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) Let u = [y, z, x] and v = [yz, zx, xy], f = xyz and g = x + y + z. Find div (grad (fg)). (10)
(b) Evaluate ∮_C F⃗ · dr⃗ counter clockwise around the boundary C of the region R by Green’s theorem, where F⃗ = [x² – y², 2x – y], and h is bounded by y = x² and y = 1/4. (10)
Q. No. 2. (a) Three forces P, Q, R, acting at a point, are in equilibrium, and the angle between P and Q is double of the angle between P and R. Prove that R² = Q(Q – P). (10)
(b) Find the centre of mass of a semi-circular lamina of radius a whose density varies as the square of the distance from the centre. (10)
Q. No. 3. (a) A particle moves in such a way that its position vector at time t is r⃗ = a cos(nt) î + b sin(nt) ĵ, where a, b, n are constants and a > b > 0. Show that the path of the particle is an ellipse of semi-major and minor axes a, b respectively, and that the field of force is directed towards the centre of the ellipse. Also find the maximum speed. (10)
(b) An aeroplane is flying with uniform speed v₀ in an arc of a vertical circle of radius a, whose centre is at a height h vertically above a point O of the ground. If a bomb is dropped from the aeroplane when at a height Y and strikes the ground at O, show that Y satisfies the equation Y²(1 + K) – 2Y(h + K) + h² – a² = 0, where K = h² + (a²v₀²) / g². (10)
Q. No. 4. (a) Solve the given initial-value problem. Give the largest interval I over which the solution is defined: x y’ + y = eˣ, y(1) = 2. (10)
(b) Find the general solution of the given higher-order differential equation: y”’ – 4y” – 5y’ = 0. (10)
Q. No. 5. (a) Find two power series solutions of the given differential equation about the ordinary point x = 0: y” – 2xy’ + y = 0. (10)
(b) Find the general solution of the given Bessel’s equation on (0, ∞): x²y” + xy’ + (9x² – 4)y = 0. (10)
Q. No. 6. (a) Find the Fourier series of the given function f(x), which is assumed to have the
period 2π. Show the details of your work.
f(x) = { x, -π < x < 0
{ π – x, 0 < x < π (10)
(b) Find u(x,t) for the string of length L=1 and c²=1 when the initial velocity is zero
and the initial deflection with small k (say, 0.01) is kx(1 – x). (10)
Q. No. 7. (a) Use the Bisection method to determine an approximation to the root of the given
function in the interval [1,2] that is accurate to at least within 10⁻⁴.
f(x) = x³ + 4x² – 10 = 0. (10)
(b) Values for f(x) = x eˣ are given in the following table. Use all the applicable three-
point and five-point formulas to approximate f'(2.0).
| x | 1.8 | 1.9 | 2.0 | 2.1 | 2.2 |
|——|———–|———–|———–|———–|———|
| f(x) | 10.889365 | 12.703199 | 14.778112 | 17.148957 | 19.85503| (10)
Q. No. 8. (a) Use the Modified Euler method to approximate the solution to each of the
following initial-value problem,
y’ = -5y + 5t² + 2t, 0 ≤ t ≤ 1, y(0) = 1/3, with h = 0.1 (10)
(b) Use a fixed-point iteration method to determine a solution accurate to within 10⁻²
for x⁴ – 3x² – 3 = 0 on [1, 2]. Use p₀ = 1. (10)
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