Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2018 Winter 4th Sem New 2140105 Numerical Methods Previous Question Paper
1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ?IV (NEW) EXAMINATION ? WINTER 2018
Subject Code:2140105 Date:22/11/2018
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.
Q.1 (a) Name five iterative methods which evaluate the root of
equations.
03
(b) Perform five iterations of Bisection method to obtain real root
of
3
10 xx ? ? ? .
04
(c) By the method of least squares, find the straight line
y ax b ?? that best fits the following data:
x 0 5 10 15 20 25
y 12 15 17 22 24 30
07
Q.2 (a) Mention atleast two difference between Newton?s forward
Interpolation and Newton?s divided difference interpolation.
03
(b) Find second degree polynomial passing through the points
(-1, 8), (0, 3), (2, 1) and (3, 12) using Lagrange interpolation.
04
(c) Obtain cubic splines for every subintervals from the
following data:
x 0 1 2 3
y 1 2 33 244
07
OR
(c) Using Newton?s Divided Difference Interpolation find f(x)
from the following table:
x 1 2 7 8
y 1 5 5 4
07
Q.3 (a) Use Gauss Elimination to solve:
x + 3y + 2z = 5
2x + 4y - 6z = -4
x + 5y + 3z = 10
03
(b) Consider following tabular values:
x 25 25.1 25.2 25.3 25.4 25.5 25.6
F(x) 3.205 3.217 3.232 3.245 3.256 3.268 3.280
Determine the area bounded by given curve and x-axis
between x = 25 to x = 25.6 by the Trapezoidal rule.
04
(c) Describe Newton-Raphson method and find root of equation
sin cos 0 x x x ?? which is near ?? correct upto 5 decimal
places.
07
OR
Q.3 (a)
Find approximate root of
3
2 1 0 xx ? ? ? starting from x0 = 1.5
to x1 = 2 by Secant method correct upto 3 decimal places.
03
(b)
Evaluate by Simpson?s
1
3
Rule,
5
0
1
54
dx
x ?
?
taking 10 equal
parts, hence obtain approximate value of loge5.
04
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1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ?IV (NEW) EXAMINATION ? WINTER 2018
Subject Code:2140105 Date:22/11/2018
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.
Q.1 (a) Name five iterative methods which evaluate the root of
equations.
03
(b) Perform five iterations of Bisection method to obtain real root
of
3
10 xx ? ? ? .
04
(c) By the method of least squares, find the straight line
y ax b ?? that best fits the following data:
x 0 5 10 15 20 25
y 12 15 17 22 24 30
07
Q.2 (a) Mention atleast two difference between Newton?s forward
Interpolation and Newton?s divided difference interpolation.
03
(b) Find second degree polynomial passing through the points
(-1, 8), (0, 3), (2, 1) and (3, 12) using Lagrange interpolation.
04
(c) Obtain cubic splines for every subintervals from the
following data:
x 0 1 2 3
y 1 2 33 244
07
OR
(c) Using Newton?s Divided Difference Interpolation find f(x)
from the following table:
x 1 2 7 8
y 1 5 5 4
07
Q.3 (a) Use Gauss Elimination to solve:
x + 3y + 2z = 5
2x + 4y - 6z = -4
x + 5y + 3z = 10
03
(b) Consider following tabular values:
x 25 25.1 25.2 25.3 25.4 25.5 25.6
F(x) 3.205 3.217 3.232 3.245 3.256 3.268 3.280
Determine the area bounded by given curve and x-axis
between x = 25 to x = 25.6 by the Trapezoidal rule.
04
(c) Describe Newton-Raphson method and find root of equation
sin cos 0 x x x ?? which is near ?? correct upto 5 decimal
places.
07
OR
Q.3 (a)
Find approximate root of
3
2 1 0 xx ? ? ? starting from x0 = 1.5
to x1 = 2 by Secant method correct upto 3 decimal places.
03
(b)
Evaluate by Simpson?s
1
3
Rule,
5
0
1
54
dx
x ?
?
taking 10 equal
parts, hence obtain approximate value of loge5.
04
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2
(c) State diagonally dominant condition and evaluate by Gauss
Siedel method:
5x + y - z = 10
2x + 4y + z = 14
x + y + 8z = 20
07
Q.4 (a) State finite difference quotients for first and second order
derivatives.
03
(b)
Solve heat equation
2
2
uu
tx
??
?
??
with u(x, 0) = 0, u(0, t) = 0 and
u(1, t) = t with k =
1
8
, h =
1
4
.
04
(c)
Solve by Runge-Kutta fourth order 2
dy
xy
dx
?? , y(0) = 1,
h = 0.1 find y (0.1) and y(0.2).
07
OR
Q.4 (a) State Gauss-Seidel method for Laplace equation. 03
(b) Discuss shooting approach for Boundary Value Problem in
brief.
04
(c)
Write two differences between finite difference method and
finite element method.
07
Q.5 (a) Evaluate (1) (1 )(1 ) 1 ? ? ? ? ? (2) E ? ? ? 03
(b)
Solve by Runge-Kutta second order 3
dy
xy
dx
?? , y(1)=1.3,
h=0.1 find y (1.2).
04
(c)
Evaluate IVP
dy
xy
dx
? , y(1) = 1 and hence find y(1.5) taking
h=0.1 by Euler?s method.
07
OR
Q.5 (a) Describe Galerkin approach in brief. 03
(b) Solve y?= x + y with boundary conditions y(0) = y(1) = 0 by
finite difference method
04
(c) Describe Rayleigh Ritz method in brief. 07
*************
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This post was last modified on 20 February 2020