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A

Section A: Long Answer Questions

Attempt any TWO questions.

3 questions·10 marks each
1long10 marks

State and prove the mean value theorem (Lagrange's). Verify it for (f(x) = \sqrt{x}) on [1,4].

mean-value-theorem
2long10 marks

Evaluate (\int \frac{dx}{(x^2 + a^2)^2}) using a suitable substitution.

integration
3long10 marks

Solve the differential equation (\frac{d^2y}{dx^2} + 4y = \cos 2x).

differential-equations
B

Section B: Short Answer Questions

Attempt any EIGHT questions.

9 questions·5 marks each
4short5 marks

Evaluate (\lim_{x \to 1}\frac{x^n - 1}{x - 1}).

limits
5short5 marks

If (y = e^{m\sin^{-1} x}), find (\frac{dy}{dx}).

differentiation
6short5 marks

Expand (\log(1 + x)) in a Maclaurin series.

series-expansion
7short5 marks

Find the asymptotes of (y = \frac{2x^2 - 1}{x^2 - 4}).

curve-tracing
8short5 marks

Evaluate (\int \frac{1}{1 + \cos x},dx).

integration
9short5 marks

Solve ((1 + x^2)\frac{dy}{dx} + 2xy = 4x^2).

differential-equations
10short5 marks

If (\vec{a} \times \vec{b} = \vec{0}) and (\vec{a}\cdot\vec{b} = 0), what can you say about (\vec{a}) and (\vec{b})?

vectors
11short5 marks

Find (\frac{\partial u}{\partial x}) if (u = x^y).

partial-derivatives
12short5 marks

Change the order of integration in (\int_0^1 \int_x^1 f(x,y),dy,dx).

multiple-integrals