Let us solve the questions one by one:

(i) For $x \in \mathbb{R}, \cosh(-x) =$

Recall the property of hyperbolic cosine:
$\cosh(-x) = \cosh(x).$
Thus, the correct answer is:
(a) $\cosh(x)$.

(ii) If $y = (ax + b)^m$, then $\frac{d^n y}{dx^n} =$

Using the general formula for $n$-th derivatives of a function of the form $(ax + b)^m$:
$\frac{d^n y}{dx^n} = \frac{m!}{(m-n)!} a^n (ax + b)^{m-n},$
where $m!$ is the factorial of $m$.

Thus, the correct answer is:
(a) $\frac{m!}{(m-n)!} a^n (ax + b)^{m-n}$.

(iii) If $f$ is continuous on $[a, b]$ and differentiable in $(a, b)$, then there exists at least one point $c \in (a, b)$ such that

$f'(c) = \frac{f(b) - f(a)}{b - a}.$

This result is the Mean Value Theorem (MVT). The given expression matches the formula:
$\frac{f(b) - f(a)}{b - a}.$

Thus, the correct answer is:
(c) $\frac{f(b) - f(a)}{b - a}$.

(iv) Evaluate $\int_{0}^{\pi/2} \sin^5 x \, dx$:

Using the reduction formula for $\int \sin^n x \, dx$:
[ \int \sin^n x , dx = \frac{n-1}{n} \int \sin