Important one mark questions

12th Standard EM

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Maths

Use blue pen only
Time : 00:20:00 Hrs
Total Marks : 25

    Part - A

    Answer all the questions

    25 x 1 = 25
  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. If A is a 3 × 3 non-singular matrix such that AAT = ATA and B = A-1AT, then BBT = 

    (a)

    A

    (b)

    B

    (c)

    I

    (d)

    BT

  3. If A = \(\left[ \begin{matrix} 3 & 5 \\ 1 & 2 \end{matrix} \right] \), B = adj A and C = 3A, then \(\frac { \left| adjB \right| }{ \left| C \right| } \)

    (a)

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

    (b)

    \(\frac { 1 }{ 9 } \)

    (c)

    \(\frac { 1 }{ 4 } \)

    (d)

    1

  4. 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)

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

  5. If A = \(\left[ \begin{matrix} 7 & 3 \\ 4 & 2 \end{matrix} \right] \), then 9I - A = 

    (a)

    A-1

    (b)

    \(\frac { { A }^{ -1 } }{ 2 } \)

    (c)

    3A-1

    (d)

    2A-1

  6. 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

  7. 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

  8. If A = \(\left[ \begin{matrix} 3 & 1 & -1 \\ 2 & -2 & 0 \\ 1 & 2 & -1 \end{matrix} \right] \) and A-1 = \(\left[ \begin{matrix} { a }_{ 11 } & { a }_{ 12 } & { a }_{ 13 } \\ { a }_{ 21 } & { a }_{ 22 } & { a }_{ 23 } \\ { a }_{ 31 } & { a }_{ 32 } & { a }_{ 33 } \end{matrix} \right] \) then the value of a23 is

    (a)

    0

    (b)

    -2

    (c)

    -3

    (d)

    -1

  9. 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

  10. If (AB)-1 = \(\left[ \begin{matrix} 12 & -17 \\ -19 & 27 \end{matrix} \right] \) and A-1 = \(\left[ \begin{matrix} 1 & -1 \\ -2 & 3 \end{matrix} \right] \), then B-1 = 

    (a)

    \(\left[ \begin{matrix} 2 & -5 \\ -3 & 8 \end{matrix} \right] \)

    (b)

    \(\left[ \begin{matrix} 8 & 5 \\ 3 & 2 \end{matrix} \right] \)

    (c)

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

    (d)

    \(\left[ \begin{matrix} 8 & -5 \\ -3 & 2 \end{matrix} \right] \)

  11. If ATA−1 is symmetric, then A2 =

    (a)

    A-1

    (b)

    (AT)2

    (c)

    AT

    (d)

    (A-1)2

  12. 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)

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

  13. If A = \(\left[ \begin{matrix} \frac { 3 }{ 5 } & \frac { 4 }{ 5 } \\ x & \frac { 3 }{ 5 } \end{matrix} \right] \) and AT = A−1 , then the value of x is

    (a)

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

    (b)

    \(\frac { -3 }{ 5 } \)

    (c)

    \(\frac { 3 }{ 5 } \)

    (d)

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

  14. If A = \(\left[ \begin{matrix} 1 & \tan { \frac { \theta }{ 2 } } \\ -\tan { \frac { \theta }{ 2 } } & 1 \end{matrix} \right] \) and AB = I , then B = 

    (a)

    \(\left( \cos ^{ 2 }{ \frac { \theta }{ 2 } } \right) A\)

    (b)

    \(\left( \cos ^{ 2 }{ \frac { \theta }{ 2 } } \right) { A }^{ T }\)

    (c)

    \(\left( \cos ^{ 2 }{ \theta } \right) I\)

    (d)

    (Sin2\(\frac { \theta }{ 2 } \))A

  15. If A = \(\left[ \begin{matrix} \cos { \theta } & \sin { \theta } \\ -\sin { \theta } & \cos { \theta } \end{matrix} \right] \) and A(adj A) =  \(\left[ \begin{matrix} k & 0 \\ 0 & k \end{matrix} \right] \) then adj (AB) is

    (a)

    0

    (b)

    sin θ

    (c)

    cos θ

    (d)

    1

  16. If A = \(\left[ \begin{matrix} 2 & 3 \\ 5 & -2 \end{matrix} \right] \) be such that λA−1 =A, then λ is

    (a)

    17

    (b)

    14

    (c)

    19

    (d)

