CSS 2016 Applied Mathematics Past Paper

Paper details

Examination

CSS

Year

2016

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 2016.

1. Q1 · Green’s, Stokes’ and Gauss theoremsf (r) r⃑ 2 (10) (a) Prove that . r = r f ( r ) + f ( r ) (b) Verify Stokes’ theorem for ⃑ = (2x – y) i – y z 2 j – y 2 z k , where S is (10) the upper half surface of the sphere x 2 + y 2 + z 2 = 1 and C is its boundary .
2. Q2 · parallel forces and couples(a) Forces P, Q, R act at a point parallel to the sides of a triangle ABC taken in the (10) same order. Show that the magnitude of the resultant force is P + Q + R – 2 Q R cos A – 2 R P cos B – 2 P Q cos C (b) Find the distance from the cusp of the centroid of the region bounded by the (10) cardioid r = a (1 + cos ).
3. Q3 · radial and transverse components of velocity and acceleration(a) A particle describes simple harmonic motion in such a way that its velocity and (10) acceleration at a point P are u and f respectively and the corresponding quantities at another point Q are v and g. Find the distance PQ. (b) Derive the radial and transverse components of velocity and acceleration of a particle. (10)
4. Q4 · Ordinary differential equations (20%)Solve the following differential equations: dy y 3 4 (10) (a) + = x y dx x 2 3 2x (b) (D – 5D + 6) y = x e (10)
5. Q5 · variation of parameters(a) Solve the differential equation using the method of variation of parameters (10) d y   + y = tan x , –2<x<2 d x2 (b) Solve the Euler – Cauchy differential equation x2 y – 3 x y + 4y = x2 ln x. (10)
6. Q6 · boundary value problems(a) Find the Fourier series of the following function: (10)  – x if –  < x < 0 f(x) =   x if 0 < x <  (b) Solve the initial - boundary value problem: (10)
7. Q7 · the fixed- point iterative method(a) Apply Newton – Raphson method to find the smaller positive root of the equation (10) x2 – 4x + 2 = 0 (b) Solve the following system of equations by Gauss – Seidel iterative method by (10) taking the initial approximation as x1 = 0, x2 = 0, x3 = 0: 5x1 + x2 – x3 = 4 x1 + 4x2 + 2x3 = 15 x1 – 2x2 + 5x3 = 12
8. Q8 · trapezoidal ruledx (a) Approximate  1 + x2 using (10) (i) Trapezoidal rule with n = 4 (ii) Simpson’s rule with n = 4 Also compare the results with the exact value of the integral. (b) Apply the improved Euler method to solve the initial – value problem: (10) y = x + y, y (0) = 0 by choosing h = 0.2 and computing y1, ..., y5.

Topic-wise Applied Mathematics past-paper analysis.