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Application of Matrices and Determinants Model Question Paper 1

12th Standard EM

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Maths

Time : 01:00:00 Hrs
Total Marks : 50
    5 x 1 = 5
  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\(\left[ \begin{matrix} 1 & -2 \\ 1 & 4 \end{matrix} \right] =\left[ \begin{matrix} 6 & 0 \\ 0 & 6 \end{matrix} \right] \), 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)

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

  3. 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

  4. 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

  5. If A is a square matrix that IAI = 2, than for any positive integer n, |An| =

    (a)

    0

    (b)

    2n

    (c)

    2n

    (d)

    n2

  6. 5 x 2 = 10
  7. If A is symmetric, prove that then adj Ais also symmetric.

  8. Prove that \(\left[ \begin{matrix} \cos { \theta } & -\sin { \theta } \\ \sin { \theta } & \cos { \theta } \end{matrix} \right] \) is orthogonal

  9. Show that the equations 3x + y + 9z = 0, 3x + 2y + 12z = 0 and 2x + y + 7z = 0 have nontrivial solutions also.

  10. Solve : 2x - y = 3, 5x + y = 4 using matrices.

  11. Solve 6x - 7y = 16, 9x - 5y = 35 using (Cramer's rule).

  12. 5 x 3 = 15
  13. Verify (AB)-1 = B-1A-1 with A = \(\left[ \begin{matrix} 0 & -3 \\ 1 & 4 \end{matrix} \right] \), B = \(\left[ \begin{matrix} -2 & -3 \\ 0 & -1 \end{matrix} \right] \).

  14. 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.

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

  16. Solve 2x - 3y = 7, 4x - 6y = 14 by Gaussian Jordan method.

  17. Solve: x + y + 3z = 4, 2x + 2y + 6z = 7, 2x + y +  z = 10.

  18. 4 x 5 = 20
  19. 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(-\(\alpha\)).

  20. If A = \(\left[ \begin{matrix} 5 & 3 \\ -1 & -2 \end{matrix} \right] \), show that A2 - 3A - 7I2 = O2. Hence find A−1.

  21. Show that the equations -2x + y + z = a, x - 2y + z = b, x + y -2z = c are consistent only if a + b + c =0.

  22. Using Gaussian Jordan method, find the values of λ and μ so that the system of equations 2x - 3y + 5z = 12, 3x + y + λz =μ, x - 7y + 8z = 17 has (i) unique solution (ii) infinite solutions and (iii) no solution.

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