Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2018 Winter 4th Sem New 2141703 Numerical Techniques And Statistical Methods Previous Question Paper
1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ?IV (NEW) EXAMINATION ? WINTER 2018
Subject Code:2141703 Date:22/11/2018
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 0.333 is the approximate value of 1/3, compute the absolute, relative and
percentage errors.
03
(b) Calculate the mean and mode from the following frequency distribution:
Marks 0-10 10-20 20-30 30-40 40-50 50-60
No. of
students
12 18 27 20 17 6
04
(c) Compute the following system of equations by using Gauss-Sedial method
correct up to 4 decimal place.
61 10 2
51 10 2
44 2 10
3 2 1
3 2 1
3 2 1
? ? ?
? ? ?
? ? ?
x x x
x x x
x x x
07
Q.2 (a) Use Langrange?s formula of interpolation and find y(9.5)
given y(7) = 3, y(8) = 1, y(9) = 1, y(10) = 9.
03
(b)
Evaluate the integral
1
2
01
dx
x
?
?
by using Trapezoidal rule with h = 0.2. Hence
obtain an approximate value of Compare the result with the exact value
obtained by a formula known from calculus.
04
(c) Apply Runge-Kutta fourth order method to find an approximate value of y for
2 . 0 ? x and 4 . 0 ? x if
2 2
2 2
y x
x y
dx
dy
?
?
? , given that where 1 ) 0 ( ? y
07
OR
(c)
Using Milne?s Predictor-Corrector method to solve the equation
2 2
y x
dx
dy
? ?
at 3 . 0 ? x , given that 1 ) 0 ( ? y .
07
Q.3 (a) Find a root of the following equations correct to three decimal places using the
method of False position method
0 2 . 1 log
10
? ? x x
03
(b) Using Newton?s interpolation formula, find the values of f (82) from the
following data.
x 80 85 90 95 100
f(x) 5026 5674 6362 7088 7854
04
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1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ?IV (NEW) EXAMINATION ? WINTER 2018
Subject Code:2141703 Date:22/11/2018
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 0.333 is the approximate value of 1/3, compute the absolute, relative and
percentage errors.
03
(b) Calculate the mean and mode from the following frequency distribution:
Marks 0-10 10-20 20-30 30-40 40-50 50-60
No. of
students
12 18 27 20 17 6
04
(c) Compute the following system of equations by using Gauss-Sedial method
correct up to 4 decimal place.
61 10 2
51 10 2
44 2 10
3 2 1
3 2 1
3 2 1
? ? ?
? ? ?
? ? ?
x x x
x x x
x x x
07
Q.2 (a) Use Langrange?s formula of interpolation and find y(9.5)
given y(7) = 3, y(8) = 1, y(9) = 1, y(10) = 9.
03
(b)
Evaluate the integral
1
2
01
dx
x
?
?
by using Trapezoidal rule with h = 0.2. Hence
obtain an approximate value of Compare the result with the exact value
obtained by a formula known from calculus.
04
(c) Apply Runge-Kutta fourth order method to find an approximate value of y for
2 . 0 ? x and 4 . 0 ? x if
2 2
2 2
y x
x y
dx
dy
?
?
? , given that where 1 ) 0 ( ? y
07
OR
(c)
Using Milne?s Predictor-Corrector method to solve the equation
2 2
y x
dx
dy
? ?
at 3 . 0 ? x , given that 1 ) 0 ( ? y .
07
Q.3 (a) Find a root of the following equations correct to three decimal places using the
method of False position method
0 2 . 1 log
10
? ? x x
03
(b) Using Newton?s interpolation formula, find the values of f (82) from the
following data.
x 80 85 90 95 100
f(x) 5026 5674 6362 7088 7854
04
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2
(c) Solve the following equations by Gauss elimination method with .
7 2 7 4 3
5 4 5 2 5
1 11 3 2
3 3 3
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
w z y x
w z y x
w z y x
w z y x
07
OR
Q.3 (a)
Compute ? ? 92 f. from the given values using Newton?s divided difference
interpolation formula.
x
8.0 9.0 9.5 11.0
f(x)= log x
2.079442 2.197225 2.251292 2.397895
03
(b) Apply modified Euler?s method to the solve initial value problem, y? = 1 ? y with
y(0) = 0.Find y(0.2) choosing h = 0.1
04
(c) Use Simpson?s 1/3 and 3/8 rule to evaluate the following integral
?
2 . 5
4
log dx x
e
and after finding the true value of the integral compare errors in the two cases.
