Table of Contents
FEDERAL PUBLIC SERVICE COMMISSION
COMPETITIVE EXAMINATION-2025 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) |
(i) Prove that $\nabla r^n = n r^{n-2} \vec{r}$, where $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$. (ii) $\vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \times \vec{b}) \times \vec{c}$, then prove that $\vec{a}$ and $\vec{c}$ are parallel. |
(10) |
| (b) | Find the area of the region that is enclosed between the curves $y = x^2$ and $y = x + 6$. | (10) |
Q. No. 2
| (a) | Find the tangential and normal components of acceleration of a point describing the ellipse
$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
with uniform speed $V$, when the particle is at $(0, b)$. |
(10) |
| (b) | Find the solution of initial value problem by separation of variables:
$\sqrt{1-y^2} dx – \sqrt{1-x^2} dy = 0, \quad y(0) = 0$
|
(10) |
Q. No. 3
| (a) | Find the general solution of the given differential equation by variation of parameters.
$3y” – 6y’ + 6y = e^x \sec x$
|
(10) |
| (b) | Find the power series solution of $(x^2 + 1)y” + xy’ – y = 0$. | (10) |
Q. No. 4
| (a) | Forces $2\vec{BC}$, $7\vec{CA}$, $3\vec{BA}$ act along the sides of a triangle $ABC$. Show that their resultant is $6\vec{DE}$, where $D$ bisects $BC$ and $E$ is a point on $CE$ such that $CE = \frac{1}{3}CA$. | (10) |
| (b) | Find the center of mass of the surface generated by the revolution of the arc of the parabola, lying between the vertex and the latus rectum, about the $x$-axis. | (10) |
Q. No. 5
| (a) | Obtain the Fourier series over the indicated interval for the given function.
$f(x) = \begin{cases} 3\pi + 2x, & -\pi < x < 0 \\ \pi + 2x, & 0 < x < \pi \end{cases}$
|
(10) |
| (b) | Solve the boundary value problem
$u_{xx} + u_{yy} = 0, \quad 0 < x < a, \quad 0 < y < b$
$u(0, y) = 0, \quad u(a, y) = 0, \quad 0 \le y \le b$
$u(x, 0) = 0, \quad u(x, b) = f(x), \quad 0 \le x \le a.$ |
(10) |
Q. No. 6
| (a) | Use Newton’s Raphson method to find the solution accurate to within $10^{-4}$ (corrected upto four decimal places) for the given problem.
$x – \cos x = 0, \quad [0, \pi/2]$
|
(10) |
| (b) | Solve the system of linear equations using Gauss Seidel method (with three digit rounding arithmetic)
$3x_1 + 4x_2 – x_3 = 8$
$5x_1 + 3x_2 + 2x_3 = 17$ $-x_1 + x_2 – 3x_3 = -8$ |
(10) |
APPLIED MATHEMATICS
Q. No. 7
| (a) | Use Euler’s method to approximate the solution of the initial value problem.
$y’ = 1 + y/x, \quad 1 \le x \le 2, \quad y(1) = 2, \text{ with } h = 0.25$
|
(10) |
| (b) | Using Green’s theorem, evaluate $\int_C \vec{F}(\vec{r}) \cdot d\vec{r}$ counter clock wise around the boundary curve $C$ of the region $R$, where $\vec{F} = \left[ \frac{1}{2}xy^4, \;\; \frac{1}{2}x^4y \right]$, $R$ the rectangle with vertices $(0, 0), (3, 0), (3, 2), (0, 2)$. | (10) |
Q. No. 8
| (a) | Evaluate the Integral $\int_{1}^{3} \frac{1}{x^2} dx$, Using Trapezoidal Rule for five points (corrected upto two decimal places). | (10) |
| (b) | Find the D’Alembert solution of the wave equation $u_{xx} = \frac{1}{c^2} u_{tt}$, subject to the Cauchy Initial conditions $u(x, 0) = f(x), \; u_t(x, 0) = g(x)$. | (10) |
5 Views