MATHEMATICS—II

Choose the correct answer (any seven):

Test the convergence of $1 + \frac{3}{7}x + \frac{3 \cdot 6}{7 \cdot 10}x^2 + \frac{3 \cdot 6 \cdot 9}{7 \cdot 10 \cdot 13}x^3 + \frac{3 \cdot 6 \cdot 9 \cdot 12}{7 \cdot 10 \cdot 13 \cdot 16}x^4 + \dots$.

Examine the convergence of the series of which the general term is $\frac{2^2 4^2 6^2 \dots (2n-2)^2}{3 \cdot 4 \cdot 5 \dots (2n-1) \cdot 2n} x^{2n}$.

State and prove the convolution theorem for Laplace transform.

Find $L^{-1} \left[ \frac{1}{(s^2+1)^2(s^2+4)} \right]$ using the convolution theorem.

Solve the differential equation $\frac{d^2y}{dx^2} + 9y = \cos 2t, y(0)=1, y(\pi/2)=-1$ using Laplace transform.

Evaluate the integral $\int_0^\infty \frac{e^{-t} \sin t}{t} \, dt$ using Laplace transform.

Expand in Fourier series $f(x) = \begin{cases} -x, & -4 \le x \le 0 \\ x, & 0 \le x \le 4 \end{cases}$ and hence deduce that $\frac{\pi^2}{8} = 1 + \frac{1}{3^2} + \frac{1}{5^2} + \frac{1}{7^2} + \dots$.

Evaluate by changing the order of integration $\int_0^\infty \int_x^\infty \frac{e^{-y}}{y} \, dy \, dx$.

Find the volume of the solid in the first octant bounded by the paraboloid $z = 36 - 4x^2 - 9y^2$.

Find the mass of a plate in the form of a quadrant of an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ whose density per unit area is given by $\rho = kxy$.

Evaluate the following integral by changing to polar coordinates: $\int_0^1 \int_0^{\sqrt{2x-x^2}} (x^2+y^2) \, dy \, dx$.

Use divergence theorem to evaluate $\iint_S \vec{A} \cdot d\vec{S}$, where $\vec{A} = 4x\hat{i} - 2y^2\hat{j} + z^2\hat{k}$ and $S$ is the surface, $x^2+y^2=4, z=0$ and $z=3$.

Find the directional derivative of $\phi(x,y,z) = x^2yz + 4xz^2$ at (1, -2, -1) in the direction $2\hat{i} - \hat{j} - 2\hat{k}$.

Evaluate $\oint_C (x^2-2xy) \, dx + (x^2y+3) \, dy$ around the boundary of the region defined by $y^2=8x$ and $x=2$ using Green's theorem.

Find (i) $\vec{A} \times (\nabla \phi)$ and (ii) $(\nabla \times \vec{A}) \times \vec{B}$, where $\vec{A} = x^2z\hat{i} + yz^3\hat{j} - 3xy\hat{k}, \vec{B} = 3x\hat{i} + 4z\hat{j} - xy\hat{k}$ and $\phi = xy^2z$.

Answer any seven. Choose the correct alternative in each.

Test the convergence of the series $\sum \left( \frac{\sqrt{n^2+1}-n}{n^p} \right)$.

Test the convergence of the series $\sum \frac{4.7.10 \dots (3n+1) x^n}{n!}$.

Find the Laplace transforms of (i) $\cos^2 t$ (ii) $\sin^2 t$.

Find the Laplace transforms of $\frac{\sinh t}{t}$.

Find the Laplace transforms of $e^{-4t} \cdot \sin ht \sin t$.

Show That $\int_0^\infty \left( \frac{\sin 2t + \sin 3t}{t e^t} \right) dt = \frac{3\pi}{4}$.

Find the Fourier series of $f(x) = x^2$ in the interval $(0, 2\pi)$ and hence deduce that $\frac{\pi^2}{12} = \frac{1}{1^2} - \frac{1}{2^2} + \frac{1}{3^2} - \frac{1}{4^2} + \dots$.

Find the Fourier series of $f(x) = (4-x^2)$ in the interval $(0, 2)$. Hence, deduce that $\frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \dots = \frac{\pi^2}{6}$.

Find the Fourier series of $f(x) = |\cos x|$ in the interval $(-\pi, \pi)$.

Find the half-range sine series of $f(x) = x, 0 < x < 1$ and $f(x) = (2-x), 1 < x < 2$ and hence deduce that $\frac{1}{1^2} + \frac{1}{3^2} + \frac{1}{5^2} + \dots = \frac{\pi^2}{8}$.

Using the transformation $x+y = u$ and $y=uv$, show that $\int_0^1 \int_0^{1-x} e^{y/(x+y)} \, dy \, dx = \frac{1}{2}(e-1)$.

Evaluate $\iiint x^2 yz \, dx \, dy \, dz$ over the region bounded by the planes $x=0, y=0, z=0$ and $x+y+z=1$.