07
Q.4 (a) Mathematics problem is given to three student s A1, A2 and A3, whose chances
of solving it are1/3, 2/5and 3/7 .What is the probability that the problem will be
solved.
03
(b) In a normal distribution 31% of the items are under 45 and 8% are over 64. Find
the parameters of the distribution.
?
?
?
t
x
dx e t f
0
2 /
2
) ( then f(0.5) = 0.19 and
f(1.4) = 0.42.
04
(c) Verify whether Poisson distribution can be assumed from the data given:
No. appeared on
dice
Frequencies
0 6
1 13
2 13
3 8
4 4
5 3
Using ?
2
? test at 5% level of significance.
d. f 1 2 3 4 5
2
05 . 0
?
3.841 5.991 7.815 9.488 11.070
07
OR
Q.4 (a) Intelligence tests were given to two groups of boys and girls
Examine if the difference between mean scores is significant at 5% level of
significance.
Mean Standard Deviation Size
Girls 75 8 60
Boys 73 10 100
03
(b) Razor blades are supplied by a manufacturing company in packets of 10. There
is a probability of 1 in 500 blades to be defective. Using Poisson distribution to
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1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ?IV (NEW) EXAMINATION ? WINTER 2018
Subject Code:2141703 Date:22/11/2018
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 0.333 is the approximate value of 1/3, compute the absolute, relative and
percentage errors.
03
(b) Calculate the mean and mode from the following frequency distribution:
Marks 0-10 10-20 20-30 30-40 40-50 50-60
No. of
students
12 18 27 20 17 6
04
(c) Compute the following system of equations by using Gauss-Sedial method
correct up to 4 decimal place.
61 10 2
51 10 2
44 2 10
3 2 1
3 2 1
3 2 1
? ? ?
? ? ?
? ? ?
x x x
x x x
x x x
07
Q.2 (a) Use Langrange?s formula of interpolation and find y(9.5)
given y(7) = 3, y(8) = 1, y(9) = 1, y(10) = 9.
03
(b)
Evaluate the integral
1
2
01
dx
x
?
?
by using Trapezoidal rule with h = 0.2. Hence
obtain an approximate value of Compare the result with the exact value
obtained by a formula known from calculus.
04
(c) Apply Runge-Kutta fourth order method to find an approximate value of y for
2 . 0 ? x and 4 . 0 ? x if
2 2
2 2
y x
x y
dx
dy
?
?
? , given that where 1 ) 0 ( ? y
07
OR
(c)
Using Milne?s Predictor-Corrector method to solve the equation
2 2
y x
dx
dy
? ?
at 3 . 0 ? x , given that 1 ) 0 ( ? y .
07
Q.3 (a) Find a root of the following equations correct to three decimal places using the
method of False position method
0 2 . 1 log
10
? ? x x
03
(b) Using Newton?s interpolation formula, find the values of f (82) from the
following data.
x 80 85 90 95 100
f(x) 5026 5674 6362 7088 7854
04
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2
(c) Solve the following equations by Gauss elimination method with .
7 2 7 4 3
5 4 5 2 5
1 11 3 2
3 3 3
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
w z y x
w z y x
w z y x
w z y x
07
OR
Q.3 (a)
Compute ? ? 92 f. from the given values using Newton?s divided difference
interpolation formula.
x
8.0 9.0 9.5 11.0
f(x)= log x
2.079442 2.197225 2.251292 2.397895
03
(b) Apply modified Euler?s method to the solve initial value problem, y? = 1 ? y with
y(0) = 0.Find y(0.2) choosing h = 0.1
04
(c) Use Simpson?s 1/3 and 3/8 rule to evaluate the following integral
?
2 . 5
4
log dx x
e
and after finding the true value of the integral compare errors in the two cases.
07
Q.4 (a) Mathematics problem is given to three student s A1, A2 and A3, whose chances
of solving it are1/3, 2/5and 3/7 .What is the probability that the problem will be
solved.
03
(b) In a normal distribution 31% of the items are under 45 and 8% are over 64. Find
the parameters of the distribution.
?
?
?
t
x
dx e t f
0
2 /
2
) ( then f(0.5) = 0.19 and
f(1.4) = 0.42.
04
(c) Verify whether Poisson distribution can be assumed from the data given:
No. appeared on
dice
Frequencies
0 6
1 13
2 13
3 8
4 4
5 3
Using ?
