North Maharashtra University 2008 B.Sc Mathematics S.Y. - MTH – 222 (B) (Numerical Analysis) - Question Paper

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NORTH MAHARASHTRA UNIVERSITY, JALGAON

Class : S.Y. B. Sc. Subject : Mathematics
Paper : MTH – 222 (B) (Numerical Analysis)

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1 : ques. of two marks
1) describe i) forward difference operator ii) backward difference
operator. obtain ? tan-1x.
2) describe shift operator E. Prove that E = one + ? .
3) describe central difference operator d and prove that d = ?E-1/2 =
?E1/2.
4) With usual notations prove that µ two =
4
1 (d two + 4).
6
5) Prove that u0 – u1 + u2 – u3 + - - - - =
2
1 u0 –
4
1 u0 +
8
1 ? 2u0 –
16
1 ? 3u0 +
- - - - .
6) provided u0 = 3, u1 = 12, u2 = 81, u3 = 200, u4 = 100, u5 = 8. obtain ? 5u0.
7) Prove that u0 +
1!
u x one +
2!
u x 2
2 + - - - = ex[u0 + ? x0 +
2!
x 2
? 2u0 + - - - -]
8) State Gauss’s forward central difference formula.
9) State Gauss’s backward central difference formula.
10) State Lagrange’s interpolation formula.
11) Using Lagrange’s interpolation formula obtain u3 if u0 = 580, u1 = 556,
u2 = 520, u4 = 385.
12) describe averaging operatorµ . Show that µ =
2
E1/2 + E-1/2 .
13) Show that two 2
2 3
d x
? x = 3.
14) Show that d = E1/2 – E-1/2 .
15) Using the method of separation of symbols prove that ux+n =
un
xC1 ? un-1 + x+1C2 ? 2un-2 + - - - -
16) provided that u0 + u8 = 1.9243, u1 + u7 = 1.9590, u2 + u6 = 1.9823, u3 +
u5 = 1.9956,. obtain u4 using ? 8u0 = 0.
17) Construct a forward difference table for the subsequent values of x, y :
x 0 five 10 15 20 25
y = f(x) six 10 13 17 23 21
18) Construct a backward difference table for the subsequent values of x, y
x 10 20 30 40 50
y = f(x) 45 65 80 92 100
19) Prove that (1 + ? )(1 – ? ) = 1
20) obtain ?
??
?
? ??
? ?
E
2
(x3).
7
21) Prove that ? logf(x) = log ??
?
??
?
+
f(x)
1 ?f(x) .
22) Prove that u3 = u2 + ? u1 + ? 2u0 + ? 3u0.
23) obtain the difference table for the data provided beneath :
x 0 one two three 4
f(x) three six 11 18 27
24) Show that ? nyx = yx+n – nC1yx+n-1 + nC2yx+n-2 + - - - - + (–1)n yx.
25) provided u0 = 1, u1 = 11, u2 = 21, u3 = 28, u4 = 29. Show that ? 4u0 = 0.
26) Form the difference table for the data :
x one two three 4
u 21 15 12 10
27) obtain
dx
dy at (2 , –2) of a curve passing through the points (0 , 2),
(2 , –2), (3 , –1) using Lagrange’s interpolation formula.
28) obtain the value of d four y2 provided beneath
x 0 one two three 4
y one two nine 28 65
29 obtain the cubic polynomial for y(0) = 1, y(1) = 0, y(2) = 1, y(3) = 10.
2 : Fill in the blanks/Multiple option ques. of 1
marks
1)
h
1
? ??
?
? ??
?
+ - - - - -
?
+
?
? +
2 3
2 3
= - - - - .
2) The value off E-nf(x) = - - - -
8
3) The value of
?
? –
?
? = - - - - -
4) If in a data 6 values are provided and 2 values are missing then
fifth differences are - - - - and 6th differences are - - - -
5) The value of
? ? ?
? ? ?
?
E
2
x4 is = - - - -
6) The value of log
? ? ?
? ? ?
?
+
( )
1 ( )
f x
f x is - - - -
7 The Lagrange’s interpolation formula is used for the arguments
which are - - - - spaced
a) equally b) distinct c) unequally d) none of these
8) one + ? = - - - -
a) E-1 b) ? c) E d) d
9) If n value of f(x) are provided then ? nf(x) is - - - - -
a) 0 b) one c) two d) n
10) The technique for computing the value of the function inside the
provided argument is called - - - -
a) interpolation b) extrapolation
c) partial fraction d) inverse interpolation

Earning:   Approval pending.