Download VTU ((Visvesvaraya Technological University) B.E/B-Tech 2019 July ( Bachelor of Engineering) First & Second Semester (1st Semester & 2nd Semester) 17 Scheme 17MAT411 Engineering Mathematics IV Question Paper
chpt
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17MAT411
Fourth Semester B.E. Degree Examination, June/Jul 2019
Engineering Mathematics - IV
Time: 3 hrs.
Max. Marks: 100
Note: Answer any FIVE full questions, choosing ONE full question from each module.
. c)
Module-1
.3
Q,
Ei,
a. If y' + y + 2x = 0 , y(0) = -1 then find y(0.1) by using Taylor's series method. Consider upto
,
,
a
E
third order derivative term. (06 Marks)
b. Find y(0. 2) by using modified Euler's method, given that y' = x + y , y(0) = 1.
,.,
Take h = 0.1and carry out two modifications at each step. (07 Marks)
I,
1
rpi c .,
? r c. If y' - , y(0) = 2 , y(0.2) = 2.0933 , y(0.4) = 2.1755 , y(0.6) = 2.2493 then find
0.0--
P..
ca ..,.
x + y
..x
E o
y(0.8) by Milne's method. (07 Marks)
si
E
,;',
.= 4
OR
r- 1
2 a. Use Taylor's series method to find y(0.1) from y' = 3x + y
2
, y(0) - I . Consider upto fourth
7.
?
1
P
derivative term.
x-
-,
y , y
(06 Marks)
b. Use Runge - Kutta method to find y(0.1) from = + (0) = -1.
= o
m y ' (07 Marks)
c
.,.
c. Use Adam - Bashforth method to find y(0.4) from y' = 1 xy , y(0) = 1 , y(0.1) = I.0025 ,
. ..,
.E. 7.
2
v, ,7:
y(0.2) = 1.0101 , y(0.3) = 1.0228. vt .,,,, (07 Marks)
U
F
Module-2
-
?
-0
?-
9
g
7
- : 3 a. Express x
3
- 5x
2
+ 6x + 1 in terms of Legendre polynomials.
' -' 8
(06 Marks)
-b
3 7' ,
,
,
b. Find y(0.1), by using Runge - Kutta method , given that y" + xy' + y = 0 , y(0) = I
5r
4, y
1
(0) = O.
-
?
8
(07 Marks)
f.:,
c
c. Solve Bessel's operation leading to J,,(x).
O
(07 Marks)
-= c*
- c)
c..
c
.
, cL, OR
?
3
= .5
.
z 4 a. Prove that J
y
, (x) -
2
sin x. (06 Marks)
,3
3 (-) 71 X
''
L--:
cz
-
th' b. Find y(0.4) by using Milne's method, given that y(0) = 1 , y'(0) = 1 , y(0.1) = 1.0998 ,
-?,
O y' (0.1) = 0.9946 , y(0.2) = 1.1987 , y'(0.2) = 0.9773 , y(0.3) = 1.2955 , y'(0.3) = 0.946.
?,,,
th -
= ..0
(07 Marks)
c. State and prove Rodrigue's formula.
CL 4)
(07 Marks)
g ?
3
c.,
Module-3
o ,-, .
o <
5 a. Derive Cauchy - Riemann equations in Cartesian coordinates. (06 Marks)
r
,
i
b. Find an analytic function f(z) = u + iv in terms of z , given that u = e
2
x(x cos 2y - y sin 2y).
(07 Marks)
f
(z-1)(z- 2)
sin Utz
-
+ cos IT Z -
g, c. Evaluate
dz , c is Iz1= 3 by residue theorem. (07 Marks)
o
OR
1 of 3
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chpt
CHIKODI
*\
LIBRARY
?
17MAT411
Fourth Semester B.E. Degree Examination, June/Jul 2019
Engineering Mathematics - IV
Time: 3 hrs.
Max. Marks: 100
Note: Answer any FIVE full questions, choosing ONE full question from each module.