2
? test at 5% level of significance.
d. f 1 2 3 4 5
2
05 . 0
?
3.841 5.991 7.815 9.488 11.070
07
OR
Q.4 (a) Intelligence tests were given to two groups of boys and girls
Examine if the difference between mean scores is significant at 5% level of
significance.
Mean Standard Deviation Size
Girls 75 8 60
Boys 73 10 100
03
(b) Razor blades are supplied by a manufacturing company in packets of 10. There
is a probability of 1 in 500 blades to be defective. Using Poisson distribution to
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3
calculate the number of packets containing no defective blade, one defective
blade, and two defective blade in a consignment of 10,000 packets.
04
(c) Fit a Binomial distribution for the given data
x 0 1 2 3 4 5 6 7 8 9
f 6 20 28 12 8 6 0 0 0 0
07
Q.5 (a) A sample of 20 items has mean 42 units and standard deviation 5 units. Test the
hypothesis that it is a random sample from a normal population with mean 45
units using t-test at 5% level of significance.
d. f 17 18 19 20 21
05 . 0
t
2.110 2.101 2.093 2.086 2.080
03
(b) The theory predicts the proportion of beans in the four groups
) , , (
4 3 2 1
G and G G G should be in the ratio 9:3:3:1. In an experiment with
1600 beans the number in the four groups was 882, 313, 287 and 118. Dose the
experiment result support the theory at 5% level of significance using test ?
2
? .
d. f 1 2 3 4 5
2
05 . 0
?
3.841 5.991 7.815 9.488 11.070
04
(c) A project schedule has the following characteristics
(1) Construct a network diagram.
(2) Determine the critical path and total project duration.
(3) Compute total float and free float for each activity.
Activity Time(days)
1-2 4
1-3 1
2-4 1
3-4 1
3-5 6
4-9 5
5-6 4
5-7 8
6-8 1
7-8 2
8-10 5
9-10 7
07
OR
Q.5 (a) Before an increase in excise duty on tea, 800 persons out of a sample 1000
persons were found to be tea drinkers. After an increase in duty, 800 people were
tea drinkers in sample 1200 using standard error of proportion, state whether there
is a significant decrease in the consumption of tea after the increase in excise
duty. ) 96 . 1 (
05 . 0
? Z
03
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1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ?IV (NEW) EXAMINATION ? WINTER 2018
Subject Code:2141703 Date:22/11/2018
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 0.333 is the approximate value of 1/3, compute the absolute, relative and
percentage errors.
03
(b) Calculate the mean and mode from the following frequency distribution:
Marks 0-10 10-20 20-30 30-40 40-50 50-60
No. of
students
12 18 27 20 17 6
04
(c) Compute the following system of equations by using Gauss-Sedial method
correct up to 4 decimal place.
61 10 2
51 10 2
44 2 10
3 2 1
3 2 1
3 2 1
? ? ?
? ? ?
? ? ?
x x x
x x x
x x x
07
Q.2 (a) Use Langrange?s formula of interpolation and find y(9.5)
given y(7) = 3, y(8) = 1, y(9) = 1, y(10) = 9.
03
(b)
Evaluate the integral
1
2
01
dx
x
?
?
by using Trapezoidal rule with h = 0.2. Hence
obtain an approximate value of Compare the result with the exact value
obtained by a formula known from calculus.
04
(c) Apply Runge-Kutta fourth order method to find an approximate value of y for
2 . 0 ? x and 4 . 0 ? x if
2 2
2 2
y x
x y
dx
dy
?
?
? , given that where 1 ) 0 ( ? y
07
OR
(c)
Using Milne?s Predictor-Corrector method to solve the equation
2 2
y x
dx
dy
? ?
at 3 . 0 ? x , given that 1 ) 0 ( ? y .
07
Q.3 (a) Find a root of the following equations correct to three decimal places using the
method of False position method
0 2 . 1 log
10
? ? x x
03
(b) Using Newton?s interpolation formula, find the values of f (82) from the
following data.
x 80 85 90 95 100
f(x) 5026 5674 6362 7088 7854
04
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2
(c) Solve the following equations by Gauss elimination method with .
7 2 7 4 3
5 4 5 2 5
1 11 3 2
3 3 3
? ? ? ?
? ? ? ?
? ? ? ?
