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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.

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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 \( [1, e] \)

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Assignment 1 includes questions such as nth differentiation of ax cos x, Rolle's theorem for sec x, expansion of ln x, and Lagrange’s mean value theorem for various functions.

This assignment involves advanced math topics such as the nth differentiation of functions, examining Rolle's theorem, expanding ln x in power series, and verifying Lagrange's mean value theorem. It is suitable for university-level students.

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