Table of Contents
FEDERAL PUBLIC SERVICE COMMISSION
COMPETITIVE EXAMINATION-2023 FOR RECRUITMENT TO POSTS IN BS-17 UNDER THE FEDERAL GOVERNMENT
APPLIED MATHEMATICS
| TIME ALLOWED: THREE HOURS | MAXIMUM MARKS = 100 |
NOTE:
- Attempt only FIVE questions in all. ALL questions carry EQUAL marks.
- All the parts (if any) of each Question must be attempted at one place instead of at different places.
- Write Q. No. in the Answer Book in accordance with Q. No. in the Q.Paper.
- No Page/Space be left blank between the answers. All the blank pages of Answer Book must be crossed.
- Extra attempt of any question or any part of the attempted question will not be considered.
- Use of Calculator is allowed.
Q. No. 1
| (a) | Forces of magnitudes $P$, $2P$, $3P$, $4P$ act respectively along the sides $AB$, $BC$, $CD$, $DA$ of a square $ABCD$, of side $a$ and forces each of magnitude $(8\sqrt{2})P$ act along the diagonals $BD$, $AC$. Find the magnitude of the resultant force and the distance of its line of action from $A$. | (10) |
| (b) | A uniform rod $AB$ of length $a$ and weight $W$ is freely hinged to a vertical wall at $A$ and is maintained in equilibrium by a light string of length $l$ fastened to $B$ and to a point $C$ at a distance $b$ vertically above $A$. Prove that the reaction at the hinge $A$ is
$W \frac{\sqrt{(a^2 + 2b^2)}}{2b}$
and find the tension in the string. |
(10) |
Q. No. 2
| (a) | Use Runge-Kutta method of order two to solve the following differential equation at $x = 1.2$ by taking $h = 0.1$
$\frac{dy}{dx} = \frac{3x + y}{x + 2y}, \quad y(1) = 1$.
|
(10) | ||||||||||||||
| (b) | Find the first and second derivatives of $f(x)$ at $x = 3$ from the following data using Newton’s forward difference interpolation formula
|
(10) |
Q. No. 3
| (a) | Find the angle between the surfaces $x^2 + y^2 + z^2 = 9$ and $z = x^2 + y^2 – 3$ at the point $(2, -1, 2)$. | (8) |
| (b) | Show that
$\nabla r^n = n r^{n-2} \vec{r}$
|
(6) |
| (c) | Find the total work done in moving particle in a force field given by $\vec{F} = 3xy\hat{i} – 5z\hat{j} + 10x\hat{k}$ along the curve $x = t^2 + 1, \; y = 2t^2, \; z = t^3$ from $t = 1$ to $t = 2$. | (6) |
Q. No. 4
| (a) | A particle $P$ moves in a plane in such a way that at any time $t$, its distance from a fixed point $O$ is $r = at + bt^2$ and the line connecting $O$ and $P$ makes an angle $\theta = ct^2$ with a fixed line $OA$. Find the radial and transverse components of the velocity and acceleration of the particle at $t = 1$. | (10) |
| (b) | Solve the following Bernoulli’s equation
$x \frac{dy}{dx} + y = \frac{1}{y^2}$
|
(10) |
Q. No. 5
| (a) | Solve the following differential equation
$x \, dy = (x \sin x – y) \, dx$
|
(10) |
| (b) | Find the general solution of the higher order differential equation
$y”’ + 8y” = -6x^2 + 9x + 2$
|
(10) |
Q. No. 6
| (a) | Find solution of $4y” + y = 0$ in the form of power series in $x$. | (10) |
| (b) | Solve the following differential equation by variation of parameters
$y” – 4y’ + 4y = (x + 1)e^{2x}$
|
(10) |
Q. No. 7
| (a) | Find real root of the equation $2x – 3\sin(x) – 5 = 0$ up to 4 decimal places by secant method. | (10) |
| (b) | Solve the following system of equations by Gauss Seidel method. Perform only five iterations.
$8x_1 – x_2 – x_3 = 6$
$x_1 + 6x_2 + x_3 = 8$ $x_1 – x_2 + 5x_3 = 5$ |
(10) |
Q. No. 8
| (a) | Expand $f(x) = \sin x, \; 0 < x < \pi$, in a Fourier cosine series. | (10) |
| (b) | Use the method of separation of variables to find the solution of the following boundary value problem
$\nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0, \quad 0 \le x \le a, \quad 0 \le y \le b$
$u_x(0,y) = 0, \quad u_x(a,y) = 0,$
$u_y(x,b) = 0, \quad u(x,0) = f(x).$ |
(10) |
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