LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
B.Sc. DEGREE EXAMINATION – MATHEMATICS
SIXTH SEMESTER – APRIL 2012
MT 6605 - NUMERICAL METHODS
Date : 20-04-2012
Time : 1:00 - 4:00
Dept. No.
Max. : 100 Marks
PART – A
Answer ALL questions:
(10 X 2 = 20 marks)
1. Explain Cramer’s rule.
2. Distinguish between Gauss Elimination and Gauss Seidel methods.
3. State the condition for convergence in Newton Raphson method.
4. What do you mean by transcendental equation?
5. Write Newton forward interpolation formula.
6. Write any two properties of divided differences.
7. Write Stirling’s formula using central difference notation.
8. Write the derivatives using Newton’s forward difference formula.
9. Define numerical integration.
10. Write Simpson’s 1/3rd and 3/8th rule.
PART – B
Answer any FIVE questions:
(5 X 8 = 40 marks)
11. Solve by Gauss elimination method: 2 x  y  4 z  12;8 x  3 y  2 z  20;4 x  11y  z  33.
12. Find the real root of x 3  9 x  1  0 correct to three significant figures using Regula falsi method.
13. Write a C program to interpolate using Newton’s forward interpolation formula.
14. Use Lagrange’s formula to find the value of y at x = 6 from the following data: x = 3, 7, 9, 10 and
the corresponding value of y = 168, 120, 72, 63.
15. Using the following data, find f’(5) by Newton’s divided difference formula:
x:
0
2
3
4
7
9
f ( x) :
4
26
58
112
466
922
16. Derive Laplace Everett’s formula.
2
17. Apply Simpson’s rule to evaluate
dx
 (1  x
0
3
)
to two decimal places by dividing the range into 4
equal parts.
18. Solve
dy
 1  y with the initial condition x = 0, y = 0 using Euler’s modified formula.
dx
PART – C
Answer any TWO questions:
(2 X 20 = 40 marks)
19. (a) Solve by Gauss Seidel method: 27 x  6 y  z  85;6 x  15 y  2 z  72; x  y  54 z  110.
(b) Find by Newton’s method the root of the equation e x  4 x , which is approximately 2, correct
to three places of decimals.
20. (a) Given x : 0
1
2
f ( x) : 2
3
12
divided difference formula.
5
14
find the cubic function of x using Newton’s
(b) Using Newton’s formula find the value of f(1.5) from the following data:
0
1
2
3
4
x:
f ( x) :
858.3 869.6 880.9 892.3 903.6
21. (a) Use Stirling’s formula to find f(1.63) given
1.50
1.60
1.70
1.80
1.90
x:
f ( x) :
17.609 20.412 23.045 25.527 27.875
(b)Given
X:
0
4
8
12
Y:
143
158
177
199
calculate y5 by Bessel’s formula.
22. (a) Apply the fourth order Runge–Kutta method, to find an approximate value of y when x = 0.2
given that y '  x  y, y (0)  1.
b
(b) Write a C program to find the value of
 ydx using Simson’s 1/3 rule.
a
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