FEDERAL PUBLIC SERVICE COMMISSION

COMPETITIVE EXAMINATION-2017

FOR RECRUITMENT TO POSTS 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) Suppose $ r(t) = (e^t cost) i + (e^t sint ) j $. Show that the angle between r and $ d^2r \over {dt^2} $ never changes. What is the angle? (10)

(b) Use divergence theorem of Gauss. Evaluate $ \int \int_S x^2dxdy+x^2ydzdz + x^2zdxdy $ where S is the closed surface consisting of the cylinder $ x^2 + y^2 = a^2, (0\le z \le b) $ and the circular disks z = 0 and z = b, $ z = b, (x^2 + y^2 \le a^2) $. (10)

Q. No. 2. (a) A 100 Kg wooden crate rests on a wooden ramp with an adjustable angle of inclination. Draw a free body diagram of the crate. If the angle of the ramp is set to 10^{6}, find the perpendicular and parallel components of the crate’s weight to the ramp. Also find the static friction force between the crate and the ramp. At what angle will the crate just begin to slip? (coefficient of static friction $ \mu_s $ between wood block and wood surface is 0.28) (10)

(b) A ladder having a uniform density and a mass m rests against a frictionless vertical wall at an angle of 60^{0}. The lower end rests on a flat surface where the coefficient of static friction is 0.40. A person of mass M=2m attempts to climb reached when the ladder begins to slip? (10)

Q. No. 3. (a) A particle moving in a straight line starts with a velocity $ \mu $ and has acceleration $ \nu $^{3}, where $ \nu $ is the velocity of the particle at time t. Find the velocity and the time as functions of the distance travelled by the particle. (10)

(b) A particle describing simple harmonic motion has velocities 5 ft/sec and 4 ft/sec when its distance from the center are 12 ft respectively. Find the time-period of motion. (10)

Q. No. 4. (a) Check for the exactness and solve the following ordinary differential equation. $ (2xy+y-tany)dx+(x^2-xtan^2y+sec^2y)dy=0 $ (10)

(b) Solve the following second order differential equation: $ {d^2y \over {dx^2}} + 4x = sec2x$ (10)

Q. No. 5. (a) Solve the initial value problem $ {d^2y \over {dx^2}} =2-6x, y'(0) = 4, y(0) = 1$ (10)

(b) Solve the following Boundary Value Problem $ y’+4y = 0, y(0) = -2, y(2\pi ) = -2$. How many solutions do you get for this problem? (10)

Q. No. 6. (a) Computer the Fourier series for the function x^{2} on the interval 0

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