ICSE Class 10

General instructions: Answer all questions. Marks are shown in brackets [ ].

Objective

1. 1.
   Construct a $3 \times 2$ matrix, whose elements $a_{ij}$ are given by $a_{ij} = i \cdot j$. Which of the following represents the correct matrix?

   1. A. $\begin{bmatrix} 1 & 2 \\ 2 & 4 \\ 3 & 6 \end{bmatrix}$
   2. B. $\begin{bmatrix} 1 & 1 \\ 2 & 2 \\ 3 & 3 \end{bmatrix}$
   3. C. $\begin{bmatrix} 1 & 2 \\ 4 & 5 \\ 6 & 7 \end{bmatrix}$
   4. D. $\begin{bmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \end{bmatrix}$
   [1]
2. 2.
   Find the values of $x$, $y$, $u$, and $v$ that satisfy the matrix equation $\begin{bmatrix} x + 5 & 3 \\ y - 2 & 5 \end{bmatrix} = \begin{bmatrix} 8 & 2u - 1 \\ 7 & v - 2 \end{bmatrix}$.

   1. A. $x = 3, y = 5, u = 1, v = 3$
   2. B. $x = 3, y = 9, u = 2, v = 7$
   3. C. $x = 5, y = 7, u = 3, v = 5$
   4. D. $x = 2, y = 8, u = 1, v = 6$
   [1]
3. 3.
   If $\left[ \begin{array}{cc} 4 & -2 \\ 4 & 0 \end{array} \right] + 3A = \left[ \begin{array}{cc} -2 & -2 \\ 1 & -3 \end{array} \right]$, which of the following matrices represents $A$?

   1. A. $\left[ \begin{array}{cc} -2 & 0 \\ -1 & -1 \end{array} \right]$
   2. B. $\left[ \begin{array}{cc} 2 & 0 \\ 1 & 1 \end{array} \right]$
   3. C. $\left[ \begin{array}{cc} -6 & -4 \\ -3 & -3 \end{array} \right]$
   4. D. $\left[ \begin{array}{cc} 2 & -4 \\ 3 & -3 \end{array} \right]$
   [1]
4. 4.
   If $ A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} $, then the value of $ A^2 $ is:

   1. A. $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$
   2. B. $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$
   3. C. $\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}$
   4. D. $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$
   [1]
5. 5.
   Given the matrix equation: $\left[ \begin{array}{cc} x & y + 2 \\ 3 & z - 1 \end{array} \right] = \left[ \begin{array}{cc} 3 & 1 \\ 3 & 2 \end{array} \right]$, find the values of $x$, $y$, and $z$.

   1. A. $x = 3$, $y = -1$, $z = 3$
   2. B. $x = 3$, $y = 1$, $z = 2$
   3. C. $x = 0$, $y = -1$, $z = 3$
   4. D. $x = 3$, $y = -1$, $z = 0$
   [1]
6. 6.
   Write the additive inverses of the following matrices: $A = \left[ \begin{array}{cc} 6 & -5 \end{array} \right]$, $B = \left[ \begin{array}{cc} -2 & 0 \\ 4 & -1 \end{array} \right]$, and $C = \left[ \begin{array}{c} -7 \\ 4 \end{array} \right]$. Which of the following correctly represents the additive inverses of $A$, $B$, and $C$?

   1. A. $A = \left[ \begin{array}{cc} -6 & 5 \end{array} \right]$, $B = \left[ \begin{array}{cc} 2 & 0 \\ -4 & 1 \end{array} \right]$, $C = \left[ \begin{array}{c} 7 \\ -4 \end{array} \right]$
   2. B. $A = \left[ \begin{array}{cc} 6 & -5 \end{array} \right]$, $B = \left[ \begin{array}{cc} -2 & 0 \\ 4 & -1 \end{array} \right]$, $C = \left[ \begin{array}{c} -7 \\ 4 \end{array} \right]$
   3. C. $A = \left[ \begin{array}{cc} -6 & -5 \end{array} \right]$, $B = \left[ \begin{array}{cc} 2 & 0 \\ 4 & 1 \end{array} \right]$, $C = \left[ \begin{array}{c} 7 \\ 4 \end{array} \right]$
   4. D. $A = \left[ \begin{array}{cc} 6 & 5 \end{array} \right]$, $B = \left[ \begin{array}{cc} -2 & 0 \\ -4 & -1 \end{array} \right]$, $C = \left[ \begin{array}{c} -7 \\ -4 \end{array} \right]$
   [1]
7. 7.
   For the matrix $A = \begin{bmatrix} 3 & 1 \\ -1 & 2 \end{bmatrix}$, which of the following correctly verifies the equation $A^2 - 5A + 7I_2 = O$?

