CSS Applied Mathematics Past Paper 2017

Instructions specific to this paper

• Attempt ONLY FIVE questions. ALL questions carry EQUAL marks
• Use of Calculator is allowed.
• , u(0,t) = 0, u(L,t) = 0
• = x 10 -1, (10)
• at x=15 for the following data;
• : 3.850 0.800 0.212

Questions

1. Q. 210 marks

   (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 100, 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 s between wood block and wood surface is 0.28) (b) A ladder having a uniform density and a mass m rests against a frictionless (10) vertical wall at an angle of 600. 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 the ladder. What fraction of the length L of the ladder will the person have reached when the ladder begins to slip?

2. Q. 310 marks

   (a) A particle moving in a straight line starts with a velocity u and has acceleration v3, where v is the velocity of the particle at time t. Find the velocity and the time as functions of the distance travelled by the particle. (b) A particle describing simple harmonic motion has velocities 5 ft/sec and 4 ft/sec (10) when its distance from the centre are 12 ft and 13 ft respectively. Find the time- period of motion.

3. Q. 410 marks

   (a) Check for the exactness and solve the following ordinary differential equation. (2xy +y –tany)dx + (x2 –xtan2y + sec2y)dy = 0 (b) Solve the following second order differential equation: (10) d2 y + 4x = sec 2x dx 2

4. Q. 510 marks

   (a) Solve the initial value problem. d2 y = 2-6x, y/(0) = 4, y(0) = 1 dx 2 (b) Solve the following Boundary Value Problem. (10) y/ + 4y = 0, y(0) = -2, y(2  ) = -2 How many solutions do you get for this problem? APPLIED MATHEMATICS

5. Q. 610 marks

   (a) Compute the Fourier series for the function x2 on the interval 0<x<L, using as a basis of function with boundary conditions u/ (0) =0 and u/ (L) = 0. Sketch the partial sums of the series for 1, 2, 3 terms. (b) Find a solution to the following partial differential equation that will also satisfy (10) the boundary conditions. u  2u  k 2 , u(x,0) = f(x) , u(0,t) = 0, u(L,t) = 0 t x

6. Q. 710 marks

   (a) Use bisection and false position methods to locate the root of f(x) = x 10 -1, between 0 and 1.3. Which of the two methods is better and why? (b) Use Lagrange interpolation polynomial of the first and second order to evaluate (10) f(x) at x=15 for the following data; x: 0 20 40 f(x): 3.850 0.800 0.212 0.800 0

7. Q. 810 marks

   (a) Use Trapezoidal Rule with 3 segments to evaluate. 0.8  2x  x e dx  0 Also find the true solution and the percentage error. (b) Consider the function f(x) = xex. Obtain approximations to f/(2) with h = 0.5 (10) using forward, backward and central difference formulas.

Standard FPSC instructions

Printed on this and every CSS paper.

• Part-II is to be attempted on the separate Answer Book.
• 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 question paper.
• No page/space be left blank between the answers. All the blank pages of the Answer Book must be crossed.
• Extra attempt of any question or any part of the attempted question will not be considered.