APPLIED MATH, PAPER-II CSS PAST PAPER 2010

FEDERAL PUBLIC SERVICE COMMISSION

COMPETITIVE EXAMINATION FOR

RECRUITMENT TO POSTS IN BPS-17 UNDER

THE FEDERAL GOVERNMENT, 2010

APPLIED MATH, PAPER-II

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TIME ALLOWED: 3 HOURS MAXIMUM MARKS:100

NOTE:

(i) Attempt FIVE question in all by selecting at least TWO questions from SECTION–A,

ONE question from SECTION–B and TWO questions from SECTION–C. All

questions carry EQUAL marks.

(ii) Use of Scientific Calculator is allowed.

SECTION – A

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Q.1. Solve the following equations:

(a) d2y/dx2 + 5 dy/dx + 6y = x (10)

(b) d2y/dx2 + 5 y x = ex (10)

Q.2. (a) Derive Cauchy Rieman partial differential equations. (10)

(b) Derive Lapace Equation. (10)

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Q.3. Solve:

(a) (2 / x2  2 / xy  2 / y 2 ) u  4 e3y (10)

(b) u” + 6u’ + 9=0; Given that u(0)=2 and u’(0)=0. (10)

SECTION – B

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Q.4. (a) Discuss the following supported by examples:

 Tensor, (5)

 ijk  lmk  (5)

 Scaler Fields for a continuously differentiable function f=f(x,y,z) (5)

(b) Can we call a vector as Tensor, discuss.

What is difference between a vector and a tensor?

What happens if we permute the subscripts of a tensor? (5)

Q.5. (a) Discuss the simplest and efficient method of finding the inverse of a square matrix aij

of order 3x3. (10)

(b) Apply any efficient method to compute the inverse of the following matrix A: (10)

A =

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1 1 4

2 10 1

25 2 1

SECTION – C

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Q.6. (a) Develop Gauss Siedal iterative Method for solving a linear system of equations A x = b,

where A is the coefficient matrix. (10)

(b) Apply Gauss Siedal iterative Method to solve the following equations: (10)

25X1 + 2X2 + X3 = 69

2X1 + 10X2 + X3 = 63

X1 + 2X2 + X3 = 43

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Q.7. (a) Derive Simpson’s Rule for finding out the integral of a function f(x) from limits x=a to x=b for

n=6 subintervals (i.e. steps). (10)

(b) Apply Simpson’s Rule for n=6 to evaluate: (10)

  

1

0

f(x)dx where f(x) 1/(1 x2).

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Q.8. (a) Derive Lagrange Interpolation Formula for 4 points: (10)

(b) A curve passes through the following points:

(0,1),(1,2),(2,5),(3,10). Apply this Lagrange Formula to interpolate the polynomial. (10)

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