SPSC 2011 Assistant Engineer / Assistant Executive Engineer (Civil) BPS-17, Engineers Cadre CCE (2011) — Applied Mathematics Paper

Assistant Engineer / Assistant Executive Engineer (Civil) BPS-17, Engineers Cadre CCE (2011) — Applied Mathematics. Combined Competitive Examination for Engineers Cadre · Saturday 26 March 2011, 2:00 pm to 5:00 pm · Max marks 100. Attempt any 05 questions.

1. 1(a) Solve the following differential equations: (12) (i) (y⁴ − 2yx³) dx + (2x²y² − x⁴) dy = 0 (ii) dy/dx − dx/dy = x/y − y/x (b) A copper ball is heated to a temperature of 200°C. Then at time t = 0 it is placed in water which is maintained at a temperature of 40°C. At the end of 3 minutes the temperature of the ball is reduced to 65°C. Find the time at which the temperature of the ball is reduced to 25°C. (08)

2. 2(a) Solve the following differential equations: (12) (i) x³y''' + 2x²y'' + 2y = 10(x + 1/x) (ii) y'' − 3y' + 2y = (1 + e⁻ˣ)⁻¹ (b) Find a solution of the equation ∂²z/∂x² = a²z given that ∂z/∂x = a sin y and ∂z/∂y = 0 when x = 0. (08)

3. 3(a) Solve the wave equation: ∂²u/∂t² = 16 ∂²u/∂y². (12) (b) Define the convergence and divergence of an infinite series. Use appropriate tests to check the convergence of the following series: (08) (i) Σ (n = 1 to ∞) (n!)² xⁿ / (2n)! (ii) Σ (n = 1 to ∞) (n!/nⁿ)ⁿ

4. 4(a) State "Convolution Property". Use this property to compute L⁻¹ [ s² / ((s² + 9)(s² + 4)) ]. (08) (b) Evaluate the following: (12) (i) L⁻¹ [ (s + 1) / (s² + 4s + 5) ] (ii) L⁻¹ [ (s² + 1) / ((s − 1)²(s + 6)) ]

5. 5(a) Evaluate the integral from (0, 2) to (2, 6) of (3x + 2y) dx + (2y − 3x) dy along the curve y = x² + 2. (05) (b) State Cauchy's Integral formula. Use this formula to evaluate ∮_C (z³ + 1)/(z − 2)³ dz, where C is the circle |z| = 5. (05) (c) State the "Residue Theorem". Hence evaluate the following integrals, where C contains all poles: (10) (i) ∮_C (z + 1) / ((z − 1)²(z − 3)) dz (ii) ∮_C 1 / ((z + 1)²(z + 4)²) dz

6. 6(a) Define derivable function. Also discuss the derivability of the function f(x) = x if 0 ≤ x ≤ 1, and f(x) = 2x − 1 if 1 < x ≤ 2, at x = 1. (05) (b) The height s (in feet) of a rocket t seconds after its launching is given by s = −t³ + 96t² + 195t + 10 (t ≥ 0). Find: (i) the velocity of the rocket when t = 0, 30, 50 seconds; (ii) the maximum height attained by the rocket. (05) (c) Evaluate any two of the following integrals: (05) (i) ∫ { n^(sin x) + (sin x)ⁿ } cos x dx (ii) ∫ sin⁻¹x dx (iii) ∫ (3x + 1)/(2x² − 2x + 3) dx (d) Evaluate the integral from 0 to π/4 of ln(1 + tan x) dx by using the properties of definite integrals. (05)

7. 7(a) Define Homogeneous function. Also show that if u = sin⁻¹( (x² + y²)/(x + y) ), then x ∂u/∂x + y ∂u/∂y = tan u. (05) (b) Apply Taylor's theorem to prove that a^(x+h) = aˣ { 1 + h log a + (h²/2!)(log a)² + (h³/3!)(log a)³ + … }. (05) (c) Evaluate any two of the following: (05) (i) ∫ (x = 0 to 1) ∫ (y = 0 to x) (x² + 4xy) dy dx (ii) ∭_S 3(x²y + y²z) dV, where S is bounded by the planes x = 1, x = 3, y = −1, y = 1, z = 2 and z = 4 (iii) ∫ (0 to ∞) x^(1/4) e^(−√x) dx, by using the definition of the Gamma function (d) Define reduction formula, and find the reduction formula for ∫ sinⁿx dx. (05)