CLASS: B.Stat.
13A/270
St. JOSEPH’S COLLEGE (AUTONOMOUS) TIRUCHIRAPPALLI – 620 002
SEMESTER EXAMINATIONS – APRIL 2013
TIME: 3 Hrs.
MAXIMUM MARKS: 100
SEM
SET
PAPER CODE
TITLE OF THE PAPER
IV
2011
11UST430301A
NUMERICAL MATHEMATICS
SECTION – A
Answer all the questions:
20 x 1 = 20
Answer in one or two sentences:
1.
What do you mean by interpolation?
2.
Give Stirling’s formula for interpolation.
3.
Define Inverse interpolation.
4.
Explain Bisection method.
5.
What do you mean by numerical integration?
Fill in the blanks:
6.
The relation between the operators  and E is _________.
7.
Newton Gauss backward difference formula is _________.
8.
Lagrange’s formula can be applied for direct interpolation and _________ interpolation.
9.
The order of convergence of Newton-Raphson method is _________.
10.
Simpson’s
1 rule is given by _________.
3
State True or False:
11.
The nth order differences of a polynomial of nth degree are constant.
12.
Stirling’s formula is the mean of the Newton Gauss forward and Newton Gauss backward
formula.
13.
Finding the value of the function corresponding to an intermediate value of the argument is the
problem of Inverse interpolation.
14.
The iterative formula for Newton-Raphson method is
x n 1  x n 
f (x n )
.

f (x n )
15.
If we put n = 1 in Newton Cote’s Quadrature formula we get Simpson’s
1 rule.
3
Choose the correct answer:
16.
Given that y0 = 1
4y is equal to
a) 0
c) 2
y1 = 4
y2 = 10
b)
d)
y3 = 20
The Newton Gauss forward can be used to interpolate in the interval
a) 0  u  1
b)  1  u  0
c) Both a) and b)
d) None of these
18.
The condition of convergence of Newton-Rapson’s method is
a)
b)
(f ( x ))
c)
2
f ( x ).f ( x )  {f ( x )}2
1
d)
Both a) and b)
None of these
19.
The divided differences are symmetrical in all their _________.
a) Arguments
b) Entries
c) Both a) and b)
d) None of these
20.
Simpson’s
a)
1 rule can be derived by putting n = _________ in Quadrature formula.
3
1
b) 2
c)
SECTION – B
3
Answer all the questions:
21.
a.
y5= 56
1
3
17.
f ( x ) f ( x )
y4 = 35
d) 6
5 x 4 = 20
Derive Newton-Gregory formula for forward interpolation.
OR
22.
b.
The population of a town in the census is given below. Estimate the population in the
year 1996 using Newton’s backward interpolation formula.
Year (x)
1961
1971
1981
1991
2001
Population
(in
46
66
81
93
101
1,000’s)
a.
Derive Newton Gauss forward formula for interpolation.
OR
b.
Find Sin(0.35) using Bessel’s formula from the following table.
x:
sinx:
23.
a.
0.1
0.2
0.3
0.4
0.5
0.0998
0.1986
0.2955
0.3894
0.4794
Explain the method of successive approximation.
OR
b.
24.
a.
Use Lagrange’s inverse interpolation formula to find the value of x when y = 20 using the
following data.
x:
1
2
3
4
y:
1
8
27
64
Explain Regula-Falsi method.
OR
25.
b.
Find the root of the equation x3 – 5x – 7 = 0, that lies between 2 and 3, using the method
of false position.
a.
Derive Simpson’s three-eighths rule.
OR
b.
Find the first and second derivatives of y = f(x) at x = 1.5
x
1.5
2.0
2.5
3.0
3.5
4.0
y 3.375 7.0 13.625 24.0 38.875 59.0
SECTION – C
Answer any FOUR questions:
26.
4 x 15 = 60
a) Derive Newton’s Divided Difference formula for interpolation.
b) Using the data given below find f(0.75)
x
:
1.00
1.25
1.50
1.75
2.00
f ( x ) : 0.3679 0.2865 0.2231 0.1738 0.1353
27.
a) Derive Bessel’s formula for interpolation.
b) Explain the appropriateness of various interpolation formulae.
28.
Find the smallest positive root of the equation 3x3 – 9x2 + 8 = 0 by the method of successive
approximation.
29.
Find the root of the equation x3 + x2 – 1 = 0
0 and 1 correct to 4 places of decimals by Newton-Raphson method.
30.
a) Derive Weddle’s rule.
b) Evaluate
1 dx

01  x
2
.
**************
that
lies
between