# Applied Mathematics CSS Paper 2020

FEDERAL PUBLIC SERVICE COMMISSION
COMPETITIVE EXAMINATION – 2020
FOR RECRUITMENT TO POST IN BS – 17
UNDER THE FEDERAL GOVERNMENT

APPLIED MATHEMATICS

TIME ALLOWED: THREE HOURS MAXIMUM MARKS = 100

NOTE:
(i) Attempt ONLY FIVE questions. ALL questions carry EQUAL marks.
(ii) All the parts (if any) of each Question must be attempted at one place instead of at different places.
(iii) Candidate must write Q. No. in the Answer Book in Accordance with Q. No. in the Q. Paper.
(iv) No Page / Space be left blank between the answers. All the blank pages of Answer Book must be crossed.
(v) Extra attempt of any question or any part of the attempted question will not be considered.
(vi) Use of Calculator is allowed.

Q. No. 1. (a) Prove that $\nabla^2r^n=n(n+1)r^{n-2}$ (10)

(b) Evaluate $\int_s\int \underline{A}.\overline{n}\space ds\space where \space \overline{A} = 18z\underline{i}-12\underline{j}+3y\underline{k}$ and S is that part of the plan $2x+3y+6z=12$ which is located in the 1st octant. (10)

Q. No. 2. A particle P of mass m slides down a friction-less inclined plane AB of an angle α with the horizontal. If it starts from rest at the top A, find (a) the acceleration (b) the velocity and (c ) the distance traveled after time t. (20)

Q. No. 3. (a) Discuss the motion of a particle moving in a straight line if it starts from rest at a distance ‘a’ from a point O and moves with an acceleration equal to k times its distance from O. (10)

(b) Find radial and traversal components of velocity and acceleration. (10)

Q. No. 4. (a) Solve ${d^2y\over dx^2}+y=Cosecx$ (10)

(b) Solve $dy+{y-Sinx\over x}dx=0$ (10)

Q. No. 5. (a) Solve the initial value problem $x(2+x){dy\over dx}+2(1+x)y = 1+3x^2, \space y(-1)=1$ (10)

(b) Find the general solution of the equation $(D^3-2D+1)y=2x^3-3x^2+4x+5$ (10)

Q. No. 6. (a) Find the Fourier series of $f(x)=\{ {x,0\lt x\lt 1 \atop 0, 1 \lt x \lt 2}$ (10)

(b) Solve the boundary value problem ${\partial^2 u \over \partial x^2} = {l \over k} {\partial u \over \partial t}$ satisfying $u(0,t) = u(l,t)=0 \space u(x,0)=lx-x^2$ (10)

Q. No. 7. (a) By using regular Falsi method, solve $Logx – Cosx=0$ (10)

(b) Find the value of f(7.5) by using Newton Gregory Backward Difference interpolation formula. (10)

Q. No. 8. (a) Applying the Taylor series method, compute $\int _a^x {Sint \over t} dt \space x = 0(0.1) l$ (10)

(b) Use fourth order RK method to solve ${dy \over dx} = t + y; y(0) = 1 \space from \space t=0 \space to \space t = 0.4 \space taking \space h=0.4$ (10)

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