Table of Contents
FEDERAL PUBLIC SERVICE COMMISSION
COMPETITIVE EXAMINATION-2024 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) | Expand a fourier series of $f(x) = x^2, \quad 1 < x < 2$ | (10) |
| (b) | Find equation of integral surface of the differential equation $2y(z – 3)p + (2x – 2)q = y(2x – 3)$ which passes through the circle $x^2 + y^2 = 2x, \quad z = 0$. | (10) |
Q. No. 2
| (a) | Solve the higher order differential equation $y”” + y” = 3x^2 + 4\sin x – 2\cos x$ | (10) |
| (b) | Solve the initial value problem $y” – 8y’ + 15y = 9xe^{2x}, \quad y(0) = 5, \; y'(0) = 10$ | (10) |
Q. No. 3
| (a) | Solve the equation $4u_{xx} + 5u_{xy} + u_{yy} + u_x + u_y = 2$, also find its canonical form. | (10) |
| (b) | Prove that $\int_{-1}^{1} x^n P_n(x) dx = \frac{2^{n+1}(n!)^2}{(2n+1)!}$, where $P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} (x^2 – 1)^n$ is Legender polynomial of degree $n$. | (10) |
Q. No. 4
| (a) | Verify the divergence theorem for $A = 4x\hat{i} – 2y^2\hat{j} + z^2\hat{k}$ taken over the region bounded by $x^2 + y^2 = 4, \; z = 0$ and $z = 3$. | (10) |
| (b) | State and prove Stoke’s theorem. | (10) |
Q. No. 5
| (a) | Using the modified Euler’s method, obtain the solution of the differential equation
$\frac{dy}{dt} = t + \sqrt{y} = f(t,y)$
with initial condition $y_0 = 1$ at $t_0 = 0$ for the range $0 \le t \le 0.6$ in step of $0.2$. |
(10) |
| (b) | Find the real roots of equation $4x + \cos x + 2 = 0$ by using Newton Raphson method, correct to four decimal places. | (10) |
Q. No. 6
| (a) | Solve the system of linear equations by Gauss-Seidel iterative method and perform the first three iterations of:
$20x + y – 2z = 17$
$3x + 20y – z = -18$ $2x – 3y + 20z = 25$ |
(10) |
APPLIED MATHEMATICS
| (b) | Solve the following Van der Pol’s equation $y” – (0.1)(1 – y^2)y’ + y = 0$, using fourth order Runge-Kutta method for $x = 0.2$, with the initial values $y(0) = 1, \; y'(0) = 0$. | (10) |
Q. No. 7
| (a) | Find the law of force for a particle moving in an orbit, $r = \frac{l}{1 – e\cos\theta}$, where $l$ is semi latus rectum and $e$ is eccentricity. | (10) |
| (b) | Prove that the speed required to project a particle from a height $h$ to fall a horizontal distance $a$ from the point of projection is at least $\sqrt{g\left(\sqrt{a^2 + h^2}\right) – h}$. | (10) |
Q. No. 8
| (a) | Find the radial and transvers components of velocity moving along a curve $ax^2 + by^2 = 1$ at any time $t$ if the polar angle $\theta = ct^2$. | (10) |
| (b) | Find the centroid of the surface formed by the revolution of the cardioide $r = a(1 + \cos\theta)$ about the initial line. | (10) |
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