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) Prove that ∇ · (rⁿ r⃗) = (n + 3)rⁿ, where r⃗ = x î + y ĵ + z k̂.
(b) If a⃗ × (b⃗ × c⃗) = (a⃗ × b⃗) × c⃗, then prove that a⃗ and c⃗ are parallel.
Find the area of the region that is enclosed between the curves y = x² and y = x + 6.
Q. No. 2 (a) Find the tangential and normal components of acceleration of a point describing
the ellipse x²/a² + y²/b² = 1 with uniform speed V, when the particle is at (0, b).
(b) Find the solution of initial value problem by separation of variables:
√(1 – y²) dx – √(1 – x²) dy = 0 (10)
Q. No. 3 (a) Find the general solution of the given differential equation by variation of parameters:
3y” – 6y’ + 6y = eˣ (10)
(b) Find the power series solution of (x + 1)y” + xy’ – y = 0.
Q. No. 4 (a) Forces 2BC, CA, AB act along the sides of a triangle ABC. Show that their resultant is 0.
Where D bisects BC and E is a point on CE such that CE = 1/3 CA. (10)
(b) Find the center of mass of the surface generated by the revolution of the arc of the parabola lying between the vertex and the latus rectum, about the x-axis. (10)
Q. No. 5 (a) Obtain the Fourier series over the indicated interval for the given function:
f(x) = { 3π + 2x, -π < x < 0
{ 2x, 0 < x < π
(b) Solve the boundary value problem:
u_xx + u_yy = 0, 0 < x < a, 0 < y < b
u(x, 0) = 0, u(a, y) = 0, 0 ≤ y ≤ b
u(0, y) = 0, u(x, b) = f(x), 0 ≤ x ≤ a
Q. No. 6 (a) Use Newton’s Raphson method to find the solution accurate to within 10⁻⁴
(corrected up to four decimal places) for the given problem:
x – cos x = 0, [0, π/2]
(b) Solve the system of linear equations using Gauss Seidel method (with three digit rounding arithmetic):
3x₁ + 4x₂ – x₃ = 8
5x₁ + 2x₂ + 2x₃ = 3
-x₁ + x₂ – 3x₃ = -8
Q. No. 7 (a) Use Euler’s method to approximate the solution of the initial value problem.
y’ = 1 + y/x, 1 ≤ x ≤ 2, y(1) = 2, with h = 0.25
(b) Using Green’s theorem, evaluate ∮ F(r) · dr counter clock wise around the (10)
boundary curve C of the region R, where F = [1/2·xy⁴, 1/2·x²·y], the rectangle
with vertices (0, 0), (3, 0), (3, 2), (0, 2). Q. No. 8 (a) Evaluate the Integral ∫13 (1 / x²) dx. Using Trapezoidal Rule for five points (10)
(corrected upto two decimal places). (b) Find the D’Alembert solution of the wave equation uₓₓ = (1 / c²)·uₜₜ subject to the (10)
Cauchy Initial conditions u(x, 0) = f(x), uₜ(x, 0) = g(x).
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