Download GTU BE/B.Tech 2019 Summer 4th Sem New 2140105 Numerical Methods Question Paper

Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2019 Summer 4th Sem New 2140105 Numerical Methods Previous Question Paper

1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ?IV(NEW) ? EXAMINATION ? SUMMER 2019
Subject Code:2140105 Date:09/05/2019
Subject Name: Numerical Methods
Time: 02:30 PM TO 05:00 PM Total Marks: 70
Instructions:
1. Attempt all questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.


MARKS

Q.1 (a) Name two interpolation methods used for unequal intervals. Also
state their formulas.
03
(b)
Perform four iterations to find a root of the equation
3
4 9 0 xx ? ? ?
using Bisection method.




xx


04
(c)
Using fourth order Runge Kutta method, find (0.1) y for differential
equation 2 , (0) 1
dy
x y y
dx
? ? ? by taking h= 0.1
07

Q.2 (a) Solve the following system by Gauss elimination method.
3 2 5, 2 4 6 4, 5 3 10 x y z x y z x y z ? ? ? ? ? ? ? ? ? ?
03
(b) Find a real root of the equation 3 cos 1 xx ?? , correct up to four
decimal places using Newton Raphson method.
04
(c) Fit a second degree polynomial using least square method to the
following data:
x 1 2 3 4 5
y
5 12 26 60 97
Also estimate y at 6 x ? .
07
OR
(c)
Fit a curve of the form
bx
y ae ? to the following data:
x 1 3 5 7 9
y
115 105 95 85 80

07
Q.3 (a) Using Newton?s forward interpolation formula, find the value of
(1.6) f .
x 1 1.4 1.8 2.2
() fx
3.49 4.82 5.96 6.5

03

(b)
Use trapezoidal rule to evaluate
2
2
0
,
2
x
dx
x ?
?
dividing the interval
into four equal parts.
04
(c) Use Gauss-Siedel method to solve the following system:
6 105, 4 8 3 155, 5 4 10 65 x y z x y z x y z ? ? ? ? ? ? ? ? ?
07






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1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ?IV(NEW) ? EXAMINATION ? SUMMER 2019
Subject Code:2140105 Date:09/05/2019
Subject Name: Numerical Methods
Time: 02:30 PM TO 05:00 PM Total Marks: 70
Instructions:
1. Attempt all questions.
2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.


MARKS

Q.1 (a) Name two interpolation methods used for unequal intervals. Also
state their formulas.
03
(b)
Perform four iterations to find a root of the equation
3
4 9 0 xx ? ? ?
using Bisection method.




xx


04
(c)
Using fourth order Runge Kutta method, find (0.1) y for differential
equation 2 , (0) 1
dy
x y y
dx
? ? ? by taking h= 0.1
07

Q.2 (a) Solve the following system by Gauss elimination method.
3 2 5, 2 4 6 4, 5 3 10 x y z x y z x y z ? ? ? ? ? ? ? ? ? ?
03
(b) Find a real root of the equation 3 cos 1 xx ?? , correct up to four
decimal places using Newton Raphson method.
04
(c) Fit a second degree polynomial using least square method to the
following data:
x 1 2 3 4 5
y
5 12 26 60 97
Also estimate y at 6 x ? .
07
OR
(c)
Fit a curve of the form
bx
y ae ? to the following data:
x 1 3 5 7 9
y
115 105 95 85 80

07
Q.3 (a) Using Newton?s forward interpolation formula, find the value of
(1.6) f .
x 1 1.4 1.8 2.2
() fx
3.49 4.82 5.96 6.5

03

(b)
Use trapezoidal rule to evaluate
2
2
0
,
2
x
dx
x ?
?
dividing the interval
into four equal parts.
04
(c) Use Gauss-Siedel method to solve the following system:
6 105, 4 8 3 155, 5 4 10 65 x y z x y z x y z ? ? ? ? ? ? ? ? ?
07






2
OR
Q.3 (a)
Evaluate (9) f by using Lagrange?s interpolation method from the
following data:
x 5 7 11 13 17
() fx
150 392 1452 2366 5202

03

(b)
Evaluate
3
0
1
1
dx
x ?
?
with 6 n ? by using Simpson?s 38 rule.
04
(c) Compute (1.5) & (1) yy ? from the following data using Cubic
Spline.
x 1 2 3
y
-8 -1 18

07

Q.4 (a) Use Taylor?s series method to find y at 0.03 x ? given that
2
1, (0) 1.
dy
x y y
dx
? ? ?
03
(b) Find the root of
10
log 1.9 0, xx?? correct up to three decimal
places with
01
34 x and x ?? using Secant method.
04
(c) Using Shooting method , Solve the boundary value problem:
, (0) 0 (1) 1.17 y y y and y ?? ? ? ?
07
OR
Q.4 (a) Solve the following system by Gauss Jordan method:
2 4, 5 9, 4 3 10 x y y z x z ? ? ? ? ? ? ? ? ? ?
03
(b)
Solve the equation y x y ???? with the boundary conditions
(0) (1) 0 yy ?? by finite difference method.
04
(c) Using Picard?s method of successive approximation, obtain a solution
up to fifth approximation of the equation , (0) 1.
dy
x y y
dx
? ? ?
07

Q.5 (a) Explain Initial value problem and boundary value problem with
example.
03
(b)
Solve
2
2
uu
tx
??
?
??
in 0 5, 0 xt ? ? ? given that
( ,0) 20, (0, ) 0, (5, ) 100 u x u t u t ? ? ? . Compute ( , ) u x t with h=1
by Crank-Nicholson method.
04
(c) Solve the boundary value problem
1
0 , (0) 0 (1)
2
y x y and y ? ? ? ? ? ? ? ? by the Rayleigh-Ritz
method.
07
OR

Q.5 (a) State the difference between finite difference method and finite
element method.
03
(b)
Discuss the concept of Laplace equation
22
22
0
uu
xy
??
??
??

04
(c) Solve the boundary value problem , (0) 0, (1) 0 y y x y y ?? ? ? ? ? ?
by the Galerkin method.
07

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