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) Prove that ∇ · [f(r)/r r⃗] = (2/r)f(r) + f'(r) (10)
(b) Verify Stokes’ theorem for A⃗ = (2x – y)î – yz²ĵ – y²zk̂, where S is the upper half surface of the sphere x² + y² + z² = 1 and C is its boundary. (10)
Q. No. 2. (a) Forces P, Q, R act at a point parallel to the sides of a triangle ABC taken in the same order. Show that the magnitude of the resultant force is
√(P² + Q² + R² – 2QR cos A – 2RP cos B – 2PQ cos C) (10)
(b) Find the distance from the cusp of the centroid of the region bounded by the cardioid r = a(1 + cos θ). (10)
Q. No. 3. (a) A particle describes simple harmonic motion in such a way that its velocity and acceleration at a point P are u and f respectively and the corresponding quantities at another point Q are v and g. Find the distance PQ. (10)
(b) Derive the radial and transverse components of velocity and acceleration of a particle. (10)
Q. No. 4. Solve the following differential equations:
(a) dy/dx + y/x = x³ y⁴ (10)
(b) (D² – 5D + 6)y = x³ e²ˣ (10)
Q. No. 5. (a) Solve the differential equation using the method of variation of parameters:
d²y/dx² + y = tan x , -π/2 < x < π/2 (10)
(b) Solve the Euler-Cauchy differential equation x²y” – 3xy’ + 4y = x² ln x. (10)
Q. No. 6. (a) Find the Fourier series of the following function: (10)
f(x) = { -x if -π < x < 0
{ x if 0 < x < π
(b) Solve the initial-boundary value problem: (10)
Q. No. 7. (a) Apply Newton-Raphson method to find the smaller positive root of the equation
x² – 4x + 2 = 0 (10)
(b) Solve the following system of equations by Gauss-Seidel iterative method by
taking the initial approximation as x₁ = 0, x₂ = 0, x₃ = 0:
5x₁ + x₂ – x₃ = 4
x₁ + 4x₂ + 2x₃ = 15
x₁ – 2x₂ + 5x₃ = 12 (10)
Q. No. 8.
(a) Approximate ∫₀¹ dx / (1 + x²) using (10)
(i) Trapezoidal rule with n = 4 (ii) Simpson’s rule with n = 4
Also compare the results with the exact value of the integral.
(b) Apply the improved Euler method to solve the initial-value problem: (10)
y’ = x + y, y(0) = 0
by choosing h = 0.2 and computing y₁, …, y₅.
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