CSS Applied Mathematics Past Paper 2016

Instructions specific to this paper

• Attempt ONLY FIVE questions. ALL questions carry EQUAL marks
• Use of Calculator is allowed.
• Trapezoidal rule with n = 4 (ii) Simpson’s rule with n = 4

Questions

1. Q. 210 marks

   (a) Forces P, Q, R act at a point parallel to the sides of a triangle ABC taken in the same order. Show that the magnitude of the resultant force is 2 2 2 P + Q + R – 2 Q R cos A – 2 R P cos B – 2 P Q cos C (b) Find the distance from the cusp of the centroid of the region bounded by the (10) cardioid r = a (1 + cos ).

2. Q. 310 marks

   (a) A particle describes simple harmonic motion in such a way that its velocity and acceleration at a point P are u and f respectively and the corresponding quantities at another point Q are v and g. Find the distance PQ. (b) Derive the radial and transverse components of velocity and acceleration of a particle. (10)

3. Q. 410 marks

   Solve the following differential equations: dy y 3 4 (a) + = x y dx x 2 3 2x (b) (D – 5D + 6) y = x e (10)

4. Q. 510 marks

   (a) Solve the differential equation using the method of variation of parameters 2 d y   + y = tan x , –2<x<2 d x2 (b) Solve the Euler – Cauchy differential equation x2 y – 3 x y + 4y = x2 ln x. (10)

5. Q. 610 marks

   (a) Find the Fourier series of the following function:  – x if –  < x < 0 f(x) =   x if 0 < x <  (b) Solve the initial - boundary value problem: (10) 1 APPLIED MATHEMATICS

6. Q. 710 marks

   (a) Apply Newton – Raphson method to find the smaller positive root of the equation x2 – 4x + 2 = 0 (b) Solve the following system of equations by Gauss – Seidel iterative method by (10) taking the initial approximation as x1 = 0, x2 = 0, x3 = 0: 5x1 + x2 – x3 = 4 x1 + 4x2 + 2x3 = 15 x1 – 2x2 + 5x3 = 12

7. Q. 810 marks

   1 dx (a) Approximate  1+x 2 using 0 (i) Trapezoidal rule with n = 4 (ii) Simpson’s rule with n = 4 Also compare the results with the exact value of the integral. (b) Apply the improved Euler method to solve the initial – value problem: (10) y = x + y, y (0) = 0 by choosing h = 0.2 and computing y1, …, y5.

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.