Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2019 Summer 4th Sem New 2141703 Numerical Techniques And Statistical Methods Previous Question Paper
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ?IV(NEW) ? EXAMINATION ? SUMMER 2019
Subject Code:2141703 Date:09/05/2019
Subject Name: Numerical Techniques & Statistical Methods
Time: 02:30 PM TO 05:30 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) If X = 3.4327, find the absolute and relative errors if :
(a) X is truncated to three decimal places.
(b) X is rounded off to three decimal places.
03
(b) Calculate mean and mode for the following data:
Class 0-10 10-20 20-30 30-40 40-50
Frequency 10 14 19 17 13
04
(c) Use Fourth order Runge-Kutta method to find (0.2) y with 0.1 h ? , given that
2 , (0) 1
dy
x y y
dx
? ? ?
07
Q.2 (a)
Using the power method, find the largest Eigen value for
12
34
A
??
?
??
??
03
(b) Apply Gauss ? Seidel iteration method to solve :
20 2 17, 3 20 18, 2 3 20 25 x y z x y z x y z ? ? ? ? ? ? ? ? ? ?
04
(c) Construct an Interpolating polynomial which takes the following values :
x
0 1 2 3 4 5 6 7
y
1 2 4 7 11 16 22 29
07
OR
(c) Obtain Cubic spline for subinterval 0 1 & 1 2 xx ? ? ? ? from the following data:
x
0 1 2 3
() fx 1 2 33 244
07
Q.3 (a) Considering following tabular values, Determine the area bounded by the given
curve and X-axis between 10 x ? to 16 x ? by Trapezoidal rule.
x
10 11 12 13 14 15 16
y
1.02 0.94 0.89 0.79 0.71 0.62 0.55
03
(b) Using Newton?s forward 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
04
(c)
Use Euler?s method to find (2) y from the differential equation 2 , (1) 1
dy
x y y
dx
? ? ?
with 0.1 h ?
07
OR
Q.3 (a)
Using Simpson?s 1/3 rule, evaluate
1
2
0
1
(1 )
dx
x ?
?
by taking 4 sub intervals.
03
(b)
Evaluate
1
2
0
e x p( ) x dx ?
?
using the Gaussian Integration formula with n = 3.
04
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1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ?IV(NEW) ? EXAMINATION ? SUMMER 2019
Subject Code:2141703 Date:09/05/2019
Subject Name: Numerical Techniques & Statistical Methods
Time: 02:30 PM TO 05:30 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) If X = 3.4327, find the absolute and relative errors if :
(a) X is truncated to three decimal places.
(b) X is rounded off to three decimal places.
03
(b) Calculate mean and mode for the following data:
Class 0-10 10-20 20-30 30-40 40-50
Frequency 10 14 19 17 13
04
(c) Use Fourth order Runge-Kutta method to find (0.2) y with 0.1 h ? , given that
2 , (0) 1
dy
x y y
dx
? ? ?
07
Q.2 (a)
Using the power method, find the largest Eigen value for
12
34
A
??
?
??
??
03
(b) Apply Gauss ? Seidel iteration method to solve :
20 2 17, 3 20 18, 2 3 20 25 x y z x y z x y z ? ? ? ? ? ? ? ? ? ?
04
(c) Construct an Interpolating polynomial which takes the following values :
x
0 1 2 3 4 5 6 7
y
1 2 4 7 11 16 22 29
07
OR
(c) Obtain Cubic spline for subinterval 0 1 & 1 2 xx ? ? ? ? from the following data:
x
0 1 2 3
() fx 1 2 33 244
07
Q.3 (a) Considering following tabular values, Determine the area bounded by the given
curve and X-axis between 10 x ? to 16 x ? by Trapezoidal rule.
x
10 11 12 13 14 15 16
y
1.02 0.94 0.89 0.79 0.71 0.62 0.55
03
(b) Using Newton?s forward 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
04
(c)
Use Euler?s method to find (2) y from the differential equation 2 , (1) 1
dy
x y y
dx
? ? ?
with 0.1 h ?
07
OR
Q.3 (a)
Using Simpson?s 1/3 rule, evaluate
1
2
0
1
(1 )
dx
x ?
?
by taking 4 sub intervals.
03
(b)
Evaluate
1
2
0
e x p( ) x dx ?
?
using the Gaussian Integration formula with n = 3.
04
2
(c)
Given that
2 2 2
2
dy
y x y
dx
?? , (0) 1, (0.1) 1.06, (0.2) 1.12, (0.3) 1.21 y y y y ? ? ? ? Evaluate
(0.4) y by Milne?s Predictor ? Corrector method.
07
Q.4 (a) There are two boxes A and B containing 4 white, 3 red and 3 white, 7 red balls
respectively. A box is chosen at random and a ball is drawn from it, If the ball is
white, find the probability that it is from box A.
03
(b) An unbiased coin is tossed 6 times. Find the probability of getting (1) exactly 4
heads (2) at least 4 heads.
04
(c) Eleven school boys were given a test in a Subject. They were given a month?s further
coaching and a second test of equal category was held at the end of it. Do the marks
give evidence that the students have benefited by extra coaching?
Boys 1 2 3 4 5 6 7 8 9 10 11
Marks 1
st
test 23 20 19 21 18 20 18 17 23 16 19
Marks 2
nd
test 24 19 22 18 20 22 20 20 23 20 17
(At 5% level of significance for n = 10 d. f. 2.228
T
t ? )
07
OR
Q.4 (a) Two people are selected at random from a group of seven men and five women.
Find the Probability that both are men or both are women.
03
(b) 100 Electric bulbs are found to be defective in a lot of 5000 bulbs. Find the
probability that at the most 3 bulbs are defective in a box of 100 bulbs.
04
(c) A dice is thrown 150 times and the following results are obtained.
Number turned up 1 2 3 4 5 6
Frequency 19 23 28 17 32 31
Test the Hypothesis that the dice is unbiased at 5% level of significance.
(At 5% level of significance for n = 5 d. f.
2
11.07
T
? ? )
07
Q.5 (a) Find the standard deviation of a group of data points:
101.8 , 103.2, 104.0, 102.5, 103.5
03
(b) Define Chi-square Test. State (a) conditions to apply test (b) application of test 04
(c) Represent the following information in form of a network. Find average duration
time or expected time of each activity and obtain the critical path.
Activity 1 -
2
2-
3
2 -
4
3 -
5
4 -
5
4 -
6
5 -
7
5-
8
7-
9
8-
9
9-
10
6-
10
Optimistic time 4 1 8 3 2 3 3 4 4 2 4 4
Most Likely time 9 5 10 6 4 7 7 8 9 6 11 7
Pessimistic time 14 18 17 8 5 10 10 9 14 10 18 9
07
OR
Q.5 (a) Compute the Median from the data:
Class 0-30 30-60 60-90 90-120 120-150 150-180
Frequency 8 13 22 27 18 7
03
(b) A bag Contains 5 white and 7 black balls. Find the expectation of a man who is
allowed to draw two balls from the bag and who is to receive one rupee for each
black ball and two rupees for each white ball.
04
(c) Draw PERT ? diagram after finding out expected time & find critical path.
Activity Sequence Optimistic Time Most Likely Time Pessimistic Time
A 1-2 7 12 13
B 1-3 7 10 12
C 2-5 8 13 15
D 3-5 10 12 22
E 5-6 10 14 18
***************
07
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This post was last modified on 20 February 2020