Table of Contents
FEDERAL PUBLIC SERVICE COMMISSION
COMPETITIVE EXAMINATION-2022 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.
- Candidate must 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) | Let $u = [y, z, x]$ and $v = [yz, zx, xy]$, $f = xyz$ and $g = x + y + z$. Find $\text{div}(\text{grad}(fg))$. | (10) |
| (b) | Evaluate $\int_C F(r) \cdot dr$ counter clockwise around the boundary $C$ of the region $R$ by Green’s theorem, where
$F = [y, -x]$, $C$ the circle $x^2 + y^2 = 1/4$
|
(10) |
Q. No. 2.
| (a) | Three forces $P, Q, R$, acting at a point, are in equilibrium, and the angle between $P$ and $Q$ is double of the angle between $P$ and $R$. Prove that $R^2 = Q(Q – P)$. | (10) |
| (b) | Find the centre of mass of a semi-circular lamina of radius $a$ whose density varies as the square of the distance from the centre. | (10) |
Q. No. 3.
| (a) | A particle moves in such a way that its position vector at time $t$ is
$r = (a \cos nt)\hat{i} + (b \sin nt)\hat{j}$,
Where $a, b, n$ are constants and $a > b > 0$. Show that the path of the particle is an ellipse of semi-major and minor axes $a, b$ respectively, and that the field of force is directed towards the centre of the ellipse. Also find the maximum speed. |
(10) |
| (b) | An aeroplane is flying with uniform speed $v_0$ in an arc of a vertical circle of radius $a$, whose centre is at a height $h$ vertically above a point $O$ of the ground. If a bomb is dropped from the aeroplane when at a height $Y$ and strikes the ground at $O$, show that $Y$ satisfies the equation
$KY^2 + Y(a^2 – 2hK) + K(h^2 – a^2) = 0$,
Where $K = h + \frac{ga^2}{2v_0^2}$. |
(10) |
Q. No. 4.
| (a) | Solve the given initial-value problem. Give the largest interval $I$ over which the solution is defined.
$xy’ + y = e^x, \quad y(1) = 2$.
|
(10) |
| (b) | Find the general solution of the given higher-order differential equation.
$y”’ – 4y” – 5y’ = 0$
|
(10) |
Q. No. 5.
| (a) | Find two power series solutions of the given differential equation about the ordinary point $x=0$.
$y” – 2xy’ + y = 0$.
|
(10) |
| (b) | Find the general solution of the given Bessel’s equation on $(0, \infty)$.
$x^2y” + xy’ + (9x^2 – 4)y = 0$
|
(10) |
Q. No. 6
| (a) | Find the Fourier series of the given function $f(x)$, which is assumed to have the period $2\pi$. Show the details of your work.
$f(x) = \begin{cases} x, & -\pi < x < 0 \\ \pi - x, & 0 < x < \pi \end{cases}$
|
(10) |
| (b) | Find $u(x,t)$ for the string of length $L=1$ and $c^2=1$ when the initial velocity is zero and the initial deflection with small $k$ (say, $0.01$) is $kx(1 – x)$. | (10) |
Q. No. 7
| (a) | Use the Bisection method to determine an approximation to the root of the given function in the interval $[1,2]$ that is accurate to at least within $10^{-4}$.
$f(x) = x^3 + 4x^2 – 10 = 0$.
|
(10) | ||||||||||||
| (b) | Values for $f(x) = xe^x$ are given in the following table. Use all the applicable three-point and five-point formulas to approximate $f'(2.0)$.
|
(10) |
Q. No. 8
| (a) | Use the Modified Euler method to approximate the solution to each of the following initial-value problem,
$y’ = -5y + 5t^2 + 2t, \quad 0 \le t \le 1, \quad y(0) = \frac{1}{3}, \text{ with } h = 0.1$
|
(10) |
| (b) | Use a fixed-point iteration method to determine a solution accurate to within $10^{-2}$ for $x^4 – 3x^2 – 3 = 0$ on $[1, 2]$. Use $p_0 = 1$. | (10) |
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