First Mid Term Model Questions

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 50
    10 x 1 = 10
  1. If A\(\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right] =\left[ \begin{matrix} 6 & 0 \\ 0 & 6 \end{matrix} \right] \), then A = 

    (a)

    \(\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right] \)

    (b)

    \(\left[ \begin{matrix} 1 & 2 \\ -1 & 4 \end{matrix} \right] \)

    (c)

    \(\left[ \begin{matrix} 4 & 2 \\ -1 & 1 \end{matrix} \right] \)

    (d)

    \(\left[ \begin{matrix} 4 & -1 \\ 2 & 1 \end{matrix} \right] \)

  2. If A, B and C are invertible matrices of some order, then which one of the following is not true?

    (a)

    adj A = |A|A-1

    (b)

    adj(AB) = (adj A)(adj B)

    (c)

    det A-1 = (det A)-1

    (d)

    (ABC)-1 = C-1B-1A-1

  3. The system of linear equations x + y + z  = 6, x + 2y + 3z =14 and 2x + 5y + λz =μ (λ, μ \(\in \) R) is consistent with unique solution if

    (a)

    λ = 8

    (b)

    λ = 8, μ ≠ 36

    (c)

    λ ≠ 8

    (d)

    none

  4. The value of (1+i)4 + (1-i)4 is

    (a)

    8

    (b)

    4

    (c)

    -8

    (d)

    -4

  5. The value of \(\frac { (cos{ 45 }^{ 0 }+isin{ 45 }^{ 0 })^{ 2 }(cos{ 30 }^{ 0 }-isin{ 30 }^{ 0 }) }{ cos{ 30 }^{ 0 }+isin{ 30 }^{ 0 } } \) is

    (a)

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

    (b)

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

    (c)

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

    (d)

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

  6. 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

  7. The quadratic equation whose roots are ∝ and β is

    (a)

    (x - ∝)(x -β) =0

    (b)

    (x - ∝)(x + β) =0

    (c)

    ∝+β=\(\frac{b}{a}\)

    (d)

    ∝.β=\(\frac{-c}{a}\)

  8. If p(x) = ax2 + bx + c and Q(x) = -ax2 + dx + c where ac ≠ 0 then p(x). Q(x) = 0 has at least _______ real roots.

    (a)

    no

    (b)

    1

    (c)

    2

    (d)

    infinite

  9. \({ tan }^{ -1 }\left( \frac { 1 }{ 4 } \right) +{ tan }^{ -1 }\left( \frac { 2 }{ 3 } \right) \)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)

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

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

    (a)

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

    (b)

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

    (c)

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

    (d)

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

  11. 5 x 2 = 10
  12. Find the adjoint of the following:
    \(\left[ \begin{matrix} -3 & 4 \\ 6 & 2 \end{matrix} \right] \)

  13. If A is a square matrix such that A3 = I, then prove that A is non-singular.

  14. Find z−1, if z=(2+3i)(1− i).

  15. Solve the equations:
    6x4-35x3+62x2-35x+6=0

  16. Evaluate \(sin\left( { cos }^{ -1 }\left( \cfrac { 1 }{ 2 } \right) \right) \)
     

  17. 5 x 3 = 15
  18. If A = \(\frac { 1 }{ 9 } \left[ \begin{matrix} -8 & 1 & 4 \\ 4 & 4 & 7 \\ 1 & -8 & 4 \end{matrix} \right] \), prove that A−1 = AT.

  19. Find the inverse of the non-singular matrix A =  \(\left[ \begin{matrix} 0 & 5 \\ -1 & 6 \end{matrix} \right] \), by Gauss-Jordan method.

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

  21. Show that the complex numbers 3 + 2i, 5i, -3 + 2i and -i form a square.

  22. Find all real numbers satisfying 4x-3(2x+2)+25=0

  23. 3 x 5 = 15
  24. If F(\(\alpha\)) = \(\left[ \begin{matrix} \cos { \alpha } & 0 & \sin { \alpha } \\ 0 & 1 & 0 \\ -\sin { \alpha } & 0 & \cos { \alpha } \end{matrix} \right] \), show that [F(\(\alpha\))]-1 = F(-\(\alpha\)).

  25. Simplify: (1+i)18

  26. Provethat \({ tan }^{ -1 }\left( \cfrac { 1-x }{ 1+x } \right) -{ tan }^{ -1 }\left( \cfrac { 1-y }{ 1+y } \right) ={ sin }^{ -1 }\left( \cfrac { y-x }{ \sqrt { 1+{ x }^{ 2 } } .\sqrt { 1+{ y }^{ 2 } } } \right) \\ \)

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