I can see the image you uploaded with a set of mathematical questions from an assignment. Let’s go through some of these questions. If you’d like me to solve a specific one, please let me know.

Here are the questions listed from the image:

Short Answer Questions:

1. Find the $n^{th}$ 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 the third degree term.

Long Answer Questions:

1. Expand $4x^2 + 7x + 5$ in powers of $(x - 3)$.
2. If $y = \ln \left( x + \sqrt{1 + x^2} \right)^2$, prove that $(1 + x^2)y_{n+2} + (2n+1)xy_{n+1} + n^2 y_n = 0$.
3. Verify Lagrange’s mean value theorem for:
   • $\sqrt{x^2 - 4}$ in $[2, 4]$
   • $\ln x$ in $[1, e]$
4. If $y = x^{n-1} \ln x$, show that $y_n = \frac{(n-1)!}{x}$.
5. If $\cos^{-1} \frac{y}{b} = \log \left( \frac{x}{n} \right)^n$, show that $x^2 y_{n+2} + (2n+1) xy_{n+1} + 2n^2 y_n = 0$.
6. If $y = a \cos \ln x + b \sin \ln x$, show that $x^2 y_{n+2} + (2n+1) xy_{n+1} + (n^2+1) y_n = 0$.
7. If $y = \sin (m \sin^{-1} x)$, show that $(1 - x^2) y_{n+2} - (2n+1) xy_{n+1} + (m^2 - n^2) y_n = 0$.

Please let me know which problem you'd like to explore, or if you'd like help with multiple questions!