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A

Section A: Long Answer Questions

Attempt any TWO questions.

3 questions·10 marks each
1long10 marks

Define limit and continuity of a function. Evaluate (\lim_{x \to 0}\frac{\sin x}{x}) and discuss the continuity of (f(x) = \frac{\sin x}{x}) at x = 0.

limits
2long10 marks

State Leibnitz's theorem for the nth derivative of a product. If (y = x^2 e^x), find (y_n).

successive-differentiation
3long10 marks

Find the area of the region bounded by the curve (y = x^2), the x-axis and the lines x = 1 and x = 3.

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{1 - \cos x}{x^2}).

limits
5short5 marks

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

differentiation
6short5 marks

State and verify Rolle's theorem for (f(x) = x^2 - 5x + 6) on [2,3].

mean-value-theorem
7short5 marks

Find the equation of the tangent to the curve (y = x^2 + 2x) at the point (1,3).

applications-of-derivatives
8short5 marks

Evaluate (\int \frac{dx}{x^2 + 6x + 13}).

integration
9short5 marks

Solve the differential equation (\frac{dy}{dx} = e^{x - y}).

differential-equations
10short5 marks

Find the cross product of (\vec{a} = \hat{i} + 2\hat{j} + \hat{k}) and (\vec{b} = 2\hat{i} + \hat{j} - \hat{k}).

vectors
11short5 marks

If (u = x^3 + y^3 + z^3 - 3xyz), find (\frac{\partial u}{\partial x}).

partial-derivatives
12short5 marks

Evaluate (\int_0^2 \int_0^3 (x + 2y),dy,dx).

multiple-integrals