Download GTU (Gujarat Technological University) BE/BTech (Bachelor of Engineering / Bachelor of Technology) 2019 Winter 4th Sem New 2141703 Numerical Techniques And Statistical Methods Previous Question Paper
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ? IV (New) EXAMINATION ? WINTER 2019
Subject Code: 2141703 Date: 07/12/2019
Subject Name: Numerical Techniques & Statistical Methods
Time: 10:30 AM TO 01:30 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) Find the relative error if the number X = 0.004997 is
(1) Truncated to three decimal digits
(2) Rounded off to three decimal digits
03
(b) Calculate mean, median and standard deviation from the following
data:
Wages
(Rs.?000):
0-10 10-20 20-30 30-40 40-50 50-60
No. of
workers:
12 17 23 39 16 03
04
(c) Explain diagonally dominant system of equations. Solve the following
system of equations by Gauss Jacobi method.
3x + 20y ? z = -18, 2x ? 3y + 20z = 25, 20x + y ? 2z = 17
07
Q.2 (a) The following data gives the velocity of a particle for 20 seconds at an
interval of 5 seconds. Find the initial acceleration using the entire data:
Time t (sec): 0 5 10 15 20
Velocity v(m/sec): 0 3 14 69 228
03
(b) A curve passes through the points (0, 18), (1, 10), (3, -18) and (6, 90).
Find the slope of the curve at x = 2 using Lagrange Interpolation
technique.
04
(c) Evaluate the integral
? (
?? ???????? ) ???? 0.5
0
Using Romberg?s method, correct to 3 decimal places.
07
OR
(c) Evaluate
? ?? ??? 2
???? 1.5
0.2
Using Gaussian quadrature formula for n = 2 and n = 3.
07
Q.3 (a) Find by Newton?s method, the real positive root of the equation
x
4
? x =10, correct to three decimal places.
03
(b) Using Taylor?s series method find approximate value of y at x = 0.2 for
the differential equation dy/dx = 2y + 3e
x
, y (0) = 0. Compare the
numerical solution obtained with the exact solution.
04
(c) The following values of x and y are given:
x: 1 2 3 4
y: 1 2 5 11
Find the cubic splines and evaluate y (1.5) and y
?
(3).
07
OR
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1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ? IV (New) EXAMINATION ? WINTER 2019
Subject Code: 2141703 Date: 07/12/2019
Subject Name: Numerical Techniques & Statistical Methods
Time: 10:30 AM TO 01:30 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) Find the relative error if the number X = 0.004997 is
(1) Truncated to three decimal digits
(2) Rounded off to three decimal digits
03
(b) Calculate mean, median and standard deviation from the following
data:
Wages
(Rs.?000):
0-10 10-20 20-30 30-40 40-50 50-60
No. of
workers:
12 17 23 39 16 03
04
(c) Explain diagonally dominant system of equations. Solve the following
system of equations by Gauss Jacobi method.
3x + 20y ? z = -18, 2x ? 3y + 20z = 25, 20x + y ? 2z = 17
07
Q.2 (a) The following data gives the velocity of a particle for 20 seconds at an
interval of 5 seconds. Find the initial acceleration using the entire data:
Time t (sec): 0 5 10 15 20
Velocity v(m/sec): 0 3 14 69 228
03
(b) A curve passes through the points (0, 18), (1, 10), (3, -18) and (6, 90).
Find the slope of the curve at x = 2 using Lagrange Interpolation
technique.
04
(c) Evaluate the integral
? (
?? ???????? ) ???? 0.5
0
Using Romberg?s method, correct to 3 decimal places.
07
OR
(c) Evaluate
? ?? ??? 2
???? 1.5
0.2
Using Gaussian quadrature formula for n = 2 and n = 3.
07
Q.3 (a) Find by Newton?s method, the real positive root of the equation
x
4
? x =10, correct to three decimal places.
03
(b) Using Taylor?s series method find approximate value of y at x = 0.2 for
the differential equation dy/dx = 2y + 3e
x
, y (0) = 0. Compare the
numerical solution obtained with the exact solution.
04
(c) The following values of x and y are given:
x: 1 2 3 4
y: 1 2 5 11
Find the cubic splines and evaluate y (1.5) and y
?
(3).
07
OR
2
Q.3 (a) For the given set of order pairs (x0, y 0), (x1, y 1), define first order forward
difference, first order backward difference and first order divided
difference.
03
(b) Using modified Euler?s method, find y (0.2), given
dy/dx = y + e
x
, y (0) = 0
04
(c) Apply Runge-Kutta method to find approximate value of y for x = 0.2,
in steps of 0.1, if dy/dx = x + y
2
, given that y = 1 when x = 0.
