Browse papers
A

Section A: Long Answer Questions

Attempt any TWO questions.

3 questions·10 marks each
1long10 marks

Define continuity of a function at a point. Discuss the continuity of (f(x) = \frac{x^2 - 1}{x - 1}) at x = 1 and redefine it to make it continuous.

continuity
2long10 marks

State and prove the fundamental theorem of integral calculus. Hence evaluate (\int_1^2 (3x^2 + 2x),dx).

integration
3long10 marks

Solve the second-order differential equation (\frac{d^2y}{dx^2} - 5\frac{dy}{dx} + 6y = e^{2x}).

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{\tan x - x}{x^3}).

limits
5short5 marks

If (y = \log(\sin x)), find (\frac{dy}{dx}).

differentiation
6short5 marks

State Taylor's theorem and write the Maclaurin series expansion of (e^x).

series-expansion
7short5 marks

Find the radius of curvature of the curve (y = x^2) at the point (1,1).

applications-of-derivatives
8short5 marks

Evaluate (\int e^x \sin x , dx).

integration
9short5 marks

Solve ((x^2 + y^2)dx - 2xy,dy = 0).

differential-equations
10short5 marks

Find the scalar triple product of (\hat{i}+\hat{j}, \hat{j}+\hat{k}, \hat{k}+\hat{i}).

vectors
11short5 marks

If (u = \log(x^2 + y^2)), show that (\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0).

partial-derivatives
12short5 marks

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

multiple-integrals