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A

Section A: Long Answer Questions

Attempt any TWO questions.

3 questions·10 marks each
1long10 marks

State and prove that a differentiable function is continuous. Examine the continuity and differentiability of f(x) = |x| at x = 0.

continuity
2long10 marks

Evaluate the integral (\int \frac{dx}{a^2 + x^2}) and (\int \frac{dx}{\sqrt{a^2 - x^2}}). Hence find (\int \frac{dx}{x^2 + 4x + 8}).

integration
3long10 marks

Find the area bounded by the parabola (y^2 = 4ax) and its latus rectum.

applications-of-integration
B

Section B: Short Answer Questions

Attempt any EIGHT questions.

9 questions·5 marks each
4short5 marks

Evaluate (\lim_{x \to 0} \frac{\sin x}{x}) using the definition of limit.

limits
5short5 marks

If (y = e^{ax}\sin bx), find (\frac{dy}{dx}).

differentiation
6short5 marks

State and verify Rolle's theorem for (f(x) = x^2 - 4x + 3) on [1,3].

mean-value-theorem
7short5 marks

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

curve-tracing
8short5 marks

Evaluate (\int_0^{\pi/2} \sin^4 x , dx) using the reduction formula.

integration
9short5 marks

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

differential-equations
10short5 marks

Find the angle between the vectors (\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}) and (\vec{b} = 3\hat{i} - 2\hat{j} + \hat{k}).

vectors
11short5 marks

If (u = x^2 + y^2), find (\frac{\partial u}{\partial x}) and (\frac{\partial u}{\partial y}).

partial-derivatives
12short5 marks

Evaluate the double integral (\int_0^1 \int_0^2 (x + y), dy, dx).

multiple-integrals