? ? ? ?
w z y x
w z y x
w z y x
w z y x
07
OR
Q.3 (a)
Compute ? ? 92 f. from the given values using Newton?s divided difference
interpolation formula.
x
8.0 9.0 9.5 11.0
f(x)= log x
2.079442 2.197225 2.251292 2.397895
03
(b) Apply modified Euler?s method to the solve initial value problem, y? = 1 ? y with
y(0) = 0.Find y(0.2) choosing h = 0.1
04
(c) Use Simpson?s 1/3 and 3/8 rule to evaluate the following integral
?
2 . 5
4
log dx x
e
and after finding the true value of the integral compare errors in the two cases.
07
Q.4 (a) Mathematics problem is given to three student s A1, A2 and A3, whose chances
of solving it are1/3, 2/5and 3/7 .What is the probability that the problem will be
solved.
03
(b) In a normal distribution 31% of the items are under 45 and 8% are over 64. Find
the parameters of the distribution.
?
?
?
t
x
dx e t f
0
2 /
2
) ( then f(0.5) = 0.19 and
f(1.4) = 0.42.
04
(c) Verify whether Poisson distribution can be assumed from the data given:
No. appeared on
dice
Frequencies
0 6
1 13
2 13
3 8
4 4
5 3
Using ?
2
? test at 5% level of significance.
d. f 1 2 3 4 5
2
05 . 0
?
3.841 5.991 7.815 9.488 11.070
07
OR
Q.4 (a) Intelligence tests were given to two groups of boys and girls
Examine if the difference between mean scores is significant at 5% level of
significance.
Mean Standard Deviation Size
Girls 75 8 60
Boys 73 10 100
03
(b) Razor blades are supplied by a manufacturing company in packets of 10. There
is a probability of 1 in 500 blades to be defective. Using Poisson distribution to
www.FirstRanker.com www.FirstRanker.com
www.FirstRanker.com
www.FirstRanker.com
3
calculate the number of packets containing no defective blade, one defective
blade, and two defective blade in a consignment of 10,000 packets.
04
(c) Fit a Binomial distribution for the given data
x 0 1 2 3 4 5 6 7 8 9
f 6 20 28 12 8 6 0 0 0 0
07
Q.5 (a) A sample of 20 items has mean 42 units and standard deviation 5 units. Test the
hypothesis that it is a random sample from a normal population with mean 45
units using t-test at 5% level of significance.
d. f 17 18 19 20 21
05 . 0
t
2.110 2.101 2.093 2.086 2.080
03
(b) The theory predicts the proportion of beans in the four groups
) , , (
4 3 2 1
G and G G G should be in the ratio 9:3:3:1. In an experiment with
1600 beans the number in the four groups was 882, 313, 287 and 118. Dose the
experiment result support the theory at 5% level of significance using test ?
2
? .
d. f 1 2 3 4 5
2
05 . 0
?
3.841 5.991 7.815 9.488 11.070
04
(c) A project schedule has the following characteristics
(1) Construct a network diagram.
(2) Determine the critical path and total project duration.
(3) Compute total float and free float for each activity.
Activity Time(days)
1-2 4
1-3 1
2-4 1
3-4 1
3-5 6
4-9 5
5-6 4
5-7 8
6-8 1
7-8 2
8-10 5
9-10 7
07
OR
Q.5 (a) Before an increase in excise duty on tea, 800 persons out of a sample 1000
persons were found to be tea drinkers. After an increase in duty, 800 people were
tea drinkers in sample 1200 using standard error of proportion, state whether there
is a significant decrease in the consumption of tea after the increase in excise
duty. ) 96 . 1 (
05 . 0
? Z
03
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4
(b) In a school heights of six randomly chosen girls are 63, 65, 68, 69 and 72 inches
and those of nine randomly chosen boys 61, 62, 65, 66, 69, 70, 71, 72 and 73
inches. Test whether the girls are taller than boys at 5% level of significance.
d. f 11 12 13 14 15
05 . 0
t
1.796 1.782 1.77 1.761 1.753
04
(c) A project has following time estimates:
(a) Draw the project network.
(b) Find the critical path and project duration.
(c) What is the probability that the project will be completed 4 days earlier
than expected? ) 4082 . 0 ) 33 . 1 ( ( ? ? Z P
Activity
Estimated durations (days)
Optimistic
) (
o
t
Most likely
) (
m
t
Pessimistic
) (
p
t
(1, 2) 1 1 7
(1,3) 1 4 7
(1,4) 2 2 1
(2, 5) 1 1 1
(3, 5) 2 5 14
(4, 6) 2 5 8
(5, 6) 3 6 15
07
*************
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This post was last modified on 20 February 2020