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12th Standard Maths English Medium Reduced Syllabus Important Questions - 2021 Part - 2

12th Standard

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Maths

Time : 02:45:00 Hrs
Total Marks : 165

    Multiple Choice Questions

    15 x 1 = 15
  1. If A = \(\left[ \begin{matrix} 2 & 0 \\ 1 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} 1 & 4 \\ 2 & 0 \end{matrix} \right] \) then |adj (AB)| = 

    (a)

    -40

    (b)

    -80

    (c)

    -60

    (d)

    -20

  2. 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

  3. The rank of the matrix \(\left[ \begin{matrix} 1 \\ \begin{matrix} 2 \\ -1 \end{matrix} \end{matrix}\begin{matrix} 2 \\ \begin{matrix} 4 \\ -2 \end{matrix} \end{matrix}\begin{matrix} 3 \\ \begin{matrix} 6 \\ -3 \end{matrix} \end{matrix}\begin{matrix} 4 \\ \begin{matrix} 8 \\ -4 \end{matrix} \end{matrix} \right] \) is

    (a)

    1

    (b)

    2

    (c)

    4

    (d)

    3

  4. z1, z2 and z3 are complex number such that z1+z2+z3=0 and |z1|=|z2|=|z3|=1 then z12+z22+z33 is

    (a)

    3

    (b)

    2

    (c)

    1

    (d)

    0

  5. If \(\cfrac { z-1 }{ z+1 } \) is purely imaginary, then |z| is

    (a)

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

    (b)

    1

    (c)

    2

    (d)

    3

  6. The polynomial x3-kx2+9x has three real zeros if and only if, k satisfies

    (a)

    |k|≤6

    (b)

    k=0

    (c)

    |k|>6

    (d)

    |k|≥6

  7. If x3+12x2+10ax+1999 definitely has a positive zero, if and only if

    (a)

    a≥0

    (b)

    a>0

    (c)

    a<0

    (d)

    a≤0

  8. The polynomial x3+2x+3 has

    (a)

    one negative and two real roots

    (b)

    one positive and two imaginary roots

    (c)

    three real roots

    (d)

    no solution

  9. If x=\(\frac{1}{5}\), the valur of cos (cos-1x+2sin-1x) is

    (a)

    \(-\sqrt { \frac { 24 }{ 25 } } \)

    (b)

    \(\sqrt { \frac { 24 }{ 25 } } \)

    (c)

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

    (d)

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

  10. \({ tan }^{ -1 }\left( \frac { 1 }{ 4 } \right) +{ tan }^{ -1 }\left( \frac { 2 }{ 3 } \right) \)is equal to

    (a)

    \(\frac { 1 }{ 2 } { cos }^{ -1 }\left( \frac { 3 }{ 5 } \right) \)

    (b)

    \(\frac { 1 }{ 2 } { sin }^{ -1 }\left( \frac { 3 }{ 5 } \right) \)

    (c)

    \(\frac { 1 }{ 2 } {tan }^{ -1 }\left( \frac { 3 }{ 5 } \right) \)

    (d)

    \({ tan}^{ -1 }\left( \frac { 1}{ 2 } \right) \)

  11. 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)

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

  12. The eccentricity of the hyperbola whose latus rectum is 8 and conjugate axis is equal to half the distance between the foci is

    (a)

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

    (b)

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

    (c)

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

    (d)

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

  13. If the normals of the parabola y2 = 4x drawn at the end points of its latus rectum are tangents to the circle (x−3)2+(y+2)2=r2 , then the value of r2 is

    (a)

    2

    (b)

    3

    (c)

    1

    (d)

    4

  14. If x+y=k is a normal to the parabola y2 =12x, then the value of k is

    (a)

    3

    (b)

    -1

    (c)

    1

    (d)

    9

  15. The locus of a point whose distance from (-2,0) is \(\frac { 2 }{ 3 } \) times its distance from the line x = \(\frac { -9 }{ 2 } \) is

    (a)

    a parabola

    (b)

    a hyperbola

    (c)

    an ellipse

    (d)

    a circle

  16. 2 Marks

    15 x 2 = 30
  17. Flnd the equation of the plane containing the line of intersection of the planes x + y + Z - 6 = 0 and
    2x + 3y + 4z + 5 = 0 and passing through the point (1, 1, 1)

