The given events \( A \) and \( B \) are such that \( P(A) = \frac{1}{4} \), \( P(B) = \frac{1}{2} \), and \( P(A \cap B) = \frac{1}{8} \); then find \( P(A' \cap B') \).
Show that the relation:
on the set \( \mathbb{Z} \) of integers is an equivalence relation.
Given:
for \( -1 < x < 1 \), prove that:
The differential coefficient of the \( \sin(x^2 + 5) \) with respect to \( x \) will be:
If
Then find \( AB \) and \( BA \).
Solve:
\[ A = \begin{bmatrix} 1 & -1 & 0 \\ 2 & 3 & -2 \\ -2 & 0 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 3 & -15 & 5 \\ -1 & 6 & -2 \\ 1 & -5 & 2 \end{bmatrix}, \]
then find the value of \( (AB)^{-1} \).
(d) If \( P(A) = \frac{3}{13} \), \( P(B) = \frac{5}{13} \), and \( P(A \cap B) = \frac{2}{13} \), find the value of \( P(B | A) \):
A die is thrown two times. It is found that the sum of the appeared numbers is 6. Find the conditional that the number 4 appeared at least one time.
There are two children in a family. If it is known that at least one child is a boy, find the that both children are boys.
Prove that the \( f(x) = x^2 \) is continuous at \( x \neq 0 \).
Differentiate the \( \sin mx \) with respect to \( x \).
The principal value of the \( \cot^{-1}\left(-\frac{1}{\sqrt{3}}\right) \) will be:
Minimize Z = 5x + 3y \text{ subject to the constraints} \[ 4x + y \geq 80, \quad x + 5y \geq 115, \quad 3x + 2y \leq 150, \quad x \geq 0, \quad y \geq 0. \]