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.
(iii)Use of Scientific Calculator is Allowed.
Q. No. 1. (a) Find the directional derivative of f(x, y, z) = x²y² + yz² at the point (2, -1, 1) in the direction of the vector î + 2ĵ + 2k̂? (10)
(b) Evaluate ∫_c (xy + y²) dx + x² dy where c is bounded by the line y = x and the curve y = x². (10)
Q. No. 2. (a) Find the constants a, b, and c so that F⃗ = (x + 2y + az) î + (bx – 3y – z) ĵ + (4x + cy + 2z) k̂ is irrotational and hence find the function ψ such that F⃗ = ∇ψ. (10)
(b) The forces F₁, F₂, F₃, F₄, F₅ and F₆ act along the sides of a regular hexagon taken in order. Verify that all the forces will be in equilibrium if, ∑ F_i = 0, and F₁ – F₄ = F₃ – F₆ = F₅ – F₂. (10)
Q. No. 3. (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 M₁, M₂ and M₃ respectively. Find the magnitude of the resultant. (10)
(b) If a particle P move with a velocity V given by V² = n²(ax² + 2bx + c). Show that P executes a simple harmonic motion. Find the centre, the amplitude and the time period of the motion? (10)
Q. No. 4. (a) What is the difference between linear differential equation and Bernoulli’s equation? Also find the solution of the following differential equation: x [dy/dx + y] = 1 – y. (10)
(b) Use the method of undetermined coefficients to solve the following differential equation: y” – 3y’ + 2y = 2x³ – 9x² + 6x. (10)
Q. No. 5. (a) Solve the equation 0=1/2 +1/4 – xsin x – (1/4)x cos 2x = 0 with x₀ = π/2. (10)
(b) Derive two point Gaussian integration formula for the following integral and use it to solve the integral: ∫₁^{1.6} 2x / (x² – 4) dx. (10)
Q. No. 6. (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. (10)
| 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 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 iterations of Gauss Seidal method. (10)
6x – 3y + z = 11
2x + y – 8z = 15
x – 7y + z = 10
Q. No. 7. (a) Define even function and odd function with examples. Verify that the Fourier series for the function (10)
f(x) = { 0 When 0 < x < π
{ 1 When π < x < 2π
is f(x) = 1/2 – 2/π (sin x + 1/3 sin 3x + 1/5 sin 5x …)
(b) Solve the following partial differential equation by using method of separable variable. (10)
∂u/∂x = 2 ∂u/∂t + u , given u(x,0) = 6e⁻³ˣ
Q. No. 8. (a) The Trapezoidal rule applied to ∫₀² f(x)dx gives the value 4, and the Simpson’s rule gives value 2, what is the value of f(1)? (10)
(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 |
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