CSS 2018 Applied Mathematics Past Paper

Paper details

Examination

CSS

Year

2018

Subject

Applied Mathematics

Paper

Single Paper

Subject type

Optional

Mode

Subjective

Questions recorded from this paper

These 8 question wordings are transcribed from CSS Vista’s structured Applied Mathematics past-paper record for CSS 2018.

1. Q1 · Vector Calculus (10%)(a) If ψ = Sin , then show that ∇2ψ + k 2ψ = 0 . (10) r − yi + xj (b) Calculate the Line Integral ∫c A.dr , where A = , and the curve C is (10) x2 + y2 given by the equations x 2 + y 2 = a 2 and Z = 0 .
2. Q2 · Statics (10%)(a) Forces of magnitude P, 2P, 3P, 4P act respectively along the sides AB, BC, CD, (10) DA of a square ABCD, of sides a, and forces each of magnitude (8 2 ) P act along the diagonals BD, AC. Find the magnitude of the resultant force and distance of its line of action from A. (b) A uniform ladder, of length 70 feet, rests against a vertical wall with which it (10) makes an angle of 450, the coefficient of friction between the ladder and the wall and the ground respectively being and . If a man, whose weight is one half that of the ladder, ascends the ladder, where will he be when the ladder slips?
3. Q3 · Tangential(a) A particle moves in a straight line with an acceleration kv3. If its initial velocity is (10) u, find the velocity and the time spent when the particle has travelled a distance x. (b) Derive the Tangential and Normal components of the velocity and acceleration. (10)
4. Q4 · Cauchy-Euler equations(a) Solve the following Cauchy- Euler Equation (10) d2y dy x2 2 − 2x − 4y = 0. dx dx (b) Convert the following Bernoulli Differential Equation into standard form and (10) then solve. dy  xy  +  = xy 2 . dx  1 − x 2 
5. Q5 · variation of parameters(a) Convert the following Ordinary Differential Equation into standard form and then (10) solve using Method of Variation of Parameters. x 2 y ′′ − 3 xy ′ + 3 y = 2 x 4 e x (b) Check whether the following Ordinary Differential Equation is an Exact Equation (10) or not. If yes, then solve. (3x 2 y + 2) dx + ( x 3 + y) dy = 0
6. Q6 · Fourier series and partial differential equations (20%)(a) Find the Fourier Series of f on the given interval. (10) − 1, − π < x < 0 f ( x) =   1, 0≤ x <π (b) Solve the following Partial Differential Equation subject to the conditions given. (10) 2 ∂ u ∂ 2u a = 2 , 0 < x< L, t > 0, ∂x2 ∂t u (0, t ) = 0, u ( L, t ) = 0, t > 0, ∂u u ( x,0) = f ( x), = g ( x)[at time t = 0], and 0 < x < L. ∂t
7. Q7 · Solution of nonlinear equations by bisection, secant and Newton-Raphson methods(a) Use Newton-Raphson method to find solution accurate to within 10-4 for the non- (10) linear equation. x 3 − 2 x 2 − 5 = 0, I = [1,4] (b) Use Lagrange Interpolating polynomial of degree two to approximate f (8.4), If (10) f (8.1) = 16.94410, f (8.3) = 17.56492, f (8.6) = 18.50515, f (8.7) = 18.82091. 1 dx
8. Q8 · trapezoidal rule(a) Approximate ∫0 1 + x 2 using Trapezoidal rule and Simpson’s rule with n=4. (10) Also compare your results with the exact value of the integral. (b) Use Euler’s method to approximate the solution of the following initial value (10) problem. 1+ y y′ = , 1 ≤ t ≤ 2, with y(1) = 2, h = 0.25 t

Topic-wise Applied Mathematics past-paper analysis.