Table of Contents
FEDERAL PUBLIC SERVICE COMMISSION
COMPETITIVE EXAMINATION-2021 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) | Evaluate the surface integral $\iint_S \vec{A} \cdot \hat{n} dS$ where $\vec{A} = z\hat{i} + x\hat{j} – 3y^2z\hat{k}$ and $S$ is the portion of the cylinder $x^2 + y^2 = 8$ lying in the first octant between $z = 0$ and $z = 4$. | (10) |
| (b) | Prove that
$\nabla f(r) = \frac{f'(r)}{r}\vec{r}$
where $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ and $r = |\vec{r}|$. |
(10) |
Q. No. 2.
| (a) | The greatest resultant that two forces can have is of magnitude $P$ and the least is of magnitude $Q$. Show that, when they act at an angle $\alpha$, their resultant is of magnitude $\sqrt{P^2 \cos^2 \frac{\alpha}{2} + Q^2 \sin^2 \frac{\alpha}{2}}$. | (10) |
| (b) | A sphere of weight $W$ and radius $a$ is suspended by a string of length $l$ from a point $P$ and a weight $w$ is also suspended from $P$ by a string sufficiently long for the weight to hang below the sphere. Show that the inclination of the first string to the vertical is
$\sin^{-1} \left[ \frac{w\alpha}{(W+w)(l+a)} \right]$.
|
(10) |
Q. No. 3.
| (a) | Show that the law of force towards the pole, of a particle describing the curve $r^n = a^n \cos n\theta$ is given by
$f = \frac{(n+1)h^2 a^{2n}}{r^{2n+3}}$.
|
(10) |
| (b) | The maximum velocity that a particle executing simple harmonic motion of amplitude $a$ attains, is $v$. If it is disturbed in such a way that its maximum velocity becomes $nv$, find the change in the amplitude and the time-period of motion. | (10) |
Q. No. 4.
| (a) | Define ordinary and singular points of the differential equation $a_2(x)y” + a_1(x)y’ + a_0(x)y = 0$. When a singular point is said to be regular and irregular? Find regular and irregular singular points of the differential equation $(x^2 – 4)^2y” + (x – 2)y’ + y = 0$. | (10) |
| (b) | Show that
$J_{3/2} = \sqrt{\frac{2}{\pi x}} \left[ \frac{\sin x}{x} – \cos x \right]$.
|
(10) |
Q. No. 5.
| (a) | Solve the equation by using method of undetermined coefficients
$y” – y’ + y = 2 \cos 3x$.
|
(10) |
| (b) | Use the method of Frobenius to find two linear independent series solutions in powers of $x$ of the DE
$x^2y” – (x^2 + x)y’ + y = 0$.
|
(10) |
Q. No. 6.
| (a) | Classify general second order partial differential equation (PDE) into elliptic, parabolic and hyperbolic form. Discuss the nature of the PDE
$(1 – x^2)u_{xx} – 2xyu_{xy} + (1 – y^2)u_{yy} = 0 \text{ at each } (x,y) \in \mathbb{R}^2$.
|
(10) |
| (b) | Use the method of separation of variables to find the solution $u(x,t) : [0,T] \times [0,L] \rightarrow \mathbb{R}$ to the initial/boundary value problem
$u_t(x,t) = u_{xx}(x,t)$ for $0 < t \le T \text{ and } 0 \le x \le L$,
$u(x,0) = f(x)$, for $0 \le x \le L$, $u(0,t) = u(L,t) = 0$, for $0 < t \le T$. where $f : [0,L] \rightarrow \mathbb{R}$ is a known function. |
(10) |
Q. No. 7.
| (a) | Use Simpson’s 3/8 rule to estimate the integral
$\int_{1}^{3} (x^3 – 2x^2 + 7x – 5)dx$.
By comparing your answer with exact value, find the error. |
(10) |
| (b) | Solve the system of equations by Jacobi iterative method.
$10x + 3y + z = 19, \quad 3x + 10y + 2z = 29, \quad x + 2y + 10z = 35$
|
(10) |
Q. No. 8.
| (a) | In the following table values of $y = x + \sin x^2$ are tabulated
Construct a difference table and estimate $f(1.04)$ and $f(1.57)$. |
(10) | ||||||||||||||||
| (b) | Use trapezoidal and Simpson’s 1/3 rules to approximate $\int_{0}^{\pi/2} \sin^2(x) dx$. Find a maximum bound for the error in each case. Compare your approximations with the actual result. | (10) |
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