07
Q.4 (a) A salesman wants to know the average number of units he sells per
sales call. He checks his past sales records and comes up with the
following probabilities:
Sales in units: 0 1 2 3 4 5
Probability: 0.15 0.20 0.10 0.05 0.30 0.20
What is the average number of units he sells per sales call?
03
(b) The mean and standard deviation of a population are 225 and 278
respectively. What can we assert with 95% confidence about the
maximum error if ?? ? = 225 and n = 100. Also, construct 95%
confidence Interval.
04
(c) Out of 800 families with 4 children each, what percentage would be
expected to have (a) 2 boys and 2 girls (b) at least one boy (c) no girls
and (d) at the most 2 girls. Assume equal probabilities for boys and
girls.
07
OR
Q.4 (a) 500 apples are taken at random from a large basket and 50 are found to
be bad. Estimate the proportion of bad apples in the basket and assign
limits within which the percentage most probably lies. (critical value at
the most is 3)
03
(b) Before an increase in excise duty on tea, 800 people out of a sample of
1000 were consumers of tea. After the increase in duty, 800 people
were consumers of tea in a sample of 1200 persons. Find whether there
is a significant decrease in the consumption of tea after the increase in
duty. Use a 1% level of significance (z at 1% level of significance is
2.580)
04
(c) Test made on the breaking strength of 10 pieces of a metal gave the
following results: 578, 572, 570, 568, 572, 570, 570, 572, 596 and 584
kg. Test at 5% Level of significance if the mean breaking strength of
the wire can be assumed as 577 kg. (for v = 9, t0.05 = 2.26)
07
Q.5 (a) In a sample of 1000 the mean is 17.5 and S.D is 2.5. In another sample
of 800 the mean is 18 and S.D. is 2.7. Assume that the sample are
independent. Discuss at 1% level of significance, whether the two
samples can have come from a population which have the same S. D.
(Z at 1% LOS is 2.58)
03
(b) The following data give the number of aircraft accidents that occurred
during the various days of a week.
Day: Mon Tue Wed Thu Fri Sat
No. of
accidents:
15 19 13 12 16 15
Using ?? 2
distribution, test whether the accidents are uniformly
distributed over the week at 95% confidence level.
(for v = 5, ?? 0.05
2
= 11.07)
04
(c) A small project is composed of 7 activities whose time estimates are
listed in the table below. Activities are identified by their beginning i
and ending j node numbers.
07
FirstRanker.com - FirstRanker's Choice
1
Seat No.: ________ Enrolment No.___________
GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER ? IV (New) EXAMINATION ? WINTER 2019
Subject Code: 2141703 Date: 07/12/2019
Subject Name: Numerical Techniques & Statistical Methods
Time: 10:30 AM TO 01:30 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) Find the relative error if the number X = 0.004997 is
(1) Truncated to three decimal digits
(2) Rounded off to three decimal digits
03
(b) Calculate mean, median and standard deviation from the following
data:
Wages
(Rs.?000):
0-10 10-20 20-30 30-40 40-50 50-60
No. of
workers:
12 17 23 39 16 03
04
(c) Explain diagonally dominant system of equations. Solve the following
system of equations by Gauss Jacobi method.
3x + 20y ? z = -18, 2x ? 3y + 20z = 25, 20x + y ? 2z = 17
07
Q.2 (a) The following data gives the velocity of a particle for 20 seconds at an
interval of 5 seconds. Find the initial acceleration using the entire data:
Time t (sec): 0 5 10 15 20
Velocity v(m/sec): 0 3 14 69 228
03
(b) A curve passes through the points (0, 18), (1, 10), (3, -18) and (6, 90).
Find the slope of the curve at x = 2 using Lagrange Interpolation
technique.
04
(c) Evaluate the integral
? (
?? ???????? ) ???? 0.5
0
Using Romberg?s method, correct to 3 decimal places.
07
OR
(c) Evaluate
? ?? ??? 2
???? 1.5
0.2
Using Gaussian quadrature formula for n = 2 and n = 3.
07
Q.3 (a) Find by Newton?s method, the real positive root of the equation
x
4
? x =10, correct to three decimal places.
03
(b) Using Taylor?s series method find approximate value of y at x = 0.2 for
the differential equation dy/dx = 2y + 3e
x
, y (0) = 0. Compare the
numerical solution obtained with the exact solution.