    21

  17. If adj A = \(\left[ \begin{matrix} 2 & 3 \\ 4 & -1 \end{matrix} \right] \) and adj B = \(\left[ \begin{matrix} 1 & -2 \\ -3 & 1 \end{matrix} \right] \) then adj (AB) is

    (a)

    \(\left[ \begin{matrix} -7 & -1 \\ 7 & -9 \end{matrix} \right] \)

    (b)

    \(\left[ \begin{matrix} -6 & 5 \\ -2 & -10 \end{matrix} \right] \)

    (c)

    \(\left[ \begin{matrix} -7 & 7 \\ -1 & -9 \end{matrix} \right] \)

    (d)

    \(\left[ \begin{matrix} -6 & -2 \\ 5 & -10 \end{matrix} \right] \)

  18. 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

  19. If xayb = em, xcyd = en, Δ1 = \(\left| \begin{matrix} m & b \\ n & d \end{matrix} \right| \), Δ2 = \(\left| \begin{matrix} a & m \\ c & n \end{matrix} \right| \), Δ3 = \(\left| \begin{matrix} a & b \\ c & d \end{matrix} \right| \), then the values of x and y are respectively,

    (a)

    e21), e31)

    (b)

    log (Δ13), log (Δ23)

    (c)

    log (Δ21), log(Δ31)

    (d)

    e(Δ13),e(Δ23)

  20. Which of the following is/are correct?
    (i) Adjoint of a symmetric matrix is also a symmetric matrix.
    (ii) Adjoint of a diagonal matrix is also a diagonal matrix.
    (iii) If A is a square matrix of order n and λ is a scalar, then adj(λA) = λn adj(A).
    (iv) A(adjA) = (adjA)A = |A| I

    (a)

    Only (i)

    (b)

    (ii) and (iii)

    (c)

    (iii) and (iv)

    (d)

    (i), (ii) and (iv)

  21. If ρ(A) = ρ([A | B]), then the system AX = B of linear equations is

    (a)

    consistent and has a unique solution

    (b)

    consistent

    (c)

    consistent and has infinitely many solution

    (d)

    inconsistent

  22. If 0 ≤ θ  ≤ π and the system of equations x + (sinθ)y - (cosθ)z = 0, (cosθ)x - y +z = 0, (sinθ)x + y - z = 0 has a non-trivial solution then θ is

    (a)

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

    (b)

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

    (c)

    \(\frac { 5\pi }{ 6 } \)

    (d)

    \(\frac { \pi }{ 4 } \)

  23. The augmented matrix of a system of linear equations is \(\left[ \begin{matrix} 1 \\ \begin{matrix} 0 \\ 0 \end{matrix} \end{matrix}\begin{matrix} 2 \\ \begin{matrix} 1 \\ 0 \end{matrix} \end{matrix}\begin{matrix} 7 \\ \begin{matrix} 4 \\ \lambda -7 \end{matrix} \end{matrix}\begin{matrix} 3 \\ \begin{matrix} 6 \\ \mu +5 \end{matrix} \end{matrix} \right] \). The system has infinitely many solutions if

    (a)

    λ = 7, μ ≠ -5

    (b)

    λ = 7, μ = 5

    (c)

    λ ≠ 7, μ ≠ -5

    (d)

    λ = 7, μ = -5

  24. Let A = \(\left[ \begin{matrix} 2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2 \end{matrix} \right] \) and 4B = \(\left[ \begin{matrix} 3 & 1 & -1 \\ 1 & 3 & x \\ -1 & 1 & 3 \end{matrix} \right] \). If B is the inverse of A, then the value of x is

    (a)

    2

    (b)

    4

    (c)

    3

    (d)

    1

  25. If A = \(\left[ \begin{matrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{matrix} \right] \), then adj(adj A) is

    (a)

    \(\left[ \begin{matrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \end{matrix} \right] \)

    (b)

    \(\left[ \begin{matrix} 6 & -6 & 8 \\ 4 & -6 & 8 \\ 0 & -2 & 2 \end{matrix} \right] \)

    (c)

    \(\left[ \begin{matrix} -3 & 3 & -4 \\ -2 & 3 & -4 \\ 0 & 1 & -1 \end{matrix} \right] \)

    (d)

    \(\left[ \begin{matrix} 3 & -3 & 4 \\ 0 & -1 & 1 \\ 2 & -3 & 4 \end{matrix} \right] \)

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