CSS Applied Mathematics Past Paper 2023

Instructions specific to this paper

• Attempt ONLY FIVE questions. ALL questions carry EQUAL marks
• Use of Calculator is allowed.
• at x = 3 from the following data using (10)
• 4.1023 5.1047 8.1971 9.1096 4.1122 6.1148

Questions

1. Q. 210 marks

   (a) Use Runge-Kutta method of order two to solve the following differential equation at x=1.2 by taking h=0.1 𝑑𝑦 3𝑥+𝑦 = 𝑥+2𝑦 𝑦(1) = 1 . 𝑑𝑥 (b) Find the first and second derivatives of f(x) at x = 3 from the following data using (10) Newton’s forward difference interpolation formula x 3 3.5 4 4.5 5 5.5 f(x) 4.1023 5.1047 8.1971 9.1096 4.1122 6.1148

2. Q. 38 marks

   (a) Find the angle between the surfaces 𝑥 2 + 𝑦 2 + 𝑧 2 = 9 and 𝑧 = 𝑥 2 + 𝑦 2 − 3 at the point (2, −1, 2). (b) Show that (6) 𝑛 𝑛−2 ∇𝑟 = 𝑛𝑟 𝑟 (c) Find the total work done in a moving particle in a force field given by (6) 𝐹 = 3𝑥𝑦 𝒊 − 5 𝑧 𝒋 + 10 𝑥 𝒌 along the curve 𝑥 = 𝑡 2 + 1, 𝑦 = 2𝑡 2 , 𝑧 = 𝑡 3 from 𝑡 = 1 to 𝑡 = 2.

3. Q. 410 marks

   (a) A particle P moves in a plane in such a way that at any time t, its distance from a fixed point O is 𝑟 = 𝑎 𝑡 + 𝑏 𝑡 2 and the line connecting O and P makes an angle 3 𝜃 = 𝑐𝑡 2 with a fixed line OA. Find the radial and transverse components of the velocity and acceleration of the particle at 𝑡 = 1. (b) Solve the following Bernouli’s equation (10) 𝑑𝑦 1 𝑥 +𝑦 = 2 𝑑𝑥 𝑦

4. Q. 510 marks

   (a) Solve the following differential equation 𝑥 𝑑𝑦 = (𝑥 𝑠𝑖𝑛𝑥 − 𝑦) 𝑑𝑥 (b) Find the general solution of the higher order differential equation (10) 𝑦 ′′′ + 8𝑦 ′′ = −6 𝑥 2 + 9𝑥 + 2 APPLIED MATHEMATICS

5. Q. 610 marks

   (a) Find solution of 4𝑦 ′′ + 𝑦 = 0 in the form of power series in x. (b) Solve the following differential equation by variation of parameters (10) 𝑦 ′′ − 4𝑦 ′ + 4𝑦 = (𝑥 + 1)𝑒 2𝑥

6. Q. 710 marks

   (a) Find real root of the equation 2𝑥 − 3 sin(𝑥) − 5 = 0 up to 4 decimal places by secant method. (b) Solve the following system of equations by Guass Seidel method. Perform only (10) five iterations. 8 x1 − x2 − x3 =6 x1 + 6 x2 + x3 = 8 x1 − x2 + 5 x3 = 5

7. Q. 810 marks

   (a) Expand 𝑓(𝑥) = sin 𝑥, 0 < 𝑥 < 𝜋, in a Fourier cosine series. (b) Use the method of separation of variables to find the solution of the following (10) boundary value problem 𝜕 2𝑢 𝜕 2𝑢 ∇2 𝑢 = + = 0, 0 ≤ 𝑥 ≤ 𝑎, 0≤𝑦≤𝑏 𝜕 𝑥2 𝜕 𝑦2 𝑢𝑥 (0, 𝑦) = 0, 𝑢𝑥 (𝑎, 𝑦) = 0, 𝑢𝑦 (𝑥, 𝑏) = 0, 𝑢(𝑥, 0) = 𝑓(𝑥).

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.