04
(c) The following values of x and y are given:
x: 1 2 3 4
y: 1 2 5 11
Find the cubic splines and evaluate y (1.5) and y
?
(3).
07
OR
2
Q.3 (a) For the given set of order pairs (x0, y 0), (x1, y 1), define first order forward
difference, first order backward difference and first order divided
difference.
03
(b) Using modified Euler?s method, find y (0.2), given
dy/dx = y + e
x
, y (0) = 0
04
(c) Apply Runge-Kutta method to find approximate value of y for x = 0.2,
in steps of 0.1, if dy/dx = x + y
2
, given that y = 1 when x = 0.
07
Q.4 (a) A salesman wants to know the average number of units he sells per
sales call. He checks his past sales records and comes up with the
following probabilities:
Sales in units: 0 1 2 3 4 5
Probability: 0.15 0.20 0.10 0.05 0.30 0.20
What is the average number of units he sells per sales call?
03
(b) The mean and standard deviation of a population are 225 and 278
respectively. What can we assert with 95% confidence about the
maximum error if ?? ? = 225 and n = 100. Also, construct 95%
confidence Interval.
04
(c) Out of 800 families with 4 children each, what percentage would be
expected to have (a) 2 boys and 2 girls (b) at least one boy (c) no girls
and (d) at the most 2 girls. Assume equal probabilities for boys and
girls.
07
OR
Q.4 (a) 500 apples are taken at random from a large basket and 50 are found to
be bad. Estimate the proportion of bad apples in the basket and assign
limits within which the percentage most probably lies. (critical value at
the most is 3)
03
(b) Before an increase in excise duty on tea, 800 people out of a sample of
1000 were consumers of tea. After the increase in duty, 800 people
were consumers of tea in a sample of 1200 persons. Find whether there
is a significant decrease in the consumption of tea after the increase in
duty. Use a 1% level of significance (z at 1% level of significance is
2.580)
04
(c) Test made on the breaking strength of 10 pieces of a metal gave the
following results: 578, 572, 570, 568, 572, 570, 570, 572, 596 and 584
kg. Test at 5% Level of significance if the mean breaking strength of
the wire can be assumed as 577 kg. (for v = 9, t0.05 = 2.26)
07
Q.5 (a) In a sample of 1000 the mean is 17.5 and S.D is 2.5. In another sample
of 800 the mean is 18 and S.D. is 2.7. Assume that the sample are
independent. Discuss at 1% level of significance, whether the two
samples can have come from a population which have the same S. D.
(Z at 1% LOS is 2.58)
03
(b) The following data give the number of aircraft accidents that occurred
during the various days of a week.
Day: Mon Tue Wed Thu Fri Sat
No. of
accidents:
15 19 13 12 16 15
Using ?? 2
distribution, test whether the accidents are uniformly
distributed over the week at 95% confidence level.
(for v = 5, ?? 0.05
2
= 11.07)
04
(c) A small project is composed of 7 activities whose time estimates are
listed in the table below. Activities are identified by their beginning i
and ending j node numbers.
07
3
Activity (i ? j) Estimated Duration (weeks)
Optimistic Most likely Pessimistic
1-2 3 5 8
1-3 2 4 8
1-4 6 8 12
2-5 5 9 12
3-5 3 5 9
4-6 3 6 10
5-6 2 4 8
(a) Draw the network for this project
(b) Determine the expected time and variance for each activity
(c) Find the critical path and the project variance.
(d)What is the probability that the project will be completed in 22
days?
OR
Q.5 (a) A little league baseball coach wants to know if his team is
representative of other teams in scoring runs. Nationally the average
number of runs scored by a little league team in a game is 5.7. He
chooses five games at random in which his team scored 5,9,4,11 and 8
runs. Is it likely that his team?s scores could have come from the
national distributions? Assume ?? = 0.05.
03
(b) In large city A, 20% of a random sample of 900 school children had
defective eye sight. In another large city B, 15% of a random sample of
1600 children had the same defect. Obtain 95% confidence limits for
the difference in the population proportions.
04
(c) An architect has been awarded a contract to prepare plans for an urban
renewal project. The job consists of the following activities and their
estimated times:
Activity Description Immediate
Predecessors
Time (days)
A Prepare preliminary
sketches
-- 2
B Outline
specifications
-- 1
C Prepare drawings A 3
D Write specifications A, B 2
E Run off prints C, D 1
F Have specification B, D 3
G Assemble bid
packages
E, F 1
(a) Draw the network diagram of activities for the project
Indicate the critical path and calculate the total float and free float for
each activity.
07
*************
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This post was last modified on 20 February 2020