Nalanda Open University 2009 B.Sc Mathematics Bachelor of Science ( Hors), Part-III Term End , -VIII - Question Paper

Niiliind;! Open University

Bachelor of Science (Mathematics Honours), Part-HI Tenn End Examination, 2009 Paper-VlLL (Mathematics)

Iiine: 3.00 Hrs. Full ]V

Answer any Five .Questions. _ULyestions carry eqiid rnftfks:

1. (a) If f (x) is a polynomial of degree n in x, theii prove that Aⁿ f_n(x) is a constant.

(b) By means of dividecfc.differences formula, find the values of f(2), f(8) and (15) following table.

larks: 75

from the

X

4

5

7

10

11

13

f(x)

48

100

294

900

1210

2028

2. (a) Obtain the estimates of the missing terms in the following table

X

2.0

2.1

2.2

2.3

2.4

2.5

2.6

y*

0.135

?

0.111

0.100

?

0.080

0.074

(b) EstablishLagrange!s intie r-po 1 ation.,formula for unequa!lilter.y'als'-: 3. (a) Using the following table, find f (5).

X

0

2

3

4

5

7

9

y=ft

4

26

58

112

?

466

922

(b) Obtain the derivatives for the Stirling interpolation formu 4. (a) Derive Simpson's one-third rule for numerical evaluation

(b) Calculate J &'^SirLXdx correct to 4 decimal points.-.,,

0,

il (a) Proyeth atthe-¹ line af^: d.iffe refi c.e^:equ atib n of order %'

P₀(x)f(x+n)tPi (x)f(x+n-1)+.....+P_n(x)f(x)=K(x)

over a set of consecutive integral values of x has one and (b) Solve the equation

^ux+2 - ^3ux+l + ^2ux = ²*

5:, Explain Modified form of Euler's method. Hence find a sc

dy r-- =x + | Vy | = f(x, y)

:4X:

with initial conditions y=l at x = 0 for the range 0 1 x 1 D

7.    Solve the equation

dy

x + y

dx

witli-initial.condition y(0)=l by R.unge"K<utta rule, from x=

8.    (a) Solve the system of eqtions by Gauss's elimination met

';xj+2x2+x3 = 8 '2xj+3x2+4x₃ = 20 4xj+3x2+2x₃ = 16 (b) Explain Gauss-Seidel. Meth.od...

a.

of integrals.

only one solution.

>lution of the equation

.6 in steps of 0.2:

=0 to x0.4 where h=0.1.. 10 d