   1. A. $\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$
   2. B. $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$
   3. C. $\begin{bmatrix} 7 & 5 \\ -5 & 3 \end{bmatrix}$
   4. D. $\begin{bmatrix} 8 & 5 \\ -5 & 3 \end{bmatrix}$
   [1]
8. 8.
   If $A = \begin{bmatrix} 2 & -2 \\ 5 & -5 \end{bmatrix}$ and $B = \begin{bmatrix} 3 & 4 \\ 3 & 4 \end{bmatrix}$, compute $AB$ and $BA$. Which conclusion about matrix multiplication is **not** supported by these results?

   1. A. $AB = O$ even though $A \neq O$ and $B \neq O$
   2. B. $AB = BA$, so matrix multiplication is commutative
   [1]
9. 9.
   Let $A = \left[ \begin{array}{cc} 0 & 1 \\ 5 & -1 \end{array} \right]$, $B = \left[ \begin{array}{cc} 1 & -3 \\ 0 & -2 \end{array} \right]$, and $C = \left[ \begin{array}{cc} 2 & -5 \\ 4 & 0 \end{array} \right]$. Find the value of $(3A + 4B - 5C)$ and select the correct matrix from the options below.

   1. A. $\left[ \begin{array}{cc} -6 & 16 \\ -5 & -11 \end{array} \right]$
   2. B. $\left[ \begin{array}{cc} 6 & -16 \\ 5 & 11 \end{array} \right]$
   3. C. $\left[ \begin{array}{cc} -6 & -16 \\ -5 & -1 \end{array} \right]$
   4. D. $\left[ \begin{array}{cc} -4 & 10 \\ -3 & -7 \end{array} \right]$
   [1]
10. 10.
    Given $A = \begin{bmatrix} 1 & 0 \\ -1 & 7 \end{bmatrix}$ and $A^2 = 8A + kI$, the value of $k$ is:

    1. A. $k = -7$
    2. B. $k = 7$
    3. C. $k = -1$
    4. D. $k = 1$
    [1]
11. 11.
    If matrix $ X = \begin{bmatrix} -3 & 4 \\ 2 & -3 \end{bmatrix} \begin{bmatrix} 2 \\ -2 \end{bmatrix} $ and $ 2X - 3Y = \begin{bmatrix} 10 \\ -8 \end{bmatrix} $, what is the value of matrix $ Y $?

    1. A. $\begin{bmatrix} -16 \\ 12 \end{bmatrix}$
    2. B. $\begin{bmatrix} -14 \\ 10 \end{bmatrix}$
    3. C. $\begin{bmatrix} 16 \\ -12 \end{bmatrix}$
    4. D. $\begin{bmatrix} -6 \\ 4 \end{bmatrix}$
    [1]
12. 12.
    Find the values of $ x $ and $ y $, if $ \begin{bmatrix} 1 & 2 \\ 3 & 3 \end{bmatrix} \begin{bmatrix} x & 0 \\ 0 & y \end{bmatrix} = \begin{bmatrix} x & 0 \\ 9 & 0 \end{bmatrix} $.

    1. A. $ x = 3 $ and $ y = 0 $
    2. B. $ x = 1 $ and $ y = 2 $
    3. C. $ x = 0 $ and $ y = 3 $
    4. D. $ x = 9 $ and $ y = 1 $
    [1]
13. 13.
    Given matrices $A = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix}$, $B = \begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix}$, $C = \begin{bmatrix} 4 & 1 \\ 1 & 5 \end{bmatrix}$, and $I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, what is the result of $A(B + C) - 14I$?

    1. A. $$\begin{bmatrix} 0 & 30 \\ 22 & 28 \end{bmatrix}$$
    2. B. $$\begin{bmatrix} 14 & 30 \\ 22 & 42 \end{bmatrix}$$
    3. C. $$\begin{bmatrix} -14 & 30 \\ 22 & 14 \end{bmatrix}$$
    4. D. $$\begin{bmatrix} 0 & 20 \\ 12 & 28 \end{bmatrix}$$
    [1]
14. 14.
    If $A = \left[ \begin{array}{rr} 7 & 3 \\ 5 & 2 \end{array} \right]$ and $B = \left[ \begin{array}{rr} 2 & 5 \\ 4 & 5 \end{array} \right]$, then the matrix $C$ such that $2A + 3C = 8B$, is

