If \( 2P(A) = P(B) = \frac{5}{13} \) and \( P(A | B) = \frac{3}{10} \), find \( P(A \cup B) \).
Find matrix \( AB \) if
\[ A = \begin{bmatrix} -1 & 2 & 3 \\ 4 & -2 & 5 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 1 \\ 6 & 7 \\ 5 & 3 \end{bmatrix} \]
Is the function \( f(x) \) defined by
\[ f(x) = \begin{cases} x + 5, & \text{if } x \leq 1 \\ x - 5, & \text{if } x > 1 \end{cases} \]
continuous at \( x = 1 \)?
Find the least value of ‘a’ for which the function \( f(x) = x^2 + ax + 1 \) is increasing on the interval \( [1, 2] \).
If \( x, y, z \) are all different and
\[ \Delta = \begin{vmatrix} x^2 & x^3 + 1 \\ y^2 & y^3 + 1 \\ z^2 & z^3 + 1 \end{vmatrix} = 0, \text{ show that } xyz = -1. \]
If \( \cos y = x \cos (a + y) \) and \( \cos a \neq \pm 1 \), prove that:
\[ \frac{dy}{dx} = \frac{\cos^2 (a + y)}{\sin a} \]
Solve by matrix method the system of equations:
\[ \begin{aligned} x - y + z &= 4 \\ 2x + y - 3z &= 0 \\ x + y + z &= 2 \end{aligned} \]
If
\[ A = \begin{bmatrix} 1 & 3 & 3 \\ 1 & 4 & 3 \\ 1 & 3 & 4 \end{bmatrix} \]
then verify that \( A(\text{adj} A) = |A| I \) and find \( A^{-1} \).
A die is thrown once. If E represents the event ‘the number obtained on the die is a multiple of 3’ and F represents the event ‘the number obtained on the die is even’, then tell whether the events E and F are independent.