Federal Public Service Commission (FPSC)
Competitive Examination for Recruitment to BPS-17 Posts under the Federal Government
Time Allowed: 3 Hours
APPLIED MATHEMATICS, PAPER-I
MAXIMUM MARKS: 100 Marks
Attempt FIVE questions in all by selecting at least TWO Questions From SECTION-A and THREE Questions from SECTION-B, All questions carry EQUAL marks.
(iii)Use of Scientific Calculator is Allowed.
SECTION-A
Q.1. Explain the following giving examples and supported by figures: (5+5+5+5)
(a) Gradient
(b) Divergence
(c) Curl
(d) Curvilinear Coordinates
Q.2. Given that A, B, C are vectors having components along axis. Prove that: (10+10)
(a)
B × C =
| i j k |
| Bx By Bz |
| Cx Cy Cz |
(b)
A × (B × C) = AxBxCx (i × k) + AyBxCy (j × k)
Q.3. (a) State and prove Stokes’ Theorem. (10)
(b) Given that
V = 4y i + x j + 2z k,
find
∬ (∇ × V) • n dσ
over the hemisphere
x² + y² + z² = a², z ≥ 0. (10)
SECTION-B
Q.4. Discuss the following systems supported by figures/diagrams:
(a)
• Equilibrium of a system of coplanar forces. (5)
• Centre of mass of right circular solid cone of height h. (5)
(b) Centre of gravity of a rigid body of any shape. (10)
Q.5. (a) What is Simple Harmonic Motion? Discuss it in detail using derivatives
with respect to time. (10)
(b) Describe the Simple Harmonic Motion of a pendulum and calculate the
time period of the motion. (10)
Q.6. (a) Derive expression for the following:
• Moment of inertia (5)
• Product of inertia (5)
(b) Calculate the moment of inertia of a solid sphere of mass
m = 37 and radius a = 15.
Derive the general expression. (10)
Q.7. (a) Explain Kepler’s Laws. (10)
(b) What is Impulsive Motion? Derive its equation. (10)
Q.8. (a) Define Work, Torque, Power and Energy. (10)
(b) A cricket ball is thrown vertically upwards. It attains the maximum
height h after t seconds. Calculate its: (10)
• Velocity of projection in vertically upward direction.
• Acceleration when it returns to the point of projection.
APPLIED MATHEMATICS, PAPER-II
MAXIMUM MARKS: 100 Marks
Attempt FIVE questions in all by selecting at least TWO Questions From SECTION-A, ONE Question from SECTION-B, and TWO Questions from SECTION-C. All questions carry EQUAL marks.
SECTION-A
Q.1. Solve the following equations:
(a)
d²y/dx² + 5(dy/dx) + 6y = x (10)
(b)
d²y/dx² + 5y/x = eˣ (10)
Q.2. (a) Derive Cauchy–Riemann Partial Differential Equations. (10)
(b) Derive Laplace Equation. (10)
Q.3. Solve:
(a)
(∂²/∂x² + 2∂²/∂x∂y + 3∂²/∂y²)u = 4e³ʸ (10)
(b)
uʺ + 6uʹ + 9 = 0,
Given that
u(0) = 2
uʹ(0) = 0 (10)
SECTION-B
Q.4. (a) Discuss the following supported by examples:
• Tensor (5)
• εijk εlmk (5)
• Scalar fields for a continuously differentiable function
f = f(x, y, z) (5)
(b) Can we call a vector as Tensor? Discuss.
What is the difference between a vector and a tensor?
What happens if we permute the subscripts of a tensor? (5)
Q.5. (a) Discuss the simplest and efficient method of finding the inverse
of a square matrix aij of order 3 × 3. (10)
(b) Apply any efficient method to compute the inverse of the following matrix A:
| 25 2 1 |
A = | 2 10 1 | (10)
| 1 1 4 |
SECTION-C
Q.6. (a) Develop Gauss–Seidel Iterative Method for solving a linear system
of equations A x = b, where A is the coefficient matrix. (10)
(b) Apply Gauss–Seidel Iterative Method to solve the following equations: (10)
25X₁ + 2X₂ + X₃ = 69
2X₁ + 10X₂ + X₃ = 63
X₁ + 2X₂ + X₃ = 43
Q.7. (a) Derive Simpson’s Rule for finding the integral of a function
f(x) from x = a to x = b for n = 6 subintervals (steps). (10)
(b) Apply Simpson’s Rule for n = 6 to evaluate: (10)
1
∫ dx
0
where
f(x) = 1/(1 + x²).
Q.8. (a) Derive Lagrange Interpolation Formula for 4 points. (10)
(b) A curve passes through the following points:
(0,1), (1,2), (2,5), (3,10).
Apply the Lagrange Formula to interpolate the polynomial. (10)
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