CSS Applied Mathematics Past Paper 2018

Instructions specific to this paper

• Attempt ONLY FIVE questions. ALL questions carry EQUAL marks
• Use of Calculator is allowed.

Questions

1. Q. 210 marks

   (a) Forces of magnitude P, 2P, 3P, 4P act respectively along the sides AB, BC, CD, DA of a square ABCD, of sides a, and forces each of magnitude (8 2 ) P act along the diagonals BD, AC. Find the magnitude of the resultant force and distance of its line of action from A. (b) A uniform ladder, of length 70 feet, rests against a vertical wall with which it (10) makes an angle of 450, the coefficient of friction between the ladder and the wall 1 1 and the ground respectively being and . If a man, whose weight is one half 3 2 that of the ladder, ascends the ladder, where will he be when the ladder slips?

2. Q. 310 marks

   (a) A particle moves in a straight line with an acceleration kv3. If its initial velocity is u, find the velocity and the time spent when the particle has travelled a distance x. (b) Derive the Tangential and Normal components of the velocity and acceleration. (10)

3. Q. 410 marks

   (a) Solve the following Cauchy- Euler Equation d2y dy x2 2 − 2x − 4y = 0. dx dx (b) Convert the following Bernoulli Differential Equation into standard form and (10) then solve. 1 dy  xy  +  = xy 2 . dx  1 − x 2 

4. Q. 510 marks

   (a) Convert the following Ordinary Differential Equation into standard form and then solve using Method of Variation of Parameters. x 2 y ′′ − 3 xy ′ + 3 y = 2 x 4 e x (b) Check whether the following Ordinary Differential Equation is an Exact Equation (10) or not. If yes, then solve. (3x 2 y + 2) dx + ( x 3 + y) dy = 0 APPLIED MATHEMATICS

5. Q. 610 marks

   (a) Find the Fourier Series of f on the given interval. − 1, − π < x < 0 f ( x) =   1, 0≤ x <π (b) Solve the following Partial Differential Equation subject to the conditions given. (10) ∂ 2u ∂ 2u a 2 = , 0 < x< L, t > 0, ∂x 2 ∂t 2 u (0, t ) = 0, u ( L, t ) = 0, t > 0, ∂u u ( x,0) = f ( x), = g ( x)[at time t = 0], and 0 < x < L. ∂t

6. Q. 710 marks

   (a) Use Newton-Raphson method to find solution accurate to within 10-4 for the non- linear equation. x 3 − 2 x 2 − 5 = 0, I = [1,4] (b) Use Lagrange Interpolating polynomial of degree two to approximate f (8.4), If (10) f (8.1) = 16.94410, f (8.3) = 17.56492, f (8.6) = 18.50515, f (8.7) = 18.82091. 1 dx

7. Q. 810 marks

   (a) Approximate ∫0 1 + x 2 using Trapezoidal rule and Simpson’s rule with n=4. Also compare your results with the exact value of the integral. (b) Use Euler’s method to approximate the solution of the following initial value (10) problem. 1+ y y′ = , 1 ≤ t ≤ 2, with y(1) = 2, h = 0.25 t

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.