. c)
Module-1
.3
Q,
Ei,
a. If y' + y + 2x = 0 , y(0) = -1 then find y(0.1) by using Taylor's series method. Consider upto
,
,
a
E
third order derivative term. (06 Marks)
b. Find y(0. 2) by using modified Euler's method, given that y' = x + y , y(0) = 1.
,.,
Take h = 0.1and carry out two modifications at each step. (07 Marks)
I,
1
rpi c .,
? r c. If y' - , y(0) = 2 , y(0.2) = 2.0933 , y(0.4) = 2.1755 , y(0.6) = 2.2493 then find
0.0--
P..
ca ..,.
x + y
..x
E o
y(0.8) by Milne's method. (07 Marks)
si
E
,;',
.= 4
OR
r- 1
2 a. Use Taylor's series method to find y(0.1) from y' = 3x + y
2
, y(0) - I . Consider upto fourth
7.
?
1
P
derivative term.
x-
-,
y , y
(06 Marks)
b. Use Runge - Kutta method to find y(0.1) from = + (0) = -1.
= o
m y ' (07 Marks)
c
.,.
c. Use Adam - Bashforth method to find y(0.4) from y' = 1 xy , y(0) = 1 , y(0.1) = I.0025 ,
. ..,
.E. 7.
2
v, ,7:
y(0.2) = 1.0101 , y(0.3) = 1.0228. vt .,,,, (07 Marks)
U
F
Module-2
-
?
-0
?-
9
g
7
- : 3 a. Express x
3
- 5x
2
+ 6x + 1 in terms of Legendre polynomials.
' -' 8
(06 Marks)
-b
3 7' ,
,
,
b. Find y(0.1), by using Runge - Kutta method , given that y" + xy' + y = 0 , y(0) = I
5r
4, y
1
(0) = O.
-
?
8
(07 Marks)
f.:,
c
c. Solve Bessel's operation leading to J,,(x).
O
(07 Marks)
-= c*
- c)
c..
c
.
, cL, OR
?
3
= .5
.
z 4 a. Prove that J
y
, (x) -
2
sin x. (06 Marks)
,3
3 (-) 71 X
''
L--:
cz
-
th' b. Find y(0.4) by using Milne's method, given that y(0) = 1 , y'(0) = 1 , y(0.1) = 1.0998 ,
-?,
O y' (0.1) = 0.9946 , y(0.2) = 1.1987 , y'(0.2) = 0.9773 , y(0.3) = 1.2955 , y'(0.3) = 0.946.
?,,,
th -
= ..0
(07 Marks)
c. State and prove Rodrigue's formula.
CL 4)
(07 Marks)
g ?
3
c.,
Module-3
o ,-, .
o <
5 a. Derive Cauchy - Riemann equations in Cartesian coordinates. (06 Marks)
r
,
i
b. Find an analytic function f(z) = u + iv in terms of z , given that u = e
2
x(x cos 2y - y sin 2y).
(07 Marks)
f
(z-1)(z- 2)
sin Utz
-
+ cos IT Z -
g, c. Evaluate
dz , c is Iz1= 3 by residue theorem. (07 Marks)
o
OR
1 of 3
Given that X
2
= (07 Marks)
Day Mon Tue Wed Thu Fri Sat
No. of accidents 14 18 12 11 15 14
0
2 2
6 a. Prove that H
T
+ , 1f(z)1
2
= 41f ' (z)1
2
.
ax
-
(06 Marks)
b. Discuss the transformation W = Z
2
. (07 Marks)
c. Find a bilinear transformation that maps the points . i o in Z ? plane into -I 1 in
W ? plane respectively. (07 Marks)
Module-4
7 a. In a sampling a large number of parts manufactured by a machine, the mean number of
defectives in a sample of 20 is 2, out of 1000 such samples , how many would be expected to
contain atleast 3 defective parts? (06 Marks)
b. If X is a normal variate with mean 30 and standard deviation 5, find the probabilities that
i) 26 < X 5_ 40 ii) X > 45 iii) 1X ? 301> 5.