Find the volume bounded by the cylinders $x^2 + y^2 = 2ax$ and $z^2 = 2ax$.

Evaluate $\int_C \vec{F} \cdot d\vec{r}$ where $\vec{F} = (x^2+y^2)\hat{i} - 2xy\hat{j}$ and $C$ is the rectangle in the xy-plane bounded by $y=0, x=a, y=b, x=0$.

Find the directional derivative of $\phi = x^2 - y^2 + 2z^2$ at the point P(1,2,3) in the direction of the line PQ where Q is the point (5,0,4). In what direction it will be maximum and find the maximum value of it.

Prove that $\text{div}(\text{grad } r^n) = n(n+1)r^{n-2}$, where $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$.

Answer any seven of the following as directed:

Discuss the convergence of the series $x + \frac{2^2 \cdot x^2}{L2} + \frac{3^3 \cdot x^3}{L3} + \frac{4^4 \cdot x^4}{L4} + \frac{5^5 \cdot x^5}{L5} + \cdots \infty$ ($x > 0$).

Prove that the series $\frac{\sin x}{1^3} - \frac{\sin 2x}{2^3} + \frac{\sin 3x}{3^3} - \cdots$ converges absolutely.

Expand $f(x) = x \sin x$ as a Fourier series in the interval $0 < x < 2\pi$.

Obtain Fourier expansion for the function $f(x) = \begin{cases} \pi+x, & \text{if } -\pi \le x \le 0 \\ \pi-x, & \text{if } 0 \le x \le \pi \end{cases}$ and $f(x+2\pi) = f(x)$.

Find half-range cosine series for the function $f(x) = x^2, 0 < x < 2$.

Find the Laplace transform of $t e^{-t} \sin 3t$.

Find the value of $L \left\{ \int_0^t \frac{e^{-t} \sin t}{t} \, dt \right\}$.

Find the inverse Laplace transform of $\frac{s+2}{s^2(s+1)(s-2)}$.

By changing the order of integration in $I = \int_0^1 \int_{x^2}^{2-x} xy \, dy \, dx$ evaluate the integral.

Evaluate $\iint r \sin \theta \, dr \, d\theta$ over the cardioid $r = a(1 - \cos \theta)$ above the initial line.

Evaluate $\iint_R (x+y)^2 \, dx \, dy$ where $R$ is the parallelogram in the xy$-plane with vertices $(1, 0), (3, 1), (2, 2), (0, 1) using the transformation $u = x+y$ and $v = x-2y$.

Find by triple integration, the volume of the sphere $x^2 + y^2 + z^2 = a^2$.

Show that $\nabla^2(r^n) = n(n+1)r^{n-2}$.

Find the work done in moving a particle in the force field $\vec{F} = 3x^2\hat{i} + (2xz-y)\hat{j} + z\hat{k}$ along the curve defined by $x^2 = 4y, 3x^3 = 8z$ from $x = 0$ to $x = 2$.

Verify Stokes' theorem for $\vec{F} = (x^2 + y^2)\hat{i} - 2xy\hat{j}$ taken around the rectangle bounded by the lines $x = \pm a, y = 0$ and $y = b$.

Evaluate $\int_S \vec{F} \cdot dS$, where $\vec{F} = 4x\hat{i} - 2y^2\hat{j} + z^2\hat{k}$ and $S$ is the surface bounding the region $x^2 + y^2 = 4, z = 0$ and $z = 3$.

Choose the correct or best alternatives (any seven):

Test for convergence the series whose nth term is $n^{\log x}$.

Test the series for convergence $1 + \frac{3}{7}x + \frac{3 \cdot 6}{7 \cdot 10}x^2 + \frac{3 \cdot 6 \cdot 9}{7 \cdot 10 \cdot 13}x^3 + \frac{3 \cdot 6 \cdot 9 \cdot 12}{7 \cdot 10 \cdot 13 \cdot 16}x^4 + \cdots$ to $\infty$.

Define absolutely and conditionally convergent series. Show that the series $1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \frac{1}{6} + \cdots$ to $\infty$ is convergent but not absolutely convergent.

Find the Fourier series expansion of the periodic function of period $2\pi$, $f(x) = x^2, -\pi \le x \le \pi$. Hence find the sum of the series $\frac{1}{1^2} - \frac{1}{2^2} + \frac{1}{3^2} - \frac{1}{4^2} + \cdots$.

Find the Fourier series for $f(x)$, where $f(x) = \begin{cases} -\pi, & -\pi < x < 0 \\ x, & 0 < x < \pi \end{cases}$. Hence deduce that $\frac{1}{1^2} + \frac{1}{3^2} + \frac{1}{5^2} + \cdots = \frac{\pi^2}{8}$.

Find the Fourier half-range even expansion of the function $f(x) = -\frac{x}{l} + 1, 0 \le x \le l$.

