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) Expand a fourier series of f(x) = x², 1 < x < 2 (10)
(b) Find equation of integral surface of the differential equation 2y(z – 3)p +
(2x – z)q = y(2x – 3) which passes through the circle x² + y² = 2x, z = 0. (10)
Q. No. 2 (a) Solve the higher order differential equation y”” + y” = 3x² + 4Sinx – 2Cosx (10)
(b) Solve the initial value problem y” – 8y’ + 15y = 9x e^{2x}, y(0) = 5, y'(0) = 10 (10)
Q. No. 3 (a) Solve the equation 4u_xx + 5u_xy + u_yy + u_x + u_y = 2, also find its canonical
form. (10)
(b) Prove that ∫_{-1}¹ xⁿ P_n(x)dx = (2^{n+1} (n!)²) / (2n+1)!, where P_n(x) = 1 / (2ⁿ n!) dⁿ/dxⁿ (x² – 1)ⁿ is
Legendre polynomial of degree n. (10)
Q. No. 4 (a) Verify the divergence theorem for A⃗ = 4x î – 2y² ĵ + z² k̂ taken over the region
bounded by x² + y² = 4, z = 0 and z = 3. (10)
(b) State and prove Stoke’s theorem. (10)
Q. No. 5 (a) Using the modified Euler’s method, obtain the solution of the differential equation
dy/dt = t + √y = f(t,y)
with initial condition y₀ = 1 at t₀ = 0 for the range 0 ≤ t ≤ 0.6 in steps of 0.2. (10)
(b) Find the real roots of equation 4x + Cosx + 2 = 0 by using Newton Raphson
method, correct to four decimal places. (10)
Q. No. 6 (a) Solve the system of linear equations by Gauss-Seidel iterative method and perform
the first three iterations of
20x + y – 2z = 17
3x + 20y – z = -18
2x – 3y + 20z = 25 (10)
(b) Solve the following Van der Pol’s equation y” – (0.1)(1 – y²)y’ + y = 0, using fourth order Runge-Kutta method for x = 0.2, with the initial values y(0) = 1, y'(0) = 0. (10)
Q. No. 7. (a) Find the law of force for a particle moving in an orbit, r = l / (1 – e cos θ), where l is semi latus rectum and e is eccentricity. (10)
(b) Prove that the speed required to project a particle from a height h to fall a horizontal distance a from the point of projection is at least √(g(√(a² + h²) – h)). (10)
Q. No. 8. (a) Find the radial and transverse components of velocity moving along a curve ax² + by² = 1 at any time t if the polar angle θ = c t². (10)
(b) Find the centroid of the surface formed by the revolution of the cardioide r = a(1 + cos θ) about the initial line. (10)
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