CSS 2022 Applied Mathematics Past Paper

Paper details

Examination

CSS

Year

2022

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 2022.

1. Q1 · Green’s, Stokes’ and Gauss theorems(a) Let u=[y, z, x] and v=[yz, zx, xy], f = xyz and g = x+ y+ z. Find div (grad ( fg)). (10) (b) Evaluate . counter clockwise around the boundary C of the region R by (10) Green’s theorem, where = ,− , h + = 1/4
2. Q3 · Statics (10%)(a) A particle moves in such a way that its position vector at time t is (10) = cos $ i + & sin $ j, Where a, b, n are constants and a>b>0. Show that the path of the particle is an ellipse of semi-major and minor axes a, b respectively, and that the field of force is directed towards the centre of the ellipse. Also find the maximum speed. (b) An aeroplane is flying with uniform speed v0 in an arc of a vertical circle of radius (10) a, whose centre is at a height h vertically above a point O of the ground . If a bomb is dropped from the aeroplane when at a height Y and strikes the ground at O, show that Y satisfies the equation )* + * − 2h) + ) h − = 0, -./ Where ) = h + . 01 /
3. Q2 · equilibrium of forces in three dimensions(a) Three forces P, Q, R, acting at a point, are in equilibrium, and the angle between P (10) and Q is double of the angle between P and R. Prove that = − . (b) Find the centre of mass of a semi-circular lamina of radius a whose density varies (10) as the square of the distance from the centre.
4. Q5 · solution in series(a) Find two power series solutions of the given differential equation about the (10) ordinary point x=0. − 2 2 + = 0. (b) Find the general solution of the given Bessel’s equation on (0, ∞). (10) + 2+ 9 −4 =0
5. Q6 · convergence of the Fourier series(a) Find the Fourier series of the given function f(x), which is assumed to have the (10) period 26. Show the details of your work. , −6 < <0 7 =8 6− , 0< <6 (b) Find u(x,t) for the string of length L=1 and c2=1 when the initial velocity is zero (10) and the initial deflection with small k (say, 0.01) is : 1 − .
6. Q7 · Numerical Methods (30%)(a) Use the Bisection method to determine an approximation to the root of the given (10) function in the interval [1,2] that is accurate to at least within 10-4. 7 = ; + 4 − 10 = 0. (b) Values for f (x) = xex are given in the following table. Use all the applicable three- (10) point and five-point formulas to approximate 7 2 (2.0). x 1.8 1.9 2.0 2.1 2.2 f(x) 10.889365 12.703199 14.778112 17.148957 19.85503
7. Q4 · Numerical solution of an ordinary differential equation(a) Solve the given initial-value problem. Give the largest interval I over which the (10) solution is defined. xy’+y = ex, y(1) = 2. (b) Find the general solution of the given higher-order differential equation. (10) − 4 22 − 5 2 = 0
8. Q8 · Euler and modified Euler methods(a) Use the Modified Euler method to approximate the solution to each of the (10) following initial-value problem, = −5 + 5 + 2 , 0 ≤ ≤ 1, 0 = , > h h = 0.1 (b) Use a fixed-point iteration method to determine a solution accurate to within 10-2 (10) for x4 -3x2 −3 = 0 on [1, 2]. Use p0 = 1.

Topic-wise Applied Mathematics past-paper analysis.