css-applied-mathematics2-2013

applied mathematics

FEDERAL PUBLIC SERVICE COMMISSION

COMPETITIVE EXAMINATION FOR RECRUITMENT TO POSTS IN BS-17 UNDER THE FEDERAL GOVERNMENT, 2013

TIME ALLOWED: THREE HOURS MAXIMUM MARKS: 100

NOTE: (i) Candidate must write Q. No. in the Answer Book in accordance with Q. No. in the Q. Paper.

(ii) Attempt FIVE questions in all by selecting TWO questions from SECTION-A and ONE

question from SECTION-B and TWO questions from SECTION-C ALL questions carry EQUAL marks.

(iii) Extra attempt of any question or any part of the attempted question will not be considered. (iv) Use of Calculator is allowed.

Q.No.1. Solve the following equations:

d3ydy

Sec2x 3 dxdx

2dyx

x3Cosy 0 (b)

ydx

Q.No.2.

(a)

(b)

Q.No.3.

(a) (b)

Q.No.4.

(a) (b)

Find the power series solution of the differential equation (1 x2)y// 2xy/ 2y 0, about the point x= 0. Solve Z(x+y)

(10)(10)

(10)(10)(5)

Z Z

Z(x y) (x2 y2).

y x

Classify the following equations:

2

2Z1 Z2 Zx 0 (i) 22

x x x y(ii)

2

2Z 2Z2 Zx 2xy y 4x2 22

x y y x2

u 2u

, 1 x 1,t 0 Solve:

t x2

u u( 1,t) (1,t)for t

> 0 u(-1, t) = u(1, t);

x x

u(

x, o) = x+1, -1< x <1.

(15)

Highlight the difference between a vector and a tensor. What happens if we

permute the subscripts of a tensor? Transform g

ab

(5)(15)

1 0 0

2 into Cartesian coordinates. r

applied mathematics

Q.No.5.

Q.No.6.

Q.No.7.

Q.No.8.

(10)(a) Workout the Christoffel symbols for the metric tensor g ab a20 0a2sin2

(b) Workout the two dimensional metric tensor for the coordinates p and q given by

(10)

1

p (xy)3

,q 2

x

y

(a) Solve the following system of equations by Jacobi iteration method: (10)

10x + y – 2z = 7.74 x + 12y +3z = 39.66 3x + 4y +15z = 54.8

(b) Solve Sinx = 1 + x3 Using Newton-Raphson method.

(10)(a) Find the root of xex

= 3 by regular falsi method correct to three decimal places.

(10)10

(b)

Evaluate dx

x

2

using (5+5) (10)

01 (i) Trapezoidal rule and (ii) Simpson’s rule.

(a)

Find the real root of the equation Cosx = 3x – 1 correct to seven decimal places (10) by the iterative method.

(b) Use Lagrange’s interpolation formula to find the value of y when x = 10, if the (10)

values

of x and y are given below:

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