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The uploaded document contains a math assignment from IILM University with a set of short and long-answer questions. Here's a breakdown of the problems:

### Short Answer Questions (15 marks):
1. Find the \(n\)th derivative of \( a \cos x \).
2. Find \( y_n \) if \( y = \tan^{-1}(x/c) \).
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. Check if Rolle’s theorem is applicable for \( f(x) = \sec x \) on the interval \( [0, 2\pi] \).
6. Expand \( \ln x \) in powers of \( (x - 1) \) up to the third-degree term.

### Long Answer Questions (35 marks):
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 \( (1 + x^2) y_{n+2} + (2n + 1) x y_{n+1} + n^2 y_n = 0 \).
3. Verify Lagrange's Mean Value Theorem for:
   - \( \sqrt{x^2 - 4} \) on \( [2, 4] \),
   - \( \ln x \) on \( [1, e] \).
4. If \( y = x^{n-1} \log x \), show that \( y_n = \frac{(n - 1)!}{x} \).
5. If \( \cos^{-1} y = \log \left( \frac{x^n}{n} \right) \), show that \( x^2 y_{n+2} + (2n + 1) x y_{n+1} + 2n^2 y_n = 0 \).
6. If \( y = a \cos(\ln x) + b \sin(\ln x) \), show that \( x^2 y_{n+2} + (2n + 1) x y_{n+1} + (n^2 + 1) y_n = 0 \).
7. If \( y = \sin(m \sin^{-1} x) \), show that \( (1 - x^2) y_{n+2} - (2n + 1) x y_{n+1} + (m^2 - n^2) y_n = 0 \).

Let me know if you want a solution or explanation for any of these problems.

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Here are 5 relative questions to think about:
1. How do you apply the formula for higher-order derivatives in trigonometric functions?
2. What conditions must be satisfied for Rolle’s theorem to hold?
3. How do you derive Maclaurin or Taylor series for a logarithmic function?
4. How do you verify the Lagrange Mean Value Theorem for specific functions on an interval?
5. What techniques are used to handle trigonometric inverses and logarithmic identities in advanced differentiation?

**Tip**: Always check the continuity and differentiability of functions when dealing with theorems like Rolle's or Lagrange's.

The uploaded document contains a math assignment from IILM University, featuring both short and long-answer questions covering topics like derivatives, Rolle's theorem, Taylor series expansion, and Lagrange's Mean Value Theorem.

Explore a detailed math assignment from IILM University that includes problems on derivatives, Taylor series, Rolle's theorem, and Lagrange's Mean Value Theorem. This assignment provides short and long-answer questions suitable for undergraduate students.

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