Given that (1)(0.8) = 0.288 , (1)(2.0) = 0.4772 , (1)(3) = 0.4987 , 4(1) = 0.3413. (07 Marks)
c. The joint density function of two continuous random variables X and Y is given by
{K xy, 0 x 5_ 4, 1 < y < 5
f(x, y)
0, otherwise
Find i) K ii) E(x) iii) E(2x + 3y). (07 Marks)
OR
8 a. Derive mean and standard deviation of the Poisson distribution. (06 Marks)
b. The joint probability distribution for two random variables X and Y as follows :
--
X
---
------___Y_ -2 -1 4 5
1 0.1 0.2 0 0.3
2 0.2 0.1 0.3 0
Find 1) Expectations of X, Y , XY SD of X and Y iii) Covariance of X, Y
iv) Correlation of X and Y. (07 Marks)
c. In a certain town the duration of shower has mean 5 minutes. What is the probability that
shower will last for i) 10 minutes or more ii) Less than 10 minutes iii) Between
10 and 12 minutes. (07 Marks)
Module-5
9 a. A group of boys and girls were given in Intelligence test. The mean score , SD score and
numbers in each group are as follows : (06 Marks)
Boys Girls
Mean 74 70
SD 8 10
X 12 10
Is the difference between the means of the two groups significant at 5% level of
significance? Given that to,05 2.086 for 20
b. The following table gives the number of accidents that take place in an industry during
various days of the week. Test if accidents are uniformly distributed over the week.
2 of 3
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chpt
CHIKODI
*\
LIBRARY
?
17MAT411
Fourth Semester B.E. Degree Examination, June/Jul 2019
Engineering Mathematics - IV
Time: 3 hrs.
Max. Marks: 100
Note: Answer any FIVE full questions, choosing ONE full question from each module.
. c)
Module-1
.3
Q,
Ei,
a. If y' + y + 2x = 0 , y(0) = -1 then find y(0.1) by using Taylor's series method. Consider upto
,
,
a
E
third order derivative term. (06 Marks)
b. Find y(0. 2) by using modified Euler's method, given that y' = x + y , y(0) = 1.
,.,
Take h = 0.1and carry out two modifications at each step. (07 Marks)
I,
1
rpi c .,
? r c. If y' - , y(0) = 2 , y(0.2) = 2.0933 , y(0.4) = 2.1755 , y(0.6) = 2.2493 then find
0.0--
P..
ca ..,.
x + y
..x
E o
y(0.8) by Milne's method. (07 Marks)
si
E
,;',
.= 4
OR
r- 1
2 a. Use Taylor's series method to find y(0.1) from y' = 3x + y
2
, y(0) - I . Consider upto fourth
7.
?
1
P
derivative term.
x-
-,
y , y
(06 Marks)
b. Use Runge - Kutta method to find y(0.1) from = + (0) = -1.
= o
m y ' (07 Marks)
c
.,.
c. Use Adam - Bashforth method to find y(0.4) from y' = 1 xy , y(0) = 1 , y(0.1) = I.0025 ,
. ..,
.E. 7.
2
v, ,7:
y(0.2) = 1.0101 , y(0.3) = 1.0228. vt .,,,, (07 Marks)
U
F
Module-2
-
?
-0
?-
9
g
7
- : 3 a. Express x
3
- 5x
2
+ 6x + 1 in terms of Legendre polynomials.
' -' 8
(06 Marks)
-b
3 7' ,
,
,
b. Find y(0.1), by using Runge - Kutta method , given that y" + xy' + y = 0 , y(0) = I
5r
4, y
1
(0) = O.
-
?
8
(07 Marks)
f.:,
c
c. Solve Bessel's operation leading to J,,(x).
O
(07 Marks)
-= c*
- c)
c..
c
.
, cL, OR
?
3
= .5
.
z 4 a. Prove that J
y
, (x) -
2
sin x. (06 Marks)
,3
3 (-) 71 X
''
L--:
cz
-
th' b. Find y(0.4) by using Milne's method, given that y(0) = 1 , y'(0) = 1 , y(0.1) = 1.0998 ,
-?,
O y' (0.1) = 0.9946 , y(0.2) = 1.1987 , y'(0.2) = 0.9773 , y(0.3) = 1.2955 , y'(0.3) = 0.946.
