University of Mumbai 2008-4th Sem B.E Electrical and Electronics Engineering Computational Mathematics - Question Paper

Computational Mathematics Sem IV June 2008

N.B. (1) Question No. 1 is compulsory

(2)    Attempt any four questions out of remaining six questions.

(3)    Assume suitable data, wherever required and justily the same.

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1. (a) Fintfme function t(x) whose first difference is 9x^? + 11x + 5.

(b)    Show that the set of functions P₀ <x) 1, P,(x) - x, P,(x) (3x² - 1) are orthogonal over ( - 1, 1).

dy

(c)    Using Taylor series find tho solution of x * x - y, y(2) 2 at x 2*1 correct to four

decimal places.

(d)    Using the method of separation of variables solve :

0 with u(x. 0) 4e~"

(a)    Obtain all possible solution of Laplace oquation in polar co-ordinates,

2 C?*U du    _

r jr + r + = * 0

dr or ao*

(b)    Find the number of students from the lollowmg data who secured marks not more than

Marks 3040 4050 5060 6070 70-B0 No. of students 35 48 70 40 22

(c) Using R-K method of fourth order, find y( 0-2) correct to four decimal places If    B

~ *xy + y², y(0) =1 with two steps.

3. (a) Find half range cosine series for the function 1(x) = (x - 1 )² in ( 0, 1). Hence deuce that 6

^{1 1} 1 2^{2 +} 3^{2 +}.....J

(b) Whnn a train is moving at 30 m/sec, steam is shut off and brakes are applied. The speed of the tram per second after V seconds is given below ;

Time (t) 0 5 10 15 20 25 30 35 40 Speed (V) 30 24 195 16 13-6 11*7 too 8-5 70

Using Simpson's rule determine the distance moved by train in 40 seconds.

(c) A tightly stretched string with fixed end points x 0 and x = L is initially at rest in its equilibrium position. If ii is set vibrating by giving to each of its point a velocity

( a )_{( 0} = 3 (Lx - x*) find y (x. t).

X	0	1	3	4
y	- 12	0	6	12

Also find the value of y at x = 2.

r ^dx

(b)    Evaluate J _4xg by Trapezoidal rule using 11 co-ordinates.

(c)    Find Founer series of the function

f(x> k + x₍ -n < x < 0 s 0 , 0 < x < *

Hence deduce that

n² 1 1 1

T ⁼ ? W ⁺.....

5. (a) Using Gauss-seidal Iteration method solve the system of equations ; 27 x + 6y - z * 85 6x + 15y 2z 72

X + y + 54z = 110 Use three Iterations.

(b) Find hall range sine series for the function f(x) x² - 2. 0 < x < 2. Honce deduce that

\c) With usual notations prove that

(I) (E + 1)82(E-1)m

6. la) Find a real positive root of the equation x³ - 7x + 5 = 0 by using bisection method correct to 3 places of decimal.

(b) Express the function :

f 1 for 0 x < *

[ 0 for x > n

as a Fourier Sine integral. Hence evaluate

1-cosxX

sin (xX) d*

k

(c) A bar AB oi length 10 cm has rts ends A and B Kept at 30 and 100 temperatures respectively until steady state condition is reached. Then the temperature at A is lowered to 20 and that at B to 40 and these temperatures are maintained. Find subsequent temperature distribution in the bar.

7. |a) Find the real root of 3x - cosx - 1 = 0 by Newton-Raphson method correct to fojr decimal places.

<b) Obtain complex form oi Fourier senes of f{x) * e** In ( 0. 2b)

\c) A rectangular ptate with insulated surfaces Is 8 cm wide and so long compared to its width that H may be considered infinite in the length without introducing an eppreciablo error. If

( ⁿ* V

the temperatures along one short edge y 0 is given by 100 sin~J* while the two

long edges x * 0 and x 8 as well as the other short edge are Kept at 0C, find Ihe steady state temperature function u(x. y).

Attachment: university-of-mumbai-2008-b-e-electrical-and-electronics-engineering-51166-1.pdf

Earning:   Approval pending.