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A

Section A: Long Answer Questions

Attempt any TWO questions.

3 questions·10 marks each
1long10 marks

Define the derivative of a function from first principles. Find the derivative of (\sin x) using the first-principle method.

differentiation
2long10 marks

Obtain the reduction formula for (\int \sin^n x , dx) and hence evaluate (\int_0^{\pi/2} \sin^5 x , dx).

integration
3long10 marks

Solve the differential equation (\frac{dy}{dx} = \frac{y}{x} + \tan\frac{y}{x}) (homogeneous).

differential-equations
B

Section B: Short Answer Questions

Attempt any EIGHT questions.

9 questions·5 marks each
4short5 marks

Evaluate (\lim_{x \to 0}\frac{\log(1 + x)}{x}).

limits
5short5 marks

If (y = x^n \log x), find (y_n).

successive-differentiation
6short5 marks

State and verify Cauchy's mean value theorem for (f(x) = x^2, g(x) = x) on [1,2].

mean-value-theorem
7short5 marks

Find the maximum and minimum values of (f(x) = \sin x + \cos x).

applications-of-derivatives
8short5 marks

Evaluate (\int \frac{x^2}{(x^2 + 1)(x^2 + 4)},dx).

integration
9short5 marks

Solve (\frac{dy}{dx} + y\cot x = \cos x).

differential-equations
10short5 marks

Find the work done by a force (\vec{F} = 2\hat{i} + 3\hat{j} + \hat{k}) in moving a particle along (\vec{d} = \hat{i} + \hat{j} + \hat{k}).

vectors
11short5 marks

If (u = \tan^{-1}(y/x)), find (\frac{\partial u}{\partial x}) and (\frac{\partial u}{\partial y}).

partial-derivatives
12short5 marks

Evaluate (\int_0^1 \int_0^{\sqrt{1 - x^2}} dy , dx).

multiple-integrals