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 ∇²rⁿ = n(n + 1)rⁿ⁻² (10)
(b) Evaluate ∬_S A⃗ · n̂ ds where A⃗ = 18z î – 12ĵ + 3y k̂ and S is that part of the plane 2x + 3y + 6z = 12 which is located in the 1ˢᵗ octant. (10)
Q. No. 2. A particle P of mass m slides down a frictionless inclined plane AB of an angle α with the horizontal. If it starts from rest at the top A, find (a) the acceleration (b) the velocity and (c) the distance travelled after time t. (20)
Q. No. 3. (a) Discuss the motion of a particle moving in a straight line if it starts from rest at a distance ‘a’ from a point O and moves with an acceleration equal to k times its distance from O. (10)
(b) Find radial and transversal components of velocity and acceleration. (10)
Q. No. 4. (a) Solve d²y/dx² + y = cosec x (10)
(b) Solve dy + ((y – sin x) / x) dx = 0 (10)
Q. No. 5. (a) Solve the initial value problem
x(2 + x) dy/dx + 2(1 + x)y = 1 + 3x², y(-1) = 1 (10)
(b) Find the general solution of the equation
(D³ – 2D + 1)y = 2x³ – 3x² + 4x + 5 (10)
Q. No. 6. (a) Find the Fourier series of f: (10)
f(x) = { x, 0 < x < 1
{ 0, 1 < x < 2
(b) Solve the boundary value problem ∂²u/∂x² = (1 / k) ∂u/∂t (10)
Satisfying u(0, t) = u(l, t) = 0 and u(x, 0) = lx – x²
Q. No. 7. (a) By using regular Falsi method, solve
Log x – Cos x = 0 (10)
(b) Find the value of f(7.5) by using Newton Gregory Backward Difference
Interpolation formula. (10)
X: 5, 6.1, 6.9, 8, 8.6
f(x): 3.49, 4.82, 5.96, 7.5, 8.2
Q. No. 8. (a) Applying the Taylor series method, compute (10)
∫₀ˣ (Sin t / t) dt for x = 0 (0.1) 1
(b) Use fourth order RK method to solve (10)
dy/dt = t + y ; y(0) = 1 from t = 0 to t = 0.4 taking h = 0.4
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