Federal Public Service Commission (FPSC)
Competitive Examination for Recruitment to BPS-17 Posts under the Federal Government
Time Allowed: 3 Hours
APPLIED MATHEMATICS
MAXIMUM MARKS: 100 Marks
Attempt ONLY FIVE questions, All questions carry EQUAL marks.
Use of Scientific Calculator is Allowed.
Q. No. 1. (a) Forces of magnitudes P, 2P, 3P, 4P act respectively along the sides AB, BC, CD, DA of a square ABCD, of side a and forces each of magnitude (8√2) P act along the diagonals BD, AC. Find the magnitude of the resultant force and the distance of its line of action from A. (10)
(b) A uniform rod AB of length a and weight W is freely hinged to a vertical wall at A and is maintained in equilibrium by a light string of length a fastened to B and to a point C at a distance b vertically above A. Prove that the reaction at the hinge A is
W √(a² + 2b²) / 2b
and find the tension in the string. (10)
Q. No. 2. (a) Use Runge-Kutta method of order two to solve the following differential equation at x=1.2 by taking h=0.1
dy/dx = (3x + y) / (x + 2y) , y(1) = 1 . (10)
(b) Find the first and second derivatives of f(x) at x = 3 from the following data using Newton’s forward difference interpolation formula: (10)
| x | 3 | 3.5 | 4 | 4.5 | 5 | 5.5 |
|——|——–|——–|——–|——–|——–|——–|
| f(x) | 4.1023 | 5.1047 | 8.1971 | 9.1096 | 4.1122 | 6.1148 |
Q. No. 3. (a) Find the angle between the surfaces x² + y² + z² = 9 and z = x² + y² – 3 at the point (2, -1, 2). (8)
(b) Show that ∇rⁿ = n rⁿ⁻² r⃗ (6)
(c) Find the total work done in a moving particle in a force field given by F⃗ = 3xy î – 5z ĵ + 10x k̂ along the curve x = t² + 1, y = 2t², z = t³ from t = 1 to t = 2. (6)
Q. No. 4. (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 t² and the line connecting O and P makes an angle θ = c t^(3/2) with a fixed line OA. Find the radial and transverse components of the velocity and acceleration of the particle at t = 1. (10)
(b) Solve the following Bernoulli’s equation:
x dy/dx + y = 1 / y² (10)
Q. No. 5. (a) Solve the following differential equation:
x dy = (x sin x – y) dx (10)
(b) Find the general solution of the higher order differential equation:
y”” + 8y” = -6x² + 9x + 2 (10)
Q. No. 6. (a) Find solution of 4y” + y = 0 in the form of power series in x. (10)
(b) Solve the following differential equation by variation of parameters:
y” – 4y’ + 4y = (x + 1)e^(2x) (10)
Q. No. 7. (a) Find real root of the equation 2x – 3 sin(x) – 5 = 0 up to 4 decimal places by secant method. (10)
(b) Solve the following system of equations by Gauss Seidel method. Perform only five iterations.
8x₁ – x₂ – x₃ = 6
x₁ + 6x₂ + x₃ = 8
x₁ – x₂ + 5x₃ = 5 (10)
Q. No. 8. (a) Expand f(x) = sin x, 0 < x < π, in a Fourier cosine series. (10)
(b) Use the method of separation of variables to find the solution of the following boundary value problem:
∇²u = ∂²u/∂x² + ∂²u/∂y² = 0, 0 ≤ x ≤ a, 0 ≤ y ≤ b
u_x(0, y) = 0, u_x(a, y) = 0,
u_y(x, b) = 0, u(x, 0) = f(x). (10)
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