Section A
Long Answer Questions. Attempt any TWO questions.
(a) Define gradient of vector function of \(f(x, y, z)\) and find the derivative of \(f(x, y, z) = x^3 - xy^2 - z^2\) at \(p_0(1, 1, 0)\) in the direction of \(\vec{V} = 2\vec{i} - \vec{j} + 6\vec{k}\).
(b) Define Volume of the solid and find the volume of the solid generated by revolving the region bounded by \(y = \sqrt{x}\) and the line \(y = 1\), \(x = 4\) about the line \(y = 1\).
Evaluate
(a) \(\displaystyle\int_{\pi/4}^{\pi/2} \dfrac{dx}{1 - \sin x}\)
(b) \(\displaystyle\int x^2 \sin x\, dx\)
(c) Solve the differential equation \(\dfrac{dy}{dx} - \dfrac{3}{x}y = x\), \(x > 0\).
(a) State Rolle's Theorem and show that \(x^3 + 3x + 1 = 0\) has exactly one real solution.
(b) Find the area of the region enclosed by the parabola \(y = 2 - x^2\) and the line \(y = -x\).
Attempt any Eight questions
[8×5=40]Section B
Short Answer Questions. Attempt any EIGHT questions.
Define absolute value function and sketch the graph of absolute value.
Find the limit of \(\displaystyle\lim_{h \to \infty} \dfrac{\sqrt{6h + 25} - 5}{h}\).
State integral test and apply it to test the convergence of the series \(\displaystyle\sum_{n=1}^{\infty} \dfrac{1}{n^2 + 1}\).
Find the Taylor's Series generated by \(f(x) = \dfrac{1}{x}\) at \(a = 2\). Where, if anywhere, does the series converge to \(\dfrac{1}{x}\)?
Define implicit differentiation and find the slope of the circle \(x^2 + y^2 = 25\) at the point \((3, -4)\).
Define partial derivative and find the value of \(\dfrac{\partial f}{\partial x}\) and \(\dfrac{\partial f}{\partial y}\) at the point \((4, -5)\) if \(f(x, y) = x^2 + 3xy + y - 1\).
Evaluate
(a) \(\displaystyle\int (\sin 2x \cos x + \cos 2x \sin x)\, dx\)
(b) \(\displaystyle\int_{\pi/4}^{\pi/2} \dfrac{dx}{1 - \sin x}\)
Determine the concavity of \(y = 3 + \sin x\) on \([0, 2\pi]\).
Test for convergence of the series \(\displaystyle\sum_{n=1}^{\infty} \left(\dfrac{1}{n+1}\right)^n\).