SPSC 2014 Assistant Engineer (Civil) BPS-17, Public Health Engineering Department (2014) — Applied Mathematics Paper

Assistant Engineer (Civil) BPS-17, Public Health Engineering Department (2014) — Applied Mathematics. Competitive examination for the post of Assistant Engineer (Civil) BPS-17 in PHED, Government of Sindh · Monday 27 January 2014, 2:00 pm to 5:00 pm · Max marks 100. Attempt any five questions. Formulas are written out in plain text.

1. 1(a) A helicopter rises vertically at the rate of 10 feet per second. There is a marker M 80 feet from the station S where the helicopter lifts off. At what rate is the distance between the helicopter and the marker changing when the helicopter is 192 feet high? (10) (b) Show that dⁿ/dxⁿ (ln x / x) = ((−1)ⁿ n! / xⁿ⁺¹) [ln x − 1 − 1/2 − 1/3 − … − 1/n]. (05) (c) Determine the vector function which has 2cos 2t i + 2sin 2t j + 4k as its derivative and 2i − 3j + k as its value at t = 0. (05)

2. 2(a) The temperature of a machine, when it is first shut down after operating, is 220°C and the temperature of the surrounding air is 30°C. After 20 minutes the temperature of the machine is 160°C. Find the temperature of the machine 30 minutes after it is shut down. (10) (b) Solve the following differential equations: (10) (i) dr/dθ + r tan θ = cos²θ; r(π/4) = 1 (ii) x + yp² = p(1 + xy), where p = dy/dx

3. 3(a) Find the real and imaginary parts of the complex function f(z) = z³ − 2z² + 5z − 6 and, using the CR-equations, prove that this function is analytic. (10) (b) Find the real part of an analytic function whose imaginary part is v = 3x²y − y³ + 4xy − 4y. (10)

4. 4(a) Evaluate the integral from (0, 1) to (2, 4) of (x² + y) dx + (y² − x) dy along the curve y = 2x + 2. (08) (The upper limit is printed as (2, 4); on y = 2x + 2 the point at x = 2 is (2, 6).) (b) State Cauchy's Integral formula. Use this formula to evaluate ∮_C eᶻ/(z − 2)³ dz, where C is the circle |z| = 3. (06) (c) State the "Residue Theorem". Hence evaluate ∮_C 1/((z − 1)²(z − 4)²) dz, where C contains all poles. (06)

5. 5(a) Solve the following differential equations: (12) (i) (2x³ − 6x²y + 3xy) dx − (2x³ − 3x²y + y³) dy = 0 (ii) dy/dx + y = xy³ (b) A cup of coffee was initially boiling at 120°C; it was placed in air at 20°C. After 10 minutes the temperature was 80°C. What will be the temperature of the coffee after another 10 minutes? (08)

6. 6(a) Solve ∂²z/∂x² − 2 ∂z/∂x + ∂z/∂y = 0 by the separation of variables method. (12) (b) Define convergent and divergent sequences with examples. Use an appropriate test to check the convergence of the following series: (08) (i) Σ (n = 1 to ∞) 1/(n(n + 1)(n + 2)) (ii) Σ (n = 1 to ∞) 2ⁿ/(n(n + 2))

7. 7(a) State Homogeneous function. If u = tan⁻¹((x³ + y³)/(x − y)), show that x ∂u/∂x + y ∂u/∂y = sin 2u. (10) (b) Apply Maclaurin's series to prove that tan⁻¹x = x − x³/3 + x⁵/5 − x⁷/7 + …, and also find the value of π. (10)