CSS Applied Mathematics Past Paper 2019

Instructions specific to this paper

• Attempt ONLY FIVE questions. ALL questions carry EQUAL marks
• Use of Calculator is allowed.
• 15.0 21.0 30.0 51.0
• =  0 When 0 < x < π
• 0.000 0.1280 0.5440 1.2960 2.4320 4.000

Questions

1. Q. 210 marks

   (a) Find the constants a, b, and c so that F=(x+2y+az) i + (bx – 3y – z ) j + (4x + cy + 2 z) k is irrotational and hence find the function ψ such that F = ∇ ψ (b) The forces F 1 ,F 2 ,F 3 ,F 4 ,F 5 and F 6 act along the sides of a regular hexagone taken (10) in order. Verify that all the forces will be in equilibrium if, ∑ F= 0, and F 1 – F 4 = F 3 –F 6 =F 5 –F 2 .

2. Q. 310 marks

   (a) A system of forces acts on a plate in the form of an equilateral triangle of side 2a. The moment of the forces about the three vertices are M 1 , M 2 and M 3 respectively. Find the magnitudes of the resultant. (b) If a particle P move with a velocity V given by V2 = n2 (ax2 + 2bx + c). Show that (10) P executes a simple harmonic motion. Find the centre, the amplitude and the time period of the motion?

3. Q. 410 marks

   (a) What is the difference between linear differential equation and Bernoulli’s equation? Also find the solution of the following differential equation.  dy  x + y = 1− y  dx  (b) Use the method of undetermined coefficient to solve the following differential (10) equation. y” – 3y’ + 2y = 2x3 – 9 x2 + 6 x

4. Q. 510 marks

   (a) Solve the equation 1 1 2 1 π 0= + x − x sin x − cos 2 x with x0 = 2 4 2 2 (b) Derive two point Gaussian integration formula for the following integral and use it (10) to solve the integral. 1.6 2x ∫ 1 x2 − 4 dx APPLIED MATHEMATICS

5. Q. 610 marks

   (a) Determine the second degree polynomials by using Newton’s method. Also estimate the value of f (0.1) and f (0.5) for the data. x 0.0 0.2 0.4 0.6 f (x) 15.0 21.0 30.0 51.0 (b) Does the dominate diagonal is necessary for finding the numerical solution of (10) system of linear equations by using Gauss Jacobi’s and Gauss Seidal methods. Explain the reason. In what conditions a numerical method is used instead of analytical method? Find the solution of the following system by performing three itrations of Gauss Seidal method. 6x – 3y + z = 11 2x + y – 8z = 15 x – 7y + z = 10

6. Q. 810 marks

   (a) The Trapezoidal rule applied to ∫0 f ( x ) dx gives the value 4, and the Simpson’s rule gives value 2, what is the value of f (1) ? (b) Find the first two derivatives at x=1.1 and x =1 from the following data table. (10) x 1 1.2 1.4 1.6 1.8 2.0 f (x) 0.000 0.1280 0.5440 1.2960 2.4320 4.000

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.