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) Evaluate the surface integral ∬_S A⃗ · n̂ dS where A⃗ = z î + x ĵ – 3y²z k̂ and S is the portion of the cylinder x² + y² = 8 lying in the first octant between z = 0 and z = 4. (10)
(b) Prove that ∇(f(r)) = (f'(r) / r) r⃗, where r⃗ = x î + y ĵ + z k̂ and r = |r⃗|. (10)
Q. No. 2. (a) The greatest resultant that two forces can have is of magnitude P and the least is of magnitude Q. Show that, when they act at an angle α, their resultant is of magnitude √(P² cos²(α/2) + Q² sin²(α/2)). (10)
(b) A sphere of weight W and radius a is suspended by a string of length l from a point P and a weight w is also suspended from P by a string sufficiently long for the weight to hang below the sphere. Show that the inclination of the first string to the vertical is sin⁻¹( w a / ((W + w)(a + l)) ). (10)
Q. No. 3. (a) Show that the law of force towards the pole, of a particle describing the curve rⁿ = aⁿ cos(nθ) is given by f = ((n + 1)h²) / (a²ⁿ r²ⁿ⁺³). (10)
(b) The maximum velocity that a particle executing simple harmonic motion of amplitude a attains, is v. If it is disturbed in such a way that its maximum velocity becomes nv. Find the change in the amplitude and the time-period of motion. (10)
Q. No. 4. (a) Define ordinary and singular points of the differential equation a₂(x)y” + a₁(x)y’ + a₀(x)y = 0. When a singular point is said to be regular and irregular? Find regular and irregular singular points of the differential equation (x² – 4)²y” + (x – 2)y’ + y = 0. (10)
(b) Show that J_3/2 = √((2) / (πx)) [ sin x / x – cos x ]. (10)
Q. No. 5. (a) Solve the equation by using method of undetermined coefficients:
y” – y’ + y = 2 cos 3x. (10)
(b) Use the method of Frobenius to find two linear independent series solutions in
powers of x of the DE:
x²y” – (x² + x)y’ + y = 0. (10)
Q. No. 6. (a) Classify general second order partial differential equation (PDE) into elliptic,
parabolic and hyperbolic form. Discuss the nature of the PDE
(1 – x²)u_xx – 2xy u_xy + (1 – y²)u_yy = 0 at each (x, y) ∈ ℝ². (10)
(b) Use the method of separation of variables to find the solution u(x, t) : [0, T] ×
[0, L] → ℝ to the initial/boundary value problem: (10)
u_t(x, t) = u_xx(x, t) for 0 < t ≤ T and 0 ≤ x ≤ L,
u(x, 0) = f(x), for 0 ≤ x ≤ L,
u(0, t) = u(L, t) = 0, for 0 < t ≤ T,
where f : [0, L] → ℝ is a known function.
Q. No. 7. (a) Use Simpson’s 3/8 rule to estimate the integral (10)
∫₁³ (x³ – 2x² + 7x – 5) dx.
By comparing your answer with exact value, find the error.
(b) Solve the system of equations by Jacobi iterative method: (10)
10x + 3y + z = 19, 3x + 10y + 2z = 29, x + 2y + 10z = 35
Q. No. 8. (a) In the following table values of y = x + sin x² are tabulated: (10)
| x | 1.0 | 1.1 | 1.2 | 1.3 | 1.4 | 1.5 | 1.6 |
|——|———|———|———|———|———|———|———|
| f(x) | 1.84147 | 2.03562 | 2.19146 | 2.29290 | 2.32521 | 2.27807 | 2.14935 |
Construct a difference table and estimate f(1.04) and f(1.57).
(b) Use trapezoidal and Simpson’s 1/3 rules to approximate ∫₀^{π/2} sin²(x) dx. Find a (10)
maximum bound for the error in each case. Compare your approximations with
the actual result.
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