Find the Laplace transform of (i) t^2 \cos at$; (ii) $e^{-4t} \frac{\sin 3t}{t}.

Find the Laplace transform of $\frac{1 - \cos t}{t^2}$.

Show that $\int_0^1 dx \int_0^1 \frac{x-y}{(x+y)^3} \, dy \neq \int_0^1 dy \int_0^1 \frac{x-y}{(x+y)^3} \, dx$.

Evaluate $\iint r^2 \cos^2 \theta \, r \, dr \, d\theta$ over the interior of the circle $r = 2a \cos \theta$.

Evaluate the triple integral $\int_1^e \int_1^{\log y} \int_1^{e^x} \log z \, dz \, dx \, dy$.

Find the length of the arc of the curve $y = e^x$ from the point $(0, 1)$ to $(1, e)$.

Find the directional derivatives of $\phi = xy^2 + yz^2$ at the point $(2, -1, 1)$ in the direction of the vector $\vec{i} + 2\vec{j} + 2\vec{k}$.

Find the magnitude of the velocity and acceleration of a particle which moves along the curve $x = 2 \sin 3t, y = 2 \cos 3t, z = 8t$ at any time $t > 0$. Find unit tangent vector to the curve.

If $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$, then prove that $\text{div}(\text{grad } r^n) = n(n+1)r^{n-2}$.

Evaluate $\int_C \vec{F} \cdot d\vec{r}$, where $\vec{F} = (x^2 + y^2)\hat{i} - 2xy\hat{j}$ and $C$ is the rectangle in the xy$-plane bounded by $y=0, x=a, y=b, x=0.

Choose the correct or best alternatives of the following (any seven):

Discuss the nature of convergency of an infinite geometric series.

Test the convergence for the series $\frac{x}{1 \cdot 2} + \frac{x^2}{3 \cdot 4} + \frac{x^3}{5 \cdot 6} + \frac{x^4}{7 \cdot 8} + \cdots$ to $\infty$.

Find the Laplace $(1 + \cos 2t)$ for the first principle.

Find the Laplace transform of $\frac{(1 - \cos t)}{t^2}$.

Find the Laplace inverse of $\frac{s^2}{(s^2+a^2)(s^2+b^2)}$.

Find the Fourier series representing the function $f(x) = x, 0 < x < 2\pi$ and sketch its graph from $x = -4\pi$ to $x = 4\pi$.

Find the Fourier series of the function defined as $f(x) = \begin{cases} x+\pi & \text{for } 0 \le x \le \pi \\ -x-\pi & \text{for } -\pi \le x < 0 \end{cases}$ and $f(x+2\pi) = f(x)$.

Expand $f(x) = e^x$ in a cosine series over $(0, 1)$.

Evaluate: $\int_0^\infty \int_x^\infty \frac{e^{-y}}{y} \, dy \, dx$.

Evaluate $\iint_R xy \, dx \, dy$, where $R$ is the quadrant of the circle $x^2+y^2=a^2$ and $x \ge 0, y \ge 0$.

Find by double integration the area enclosed by the pair of curves $y = 2-x$ and $y^2 = 2(2-x)$.

Compute $\iiint \frac{dx \, dy \, dz}{(x+y+z+1)^3}$ if the region of integration is bounded by the coordinates plane and plane $x+y+z=1$.

A particle moves along a plane curve such that its linear velocity is perpendicular to the radius vector. Show that the path of the particle is a circle.

Show that $\nabla \left( \frac{\vec{a} \cdot \vec{r}}{r^n} \right) = \frac{\vec{a}}{r^n} - \frac{n(\vec{a} \cdot \vec{r})}{r^{n+2}} (\vec{r})$ where $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}, r = |\vec{r}|$, $\vec{a}$ is a constant vector.

If $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$, then prove that $\text{div}(\text{grad } r^n) = n(n+1)r^{n-2}$.

Verify Green's theorem for $\oint_C (2xy \, dx - y^2 \, dy)$, where $C$ is the boundary of the region bounded by the ellipse $3x^2 + 4y^2 = 12$.

Select the correct or best alternatives of any seven of the following:

Prove that the series $\frac{1}{1^P} + \frac{1}{2^P} + \frac{1}{3^P} + \cdots \infty$ is convergent if $p > 1$ and divergent if $p \le 1$.

Test the series for convergence $\frac{1}{2}x + x^2 + \frac{9}{8}x^3 + x^4 + \frac{25}{32}x^5 + \cdots \infty$.

Find the Laplace transform by definition of function $f(t) = \begin{cases} \cos t & 0 < t < \pi \\ \sin t & t > \pi \end{cases}$.

If $J_0(t) = \sum_{r=0}^\infty \frac{(-1)^r}{(r!)^2} \left(\frac{t}{2}\right)^{2r}$, then prove that $L\{J_0(t)\} = \frac{1}{\sqrt{1+s^2}}$.