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 ∇ f(r) = (f'(r) / r) r⃗, and evaluate U for ∇ U = 2r⁴ r⃗. (8)
(b) If A⃗ and B⃗ are differentiable vector functions of position (x, y, z), find ∇ · (A⃗ × B⃗). (7)
(c) If A⃗ = x²yz î – 2xz³ ĵ + xz² k̂ and B⃗ = 2z î + y ĵ – x² k̂, find ∂²/∂x∂y (A⃗ × B⃗). (5)
Q. No. 2 (a) What is the maximum range possible for a projectile fired from a cannon having (10)
muzzle velocity 1 mile/sec? What is the height reached in this case?
(b) Find the radial and transverse components of the velocity of the particle moving (10)
along the curve ax² + by² = 1 at any time t, if the polar angle θ = c t².
Q. No. 3 (a) Verify that the linear differential equation (5)
(1 + x²) dy/dx + 2xy = e^(x²)
can be solved by setting y = v e^(x²). Find the equation satisfied by v.
(b) Explain how a first-order linear differential equation can be used to model (5)
pollution accumulation in a lake with constant inflow and outflow rates.
(c) Find the solution of the following differential equation (10)
y”’ + 8y = 2x – 5 + 8e^(-2x) ; with the conditions
y(0) = -5, y'(0) = 3, y”(0) = -4
Q. No. 4 (a) Find the general solution of the following differential equation on interval (0, ∞) (10)
16x²y” + 16xy’ + (16x² – 1)y = 0
(b) Solve the following PDEs using transformation (10)
u_t + u_x + u = 0 ; for x > 0, t > 0,
u(0, t) = sin(t); u(x, 0) = 0.
Q. No. 5 (a) Model the one-dimensional wave equation. (8)
(b) A stretched string of length L is lying along x-axis and is fixed at both ends x = 0 (12)
and x = L. Find the deflection u(x, t) of the string at any time t, if initial
displacement u(x, 0) = f(x) = Lx – x² and the initial velocity is
u_t(x, 0) = g(x) = 4.
Q. No. 6 (a) Find the Fourier series of the function (10)
f(x) = (1 – |x|/π) H(1 – |x|/π),
Where H(x) is the Heaviside unit step function defined by
H(x) = { 1, for x > 0
{ 0, for x < 0
(b) Evaluate the integral I = ∫₀¹ (1 / (1 + x⁴)) dx by using
(i) Trapezoidal rule
(ii) Simpson’s 1/3 rule by taking h = 1/4. (10)
Q. No. 7. (a) Use the Modified Euler method to solve the IVP:
y’ = y – x² + 1 ; y(0) = 0.5
for 0 ≤ x ≤ 2 with step size h = 0.5.
Compare with the exact solution. (10)
(b) Implement appropriate technique to solve the following linear system:
3x₁ – x₂ + x₃ = 2
-x₁ + 3x₂ – x₃ = 7
x₁ + x₂ – 3x₃ = -7 (10)
Q. No. 8. (a) Set up Newton’s scheme of iteration for finding the square root of a (10)
positive number N and evaluate √14.
(b) Find the first four iterations of the equation f(x) = x – 0.8 – 0.2 sin x in the (10)
interval [0, π/2] using Newton-Raphson Method.
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