    1. A. $\left[ \begin{array}{cc} \frac{7}{3} & \frac{21}{3} \\ \frac{4}{3} & \frac{21}{3} \end{array} \right]$
    2. B. $\left[ \begin{array}{cc} 2 & 12 \\ 22 & 24 \end{array} \right]$
    3. C. $\left[ \begin{array}{cc} 7 & 17 \\ 11 & 16 \end{array} \right]$
    4. D. $\left[ \begin{array}{cc} \frac{7}{3} & \frac{34}{3} \\ \frac{22}{3} & \frac{36}{3} \end{array} \right]$
    [1]
15. 15.
    If $ A = \begin{bmatrix} 2 & 1 \\ 1 & 3 \end{bmatrix} $ and $ B = \begin{bmatrix} 3 \\ -11 \end{bmatrix} $, find the matrix $X$ such that $AX = B$. Which of the following represents the matrix $X$?

    1. A. $\begin{bmatrix} 4 \\ -5 \end{bmatrix}$
    2. B. $\begin{bmatrix} 3 \\ -4 \end{bmatrix}$
    3. C. $\begin{bmatrix} -5 \\ 4 \end{bmatrix}$
    4. D. $\begin{bmatrix} 1 \\ -3 \end{bmatrix}$
    [1]
16. 16.
    If $2 \begin{bmatrix} 3 & x \\ 0 & 1 \end{bmatrix} + 3 \begin{bmatrix} 1 & 3 \\ y & 2 \end{bmatrix} = \begin{bmatrix} z & -7 \\ 15 & 8 \end{bmatrix}$, which of the following sets of values for $x$, $y$, and $z$ satisfies the equation?

    1. A. $x = -5$, $y = 5$, $z = 9$
    2. B. $x = 5$, $y = -5$, $z = 9$
    3. C. $x = -5$, $y = 5$, $z = -9$
    4. D. $x = 5$, $y = 5$, $z = 9$
    [1]
17. 17.
    If $ P = \begin{bmatrix} 2 & 6 \\ 3 & 9 \end{bmatrix} $ and $ Q = \begin{bmatrix} 3 & x \\ y & 2 \end{bmatrix} $, find the values of $ x $ and $ y $ such that $ PQ = \text{null matrix} $. Which of the following pairs is correct?

    1. A. $x = -2$, $y = -3$
    2. B. $x = 2$, $y = 3$
    3. C. $x = -6$, $y = -1$
    4. D. $x = 6$, $y = 1$
    [1]
18. 18.
    Determine which of the following confirms that $\mathbf{X} = \begin{bmatrix} 2  3 \\ 3  2 \end{bmatrix}$ is the solution of the matrix equation $\mathbf{X}^2 - 4\mathbf{X} - 5\mathbf{I} = 0$.

    1. A. $\mathbf{X}^2 - 4\mathbf{X} - 5\mathbf{I} = \begin{bmatrix} 0  0 \\ 0  0 \end{bmatrix}$
    2. B. $\mathbf{X}^2 - 4\mathbf{X} - 5\mathbf{I} = \begin{bmatrix} 5  0 \\ 0  5 \end{bmatrix}$
    3. C. $\mathbf{X}^2 - 4\mathbf{X} - 5\mathbf{I} = \begin{bmatrix} 13  12 \\ 12  13 \end{bmatrix}$
    4. D. $\mathbf{X}^2 - 4\mathbf{X} - 5\mathbf{I} = \begin{bmatrix} 0  12 \\ 12  0 \end{bmatrix}$
    [1]
19. 19.
    Given $A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$ and $I$ as the unit matrix of order 2, which of the following correctly represents $(aI + bA)^3$?

    1. A. $a^3I + 3a^2bA$
    2. B. $aI + 3ab^2A$
    3. C. $a^3I + b^3A$
    4. D. $3aI + 3bA$
    [1]
20. 20.
    If $ A = \begin{bmatrix} a & 0 \\ 0 & 2 \end{bmatrix} $, $ B = \begin{bmatrix} 0 & -b \\ 1 & 0 \end{bmatrix} $, $ M = \begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix} $ and $ BA = M^2 $, find the values of $ a $ and $ b $. What are the correct values?

    1. A. $a = 2$, $b = 2$
    2. B. $a = 1$, $b = 1$
    3. C. $a = 2$, $b = -2$
    4. D. $a = -2$, $b = 2$
    [1]