    ()

    x = -1 is one root

  18. Find the intervals of increasing and decreasing function for f(x) =x3 + 2x2 - 1.

  19. Find the point on the parabola y2=18x at which the ordinate increases at twice the rate of the abscissa.

  20. Evaluate the following limits, if necessary using L’Hopitalrule
    (i) \(\underset { x\rightarrow 2 }{ lim } \cfrac { sin\pi x }{ 2-x } \) 
    (ii) \(\cfrac { lim }{ x\rightarrow 2 } \cfrac { { x }^{ n }-{ a }^{ n } }{ x-2 } \) 
    (iii) \(\underset { x\rightarrow \infty }{ lim } \cfrac { sin\frac { 2 }{ x } }{ \frac { 1 }{ x } } \)
    (iv) \(\underset { x\rightarrow \infty }{ lim } \cfrac { { x }^{ 2 } }{ { e }^{ x } } \)

  21. Prove that the function f(x)=2x2+3x is strictly increasing on \(\left[ -\cfrac { 1 }{ 2 } ,\cfrac { 1 }{ 2 } \right] \)

  22. Prove that the function f(x)=e-x is strictky increasing on [0,1]

  23. IF u(x, y) = x2 + 3xy + y2, x, y, ∈ R, find tha linear appraoximation for u at (2, 1) 

  24. If \(u=log\sqrt { { x }^{ 2 }+{ y }^{ 2 } } \) then prove that \(\left( \cfrac { \vartheta u }{ \vartheta x } \right) +\left( \cfrac { \vartheta u }{ \vartheta y } \right) =\cfrac { 1 }{ { x }^{ 2 }+{ y }^{ 2 } } \)

  25. If \(u={ e }^{ \frac { x }{ y } }sin\left( \cfrac { x }{ y } \right) +{ e }^{ \frac { y }{ x } }cos\left( \cfrac { y }{ x } \right) \) ,then prove that \(x\cfrac { \vartheta u }{ \vartheta x } +y\cfrac { \vartheta u }{ \vartheta y } =0\)

  26. If y = sin x and x changes from \(\cfrac { \pi }{ 2 } to\cfrac { 22 }{ 14 } \) what is the approximate change in y.

  27. Calculate df for \(f=\sqrt { 2x+5 } \) when x = 22 and dx = 3.

  28. Evaluate \(\int _{ 1 }^{ 2 }{ \frac { 3x }{ { 9x }^{ 2 }-1 } dx } \)

  29. Evaluate \(\int _{ 0 }^{ \frac { \pi }{ 2 } }{ { e }^{ 2x }cosxdx } \)

  30. Find the area of the region bounded by the curve \(\sqrt { x } +\sqrt { y } =\sqrt { a } \) (x,y>0) and the co-ordinate axes.

  31. Determine the order and degree of \(\cfrac { \left[ 1+\left( \frac { dy }{ dx } \right) ^{ 2 } \right] ^{ \frac { 3 }{ 2 } } }{ \frac { { d }^{ 2 }y }{ { dx }^{ 2 } } } =k\)

  32. 3 Marks

    15 x 3 = 45
  33. Given A = \(\left[ \begin{matrix} 1 & -1 \\ 2 & 0 \end{matrix} \right] \), B = \(\left[ \begin{matrix} 3 & -2 \\ 1 & 1 \end{matrix} \right] \) and C = \(\left[ \begin{matrix} 1 & 1 \\ 2 & 2 \end{matrix} \right] \), find a matrix X such that AXB = C.

  34. Find the inverse of each of the following by Gauss-Jordan method:
    \(\left[ \begin{matrix} 2 & -1 \\ 5 & -2 \end{matrix} \right] \)
     

  35. Solve the following system of linear equations by matrix inversion method:
    2x − y = 8, 3x + 2y = −2

  36. If \(cos\alpha +cos\beta +cos\gamma =sin\alpha +sin\beta +sin\gamma =0\) then show that 
    (i) \(cos3\alpha +cos3\beta +cos3\gamma =3cos(\alpha +\beta +\gamma )\)
    (ii) \(sin3\alpha +sin3\beta +sin3\gamma +sin3\gamma =3sin\left( \alpha +\beta +\gamma \right) \)

