Let us go through the problems and solve them one by one.

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

The hyperbolic cosine function has the property:
$\cosh(-x) = \cosh(x)$ This is because $\cosh(x) = \frac{e^x + e^{-x}}{2}$, and replacing $x$ with $-x$ doesn't change the expression.

Correct Answer: (a) $\cosh(x)$

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

To differentiate repeatedly, use the generalized Leibniz rule for differentiation:
$\frac{d^n}{dx^n} \big( (ax+b)^m \big) = \frac{m!}{(m-n)!} a^n (ax+b)^{m-n}$ This formula assumes $m > n$.

Correct Answer: (c) $\frac{m!}{(m-n)!} a^n (ax+b)^{m-n}$

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

This is a direct application of the Mean Value Theorem (MVT), which states:
$f'(c) = \frac{f(b) - f(a)}{b - a}$ where $c \in (a, b)$.

Correct Answer: (c) $\frac{f(b) - f(a)}{b - a}$

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

Use the reduction formula for powers of sine:
$\int \sin^n(x) \, dx = \frac{n-1}{n} \int \sin^{n-2}(x) \, dx$ For $n = 5$, iteratively applying this yields:
$\int_{0}^{\pi/2} \sin^5(x) \, dx = \frac{4}{5} \cdot \frac{2}{3} \cdot \frac{\pi}{2} = \frac{8}{15}$

Correct Answer: (d) $\frac{8}{15}$

v) The process of determining the length of the arc of plane curves is known as:

The length of a curve $y = f(x)$ is calculated using the arc-length formula:
$L = \int \sqrt{1 + \left( \frac{dy}{dx} \right)^2} dx$ This process is called rectification.

Correct Answer: (a) Rectification

Would you like a detailed breakdown of any of these solutions? Let me know!

Related Questions:

1. What are the properties of hyperbolic functions similar to $\cosh(-x) = \cosh(x)$?
2. Can you explain the general process of using the Leibniz rule for higher-order derivatives?
3. What are the assumptions for the Mean Value Theorem to hold?
4. How do we simplify powers of $\sin(x)$ and $\cos(x)$ during integration?
5. What other applications use rectification in geometry?

Tip: For definite integrals of $\sin^n(x)$ or $\cos^n(x)$, familiarize yourself with reduction formulas—they save a lot of time!