CSS Applied Mathematics Past Paper 2022

Instructions specific to this paper

• Attempt ONLY FIVE questions. ALL questions carry EQUAL marks
• Use of Calculator is allowed.
• , which is assumed to have the (10)
• = xex are given in the following table. Use all the applicable three- (10)
• 10.889365 12.703199 14.778112 17.148957 19.85503

Questions

1. Q. 210 marks

   (a) Three forces P, Q, R, acting at a point, are in equilibrium, and the angle between P and Q is double of the angle between P and R. Prove that = − . (b) Find the centre of mass of a semi-circular lamina of radius a whose density varies (10) as the square of the distance from the centre.

2. Q. 310 marks

   (a) A particle moves in such a way that its position vector at time t is = cos $ i + & sin $ j, Where a, b, n are constants and a>b>0. Show that the path of the particle is an ellipse of semi-major and minor axes a, b respectively, and that the field of force is directed towards the centre of the ellipse. Also find the maximum speed. (b) An aeroplane is flying with uniform speed v0 in an arc of a vertical circle of radius (10) a, whose centre is at a height h vertically above a point O of the ground . If a bomb is dropped from the aeroplane when at a height Y and strikes the ground at O, show that Y satisfies the equation )* + * − 2ℎ) + ) ℎ − = 0, -./ Where ) = ℎ + . 01 /

3. Q. 410 marks

   (a) Solve the given initial-value problem. Give the largest interval I over which the solution is defined. xy’+y = ex, y(1) = 2. (b) Find the general solution of the given higher-order differential equation. (10) 222 − 4 22 − 5 2 = 0

4. Q. 510 marks

   (a) Find two power series solutions of the given differential equation about the ordinary point x=0. 22 − 2 2 + = 0. (b) Find the general solution of the given Bessel’s equation on (0, ∞). (10) 22 + 2+ 9 −4 =0 APPLIED MATHEMATICS

5. Q. 610 marks

   (a) Find the Fourier series of the given function f(x), which is assumed to have the period 26. Show the details of your work. , −6 < <0 7 =8 6− , 0< <6 (b) Find u(x,t) for the string of length L=1 and c2=1 when the initial velocity is zero (10) and the initial deflection with small k (say, 0.01) is : 1 − .

6. Q. 710 marks

   (a) Use the Bisection method to determine an approximation to the root of the given function in the interval [1,2] that is accurate to at least within 10-4. 7 = ; + 4 − 10 = 0. (b) Values for f (x) = xex are given in the following table. Use all the applicable three- (10) point and five-point formulas to approximate 7 2 (2.0). x 1.8 1.9 2.0 2.1 2.2 f(x) 10.889365 12.703199 14.778112 17.148957 19.85503

7. Q. 810 marks

   (a) Use the Modified Euler method to approximate the solution to each of the following initial-value problem, 2 1 = −5 + 5 + 2 , 0 ≤ ≤ 1, 0 = , > ℎ ℎ = 0.1 3 (b) Use a fixed-point iteration method to determine a solution accurate to within 10-2 (10) for x4 -3x2 −3 = 0 on [1, 2]. Use p0 = 1.

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.