{ 1 } \right\| =\left| { z }_{ 2 } \right| =\left| { z }_{ 3 } \right| =r>0\) and z1+z2+z3 \(\neq \) 0 prove that \(\left| \cfrac { { z }_{ 1 }{ z }_{ 2 }+{ z }_{ 2 }{ z }_{ 3 }+{ z }_{ 3 }{ z }_{ 1 } }{ { z }_{ 1 }+{ z }_{ 2 }+{ z }_{ 3 } } \right| \) =r

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