Term 1 Model Question Paper

12th Standard EM

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Maths

Time : 02:00:00 Hrs
Total Marks : 60

    Part- A

    10 x 1 = 10
  1. If |adj(adj A)| = |A|9, then the order of the square matrix A is

    (a)

    3

    (b)

    4

    (c)

    2

    (d)

    5

  2. The number of solutions of the system of equations 2x+y = 4, x - 2y = 2, 3x + 5y = 6 is

    (a)

    0

    (b)

    1

    (c)

    2

    (d)

    infinitely many

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

    (a)

    1+ i

    (b)

    i

    (c)

    1

    (d)

    0

  4. According to the rational root theorem, which number is not possible rational root of 4x7+2x4-10x3-5?

    (a)

    -1

    (b)

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

    (c)

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

    (d)

    5

  5. If sin-1 x+sin-1 y+sin-1 z=\(\frac{3\pi}{2}\), the value of x2017+y2018+z2019\(-\frac { 9 }{ { x }^{ 101 }+{ y }^{ 101 }+{ z }^{ 101 } } \)is

    (a)

    0

    (b)

    1

    (c)

    2

    (d)

    3

  6. \({ tan }^{ -1 }\left( \cfrac { 1 }{ 4 } \right) +{ tan }^{ -1 }\left( \cfrac { 2 }{ 11 } \right) \) =

    (a)

    0

    (b)

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

    (c)

    -1

    (d)

    none

  7. The radius of the circle3x2+by2+4bx−6by+b2 =0 is

    (a)

    1

    (b)

    3

    (c)

    \( \sqrt {10}\)

    (d)

    \( \sqrt {11}\)

  8. 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

  9. If \(\vec { a } \times (\vec { b } \times \vec { c } )=(\vec { a } \times \vec { b } )\times \vec { c } \) where \(\vec { a } ,\vec { b } ,\vec { c } \) are any three vectors such that \(\vec { a } ,\vec { b } \) \(\neq \) 0 and  \(\vec { a } .\vec { b } \) \(\neq \) 0 then \(\vec { a } \) and \(\vec { c } \) are

    (a)

    perpendicular

    (b)

    parallel

    (c)

    inclined at an angle \(\frac{\pi}{3}\)

    (d)

    inclined at an angle  \(\frac{\pi}{6}\)

  10. If the length of the perpendicular from the origin to the plane 2x + 3y + λz =1, λ > 0 is \(\frac{1}{5}\) then the value of λ is

    (a)

    \(2\sqrt { 3 } \)

    (b)

    \(3\sqrt { 2 } \)

    (c)

    0

    (d)

    1

  11. Part - B

    5 x 1 = 5
  12. (adj A)T

  13. (1)

    2

  14. Im(z)

  15. (2)

    \(\frac { z-\bar { z } }{ 2 } \)

  16. \(\left| -\sqrt { 3 } +i \right| \)

  17. (3)

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

  18. p(x)=xn.p\(\left( \frac { 1 }{ x } \right) \)

  19. (4)

    Reciprocal equation of type I

  20. \({ sin }^{ -1 }\left( sin\cfrac { 2\pi }{ 3 } \right) \)

  21. (5)

    adj (AT)

    Part - C

    6 x 2 = 12
  22. Prove that \(\left[ \begin{matrix} \cos { \theta } & -\sin { \theta } \\ \sin { \theta } & \cos { \theta } \end{matrix} \right] \) is orthogonal

  23. Find the following \(\left| \cfrac { 2+i }{ -1+2i } \right| \)
     

  24. Construct a cubic equation with roots 1,2, and 3

  25. Solve tan-1 \(\left( \frac { 1-x }{ 1+x } \right) =\frac { 1 }{ 2 } { tan }^{ -1 }\) x for x>0

  26. Find the general equation of the circle whose diameter is the line segment joining the points (−4,−2)and (1,1).

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

  28. Part - D

    6 x 3 = 18
  29. If A = \(\left[ \begin{matrix} 3 & 2 \\ 7 & 5 \end{matrix} \right] \) and B = \(\left[ \begin{matrix} -1 & -3 \\ 5 & 2 \end{matrix} \right] \), verify that (AB)-1 = B-1A-1

  30. If z1=3,z2=-7i, and z3=5+4i, show that z1(z2+z3)=z1z2+z1z3

  31. If z=2−2i, find the rotation of z by θ radians in the counter clockwise direction about the origin when \(\theta =\cfrac { \pi }{ 3 } \).

  32. Solve the equation x4-9x2+20=0.

  33. Solve sin-1 x> cos-1x

  34. Find the centre and radius of the circle3x2+(a+1)y2+6x−9y+a+4=0.

  35. Part- E

    3 x 5 = 15
  36. If A = \(\left[ \begin{matrix} 4 & 3 \\ 2 & 5 \end{matrix} \right] \), find x and y such that A2 + xA + yI2 = O2. Hence, find A-1.

  37. Find all cube roots of \(\sqrt { 3 } +i\)

  38. ABCD is a quadrilateral with \(\overset { \rightarrow }{ AB } =\overset { \rightarrow }{ \alpha } \) and \(\overset { \rightarrow }{ AD } =\overset { \rightarrow }{ \beta } \) and \(\overset { \rightarrow }{ AC } =2\overset { \rightarrow }{ \alpha } +3\overset { \rightarrow }{ \beta } \). If. the area of the quadrilateral is λ times the area of the parallelogram with \(\overset { \rightarrow }{ AB } \) and \(\overset { \rightarrow }{ AD } \) as adjacent sides, then prove that \(\lambda =\frac { 5 }{ 2 } \)

    ()

    plane

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