I see you've uploaded an assignment with a set of short and long-answer mathematical questions. Let me help you with any specific problem you'd like to solve or explain from the list.

Could you please specify which question you'd like to start with?

Here are some of the questions from the image for reference:

Short Answer Questions:

1. $n^{th}$ differentiation of $a^x \cos{x}$
2. Find $y_n$, if $y = \tan^{-1} \left( \frac{x}{c} \right)$
3. Find $y_n$, if $y = x^2 \sin x$ at $x = 0$
4. Find $y_n$, if $e^{ax} \sin^2 x \sin 2x$
5. Examine if Rolle's theorem is applicable for $f(x) = \sec x$ in $[0, 2\pi]$
6. Expand $\ln x$ in powers of $(x-1)$ up to third degree

Long Answer Questions:

1. Expand $4x^2 + 7x + 5$ in powers of $(x-3)$
2. Prove $(1 + x^2) y_{n+2} + (2n+1)x y_{n+1} + n^2 y_n = 0$, if $y = \ln \left( x + \sqrt{1 + x^2} \right)^2$
3. Verify Lagrange’s mean value theorem for:
   • $\sqrt{x^2 - 4}$ in $[2, 4]$
   • $\ln x$ in $[1, e]$

Let me know which one you'd like help with!