?,,,
th -
= ..0
(07 Marks)
c. State and prove Rodrigue's formula.
CL 4)
(07 Marks)
g ?
3
c.,
Module-3
o ,-, .
o <
5 a. Derive Cauchy - Riemann equations in Cartesian coordinates. (06 Marks)
r
,
i
b. Find an analytic function f(z) = u + iv in terms of z , given that u = e
2
x(x cos 2y - y sin 2y).
(07 Marks)
f
(z-1)(z- 2)
sin Utz
-
+ cos IT Z -
g, c. Evaluate
dz , c is Iz1= 3 by residue theorem. (07 Marks)
o
OR
1 of 3
Given that X
2
= (07 Marks)
Day Mon Tue Wed Thu Fri Sat
No. of accidents 14 18 12 11 15 14
0
2 2
6 a. Prove that H
T
+ , 1f(z)1
2
= 41f ' (z)1
2
.
ax
-
(06 Marks)
b. Discuss the transformation W = Z
2
. (07 Marks)
c. Find a bilinear transformation that maps the points . i o in Z ? plane into -I 1 in
W ? plane respectively. (07 Marks)
Module-4
7 a. In a sampling a large number of parts manufactured by a machine, the mean number of
defectives in a sample of 20 is 2, out of 1000 such samples , how many would be expected to
contain atleast 3 defective parts? (06 Marks)
b. If X is a normal variate with mean 30 and standard deviation 5, find the probabilities that
i) 26 < X 5_ 40 ii) X > 45 iii) 1X ? 301> 5.
Given that (1)(0.8) = 0.288 , (1)(2.0) = 0.4772 , (1)(3) = 0.4987 , 4(1) = 0.3413. (07 Marks)
c. The joint density function of two continuous random variables X and Y is given by
{K xy, 0 x 5_ 4, 1 < y < 5
f(x, y)
0, otherwise
Find i) K ii) E(x) iii) E(2x + 3y). (07 Marks)
OR
8 a. Derive mean and standard deviation of the Poisson distribution. (06 Marks)
b. The joint probability distribution for two random variables X and Y as follows :
--
X
---
------___Y_ -2 -1 4 5
1 0.1 0.2 0 0.3
2 0.2 0.1 0.3 0
Find 1) Expectations of X, Y , XY SD of X and Y iii) Covariance of X, Y
iv) Correlation of X and Y. (07 Marks)
c. In a certain town the duration of shower has mean 5 minutes. What is the probability that
shower will last for i) 10 minutes or more ii) Less than 10 minutes iii) Between
10 and 12 minutes. (07 Marks)
Module-5
9 a. A group of boys and girls were given in Intelligence test. The mean score , SD score and
numbers in each group are as follows : (06 Marks)
Boys Girls
Mean 74 70
SD 8 10
X 12 10
Is the difference between the means of the two groups significant at 5% level of
significance? Given that to,05 2.086 for 20
b. The following table gives the number of accidents that take place in an industry during
various days of the week. Test if accidents are uniformly distributed over the week.
2 of 3
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c. Find the unique fixed probability vector for the regular stochastic matrix.
0 1 0
A= 1/6 1/2 1/3 (07 Marks)
0 2/3 1/3
OR
10 a. Define the following terms :
i) Type I error and type 1I error.
ii) Transient state.
iii) Absorbing state. (06 Marks)
b. A certain stimulus administered to each of the 12 patients resulted in the following increases
of blood pressure : 5, 2, 8, -1, 3, 0, -2, 1, 5, 0, 4, 6. Can it be concluded that the stimulus will
be general be accompanied by an increase in blood pressure. Given that t
o
.
05
= 2.2 for
(07 Marks)
. Find the corresponding stationary probability vector. (07 Marks)
11 d.f.
0 2/3 1/3
c. If P = 1 / 2 0 1/ 2
1/2 1 / 2 0
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This post was last modified on 01 January 2020