CSS 2017 Applied Mathematics Past Paper

Paper details

Examination

CSS

Year

2017

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 2017.

1. Q1 · Vector Calculus (10%)(a) Suppose r(t) = (et cost) iˆ + (et sint) ĵ. Show that the angle between r and (10) dt 2 never changes. What is the angle? (b) Using divergence theorem of Gauss, Evaluate ∫∫ x2dxdy +x2ydzdx +x2zdxdy (10) where S is the closed surface consisting of the cylinder x2 + y2 = a2, (0 ≤ z ≤ b) and the circular disks z = 0 and z = b, (x2 + y2 ≤ a2).
2. Q2 · Statics (10%)(a) A 100 Kg wooden crate rests on a wooden ramp with an adjustable angle of (10) inclination. Draw a free body diagram of the crate. If the angle of the ramp is set to 100, find the perpendicular and parallel components of the crate’s weight to the ramp. Also find the static friction force between the crate and the ramp. At what angle will the crate just begin to slip? (coefficient of static friction s between wood block and wood surface is 0.28) (b) A ladder having a uniform density and a mass m rests against a frictionless (10) vertical wall at an angle of 600. The lower end rests on a flat surface where the coefficient of static friction is 0.40. A person of mass M=2m attempts to climb the ladder. What fraction of the length L of the ladder will the person have reached when the ladder begins to slip?
3. Q3 · simple harmonic motion(a) A particle moving in a straight line starts with a velocity u and has acceleration (10) v3, where v is the velocity of the particle at time t. Find the velocity and the time as functions of the distance travelled by the particle. (b) A particle describing simple harmonic motion has velocities 5 ft/sec and 4 ft/sec (10) when its distance from the centre are 12 ft and 13 ft respectively. Find the time- period of motion.
4. Q7 · Equations of first order(a) Use bisection and false position methods to locate the root of f(x) = x 10 -1, (10) between 0 and 1.3. Which of the two methods is better and why? (b) Use Lagrange interpolation polynomial of the first and second order to evaluate (10) f(x) at x=15 for the following data; x: 0 20 40 f(x): 3.850 0.800 0.212 0.800 0
5. Q4 · Ordinary differential equations (20%)(a) Check for the exactness and solve the following ordinary differential equation. (10) (2xy +y –tany)dx + (x2 –xtan2y + sec2y)dy = 0 (b) Solve the following second order differential equation: (10) d2 y + 4x = sec 2x dx 2
6. Q6 · solution in series(a) Compute the Fourier series for the function x2 on the interval 0<x<L, using as a (10) basis of function with boundary conditions u/ (0) =0 and u/ (L) = 0. Sketch the partial sums of the series for 1, 2, 3 terms. (b) Find a solution to the following partial differential equation that will also satisfy (10) the boundary conditions. u  2u  k 2 , u(x,0) = f(x) , u(0,t) = 0, u(L,t) = 0 t x
7. Q5 · boundary value problems(a) Solve the initial value problem. (10) d2 y = 2-6x, y/(0) = 4, y(0) = 1 dx 2 (b) Solve the following Boundary Value Problem. (10) y/ + 4y = 0, y(0) = -2, y(2  ) = -2 How many solutions do you get for this problem?
8. Q8 · trapezoidal rule(a) Use Trapezoidal Rule with 3 segments to evaluate. (10) 0.8  2x  x e dx  Also find the true solution and the percentage error. (b) Consider the function f(x) = xex. Obtain approximations to f/(2) with h = 0.5 (10) using forward, backward and central difference formulas.

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