  37. Obtain the Cartesian equation for the locus of z=x+iy in
    |z-4|2-|z-1|2=16

  38. Find solution, if any, of the equation 2cos2x-9cosx+4=0

  39. Find the domain of sin−1(2−3x2)

  40. Find the value of
    \(sin\left( { tan }^{ -1 }\left( \frac { 1 }{ 2 } \right) -{ cos }^{ -1 }\left( \frac { 4 }{ 5 } \right) \right) \)

  41. Find the value of 
     \(tan\left[ \frac { 1 }{ 2 } { sin }^{ -1 }\left( \frac { 2a }{ 1+{ a }^{ 2 } } \right) +\frac { 1 }{ 2 } { cos }^{ -1 }\left( \frac { 1-{ a }^{ 2 } }{ 1+{ a }^{ 2 } } \right) \right] \)

  42. Find the equation of the hyperbola in each of the cases given below:
    (i) foci(±2,0), eccentricity =\(\frac { 3 }{ 2 } \)
    (ii) Centre (2,1) , one of the foci (8,1) and corresponding directrix x = 4.
    (iii) passing through(5,−2)and length of the transverse axis along x axis and of length 8 units.

  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. If the straight line joining the points (2, 1, 4) and (a−1, 4, −1) is parallel to the line joining the points (0, 2, b −1) and (5, 3,  −2) , find the values of a and b.

  45. Find the coordinates of the foot of the perpendicular drawn from the point (-1, 2, 3) to the straight line \(\vec { r } =(\hat { i } -4\hat { j } +3\hat { k } )+t(2\hat { i } +3\hat { j } +\hat { k } )\). Also, find the shortest distance from the point to the straight line.

  46. Find the parametric form of vector equation and Cartesian equations of a straight line passing through (5, 2,8) and is perpendicular to the straight lines 
    \(\vec { r } =(\hat { i } +\hat { j } -\hat { k } )+s(2\hat { i } -2\hat { j } +\hat { k } )\)
    \(\vec { r } =(\hat { 2i } -\hat { j } -3\hat { k } )+t(\hat { i } +2\hat { j } +2\hat { k } )\)
    .

  47. Find the values in the interval \((\frac{1}{2},2)\) satisfied by the Rolle's theorem for the function \(f(x)=x+\frac{1}{x}, x\in[\frac{1}{2},2]\)

  48. 5 Marks

    15 x 5 = 75
  49. Prove that the semi-vertical angle of a cone of maximum volume and of given slant height is tan-1(\(\sqrt { 2 } \)).

  50. If w=u2ev where \( u=\cfrac { x }{ y } \) and v=logx. Find \(\cfrac { \partial w }{ \partial x } \) and \(\cfrac { \partial w }{ \partial y } \) 

  51. Find the area bounded by the curves y=|x|-1 and y=-|x|+1

  52. Find the area of the region bounded by a2y2=a2(a2-x2)

  53. AOB is the positive quadrant of the ellipse \(\cfrac { { x }^{ 2 } }{ { a }^{ 2 } } +\cfrac { { y }^{ 2 } }{ { b }^{ 2 } } =1\) where OA=a and OB=b.Find the area between the arc AB and chord AB of the elipse.

  54. A population grows at the rate of 2% per year. How long does it take for the population to double?

  55. Solve : \({ e }^{ \frac { dy }{ dx } }=x+1,y(0)=5\)

  56. Solve :(x2+xy)dy=(x2+y2)dx

  57. Solve :\(x\cfrac { dy }{ dx } sin\left( \cfrac { y }{ x } \right) +x-ysin\left( y\cfrac { y }{ x } \right) =,y(1)=\cfrac { \pi }{ 2 } \)

  58. Two cards are drawn simultaneously from a well shuffled pack of 52 cards. Find the mean and variance of the number of red cards.

  59. Suppose that a pair of fair dice are tossed and let random variable X denote the sum of outcomes. Find the mean and variance of the probability distribution of X.

  60. The p.d.f of a continuous random variable X is \(f(x)=\begin{cases} a+b{ x }^{ 2 },0\le x\le 1 \\ 0,otherwise \end{cases}\) where a and b are some constants. Find
    (a) a and b if \(E(x)=\cfrac { 3 }{ 5 } \) 
    (b) Var (X)

  61. Verify (p ∧ -p) ∧ (~q ∧ p) is a tautlogy, contradiction or contingency.

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

  63. Prove by using truth table \(\sim (pV(qVr)\equiv \left( \sim p \right) \wedge \left( \sim q\wedge \sim r \right) \) 

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