Download VTU ((Visvesvaraya Technological University) B.E/B-Tech 2019 July ( Bachelor of Engineering) First & Second Semester (1st Semester & 2nd Semester) 15 Scheme 15MAT41 Engineering Mathematics IV Question Paper
USIV5
15MAT41
Fourth Semester B.E. Degree Examination, June/July 2019
Engineering Mathematics - IV
Time: 3 hrs.
Max. Marks: 80
Note: Answer any FIVE full questions, choosing ONE full question front each module,
ti
U
. -
tj
Module-I
P
1 a. Employ Taylor's series method, find y(0.1) considering upto third degree term if y(x) cl
-0
satisfies the equation ?
dy
= x - y` , y(0) = 1.
,.. -
,,,,
(05 Marks)
O dx
P
,--:
t.0 .
?
.c b. Using Runge-Kutta method of fourth order, find y(0.1) for the equation ?
dy
=
y - x
,,, .. ?
dx y + x '
cd ?
q '='
y(0) = 1 taking h = 0.1.
(05 Marks)
,
......,
c
,,
.c
II
dv
C.
Apply Milne's method to compute y(1.4) correct to four decimal places given --=, = x
-
,
+ y ?
dx 2
and following the data : y(1) = 2, y(1.1) = 2.2156, y(1.2) = 2.4649, y(1.3) = 2.7514. ,
, ''''
a
. - (06 Marks)
?.'
OR
. ,-.
.,..
?
2 a.
Use Taylor's series method to find y(4.1) given that (x
2
+ y)y' = 1 and y(4) = 4. (05 Marks)
, cz
..,.
h. Find y at x = 0.8, given y' = x - i and y(0) = 0, y(0.2) = 0.02, y(0.4) = 0.0795,
U
=
1 '6
y(0.6) = 0.1762. Using Adams - Bashforth method. Apply the corrector formula. (05 Marks)
O
7
0
CI CZ c.
Using Modified Euler's method find y at x = 0.1 given y --- 3x + Y with y(0) = 1 taking
.1) c
???-??
2
, -
... ,t,
-ti
,.. _ h = 0.1. (06 Marks)
csi
>
,
.
..
.0
,
7
Module-2
,,
? 3 a.
Obtain the solution of the equation 2y" = 4x + y' with initial conditions y(1) = 2,
O .
O c.
d
y(1.1) - 2.2156, y(1.2) = 2.4649, y(1.3) = 2.7514 and y(1) = 2, y'(1.1) = 2.3178,
, ,
....?
3
, ?,. ..z_ ,;,
0
y(1_2) = 2.6725, y'(1 _3) = 3.0657 by computing y(1.4) applying Milne's method. (05 Marks
;...
t
,
,
1
ca -,
b. If a and r3 are two distinct roots ofJ,,(x) = 0 then prove that i x.1?(ax),I
n
( x)dx = 0 if a # 0.
O
?
--
D
,,,?
to
(05 Marks)
c. J .i4)
E
O cL)
C.
Show that .1
1
,(x) =
1
cos x
-,
(06 Marks)
2
0-
. -
7X
,- _,
ad
- ,-,i
0
4
z
-
a_
OR
Given Y
r
- xy' - y = 0 with the initial conditions y(0) = 1, y'(0) = 0. Compute y(0.2) and
y(0.2) by taking h = 0.2 using Runge - Kutta method of fourth order. (05 Marks)
b. If ,('+.2x
2
- x + 1 = aPo(x) + bPi(x) + cP2(x) + dP3(x) then, find the values of a, b, c, d.
F.
(05 Marks)
c. Derive Rodrigue's formula
1 d" 2
1
y
(06 Marks)
p
n
) =
2'n! dx"
1 of 3
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USIV5
15MAT41
Fourth Semester B.E. Degree Examination, June/July 2019
Engineering Mathematics - IV
Time: 3 hrs.
Max. Marks: 80
Note: Answer any FIVE full questions, choosing ONE full question front each module,
ti
U
. -
tj
Module-I
P
1 a. Employ Taylor's series method, find y(0.1) considering upto third degree term if y(x) cl
-0
satisfies the equation ?
dy
= x - y` , y(0) = 1.
,.. -
,,,,
(05 Marks)
O dx
P
,--:
t.0 .
?
.c b. Using Runge-Kutta method of fourth order, find y(0.1) for the equation ?
dy
=
y - x
,,, .. ?
dx y + x '
cd ?
q '='
y(0) = 1 taking h = 0.1.
(05 Marks)
,
......,
c
,,
.c
II
dv
C.
Apply Milne's method to compute y(1.4) correct to four decimal places given --=, = x
-
,
+ y ?
dx 2
and following the data : y(1) = 2, y(1.1) = 2.2156, y(1.2) = 2.4649, y(1.3) = 2.7514. ,
, ''''
a
. - (06 Marks)
?.'
OR
. ,-.
.,..
?
2 a.
Use Taylor's series method to find y(4.1) given that (x
2
+ y)y' = 1 and y(4) = 4. (05 Marks)
, cz
..,.
h. Find y at x = 0.8, given y' = x - i and y(0) = 0, y(0.2) = 0.02, y(0.4) = 0.0795,
U
=
1 '6
y(0.6) = 0.1762. Using Adams - Bashforth method. Apply the corrector formula. (05 Marks)
O
7
0
CI CZ c.
Using Modified Euler's method find y at x = 0.1 given y --- 3x + Y with y(0) = 1 taking
.1) c
???-??
2
, -
... ,t,
-ti
,.. _ h = 0.1. (06 Marks)
csi
>
,
.
..
.0
,
7
Module-2
,,
? 3 a.
Obtain the solution of the equation 2y" = 4x + y' with initial conditions y(1) = 2,
O .
O c.
d
y(1.1) - 2.2156, y(1.2) = 2.4649, y(1.3) = 2.7514 and y(1) = 2, y'(1.1) = 2.3178,
, ,
....?
3
, ?,. ..z_ ,;,
0
y(1_2) = 2.6725, y'(1 _3) = 3.0657 by computing y(1.4) applying Milne's method. (05 Marks
;...
t
,
,
1
ca -,
b. If a and r3 are two distinct roots ofJ,,(x) = 0 then prove that i x.1?(ax),I
n
( x)dx = 0 if a # 0.
O
?
--
D
,,,?
to
(05 Marks)
c. J .i4)
E
O cL)
C.
Show that .1
1
,(x) =
1
cos x
-,
(06 Marks)
2
0-
. -
7X
,- _,
ad
- ,-,i
0
4
z
-
a_
OR
Given Y
r
- xy' - y = 0 with the initial conditions y(0) = 1, y'(0) = 0. Compute y(0.2) and
y(0.2) by taking h = 0.2 using Runge - Kutta method of fourth order. (05 Marks)
b. If ,('+.2x
2
- x + 1 = aPo(x) + bPi(x) + cP2(x) + dP3(x) then, find the values of a, b, c, d.
F.
(05 Marks)
c. Derive Rodrigue's formula
1 d" 2
1
y
(06 Marks)
p
n
) =
2'n! dx"
1 of 3
Module-3
5 a. State and prove Cauchy-Reimann equation in polar form. (05 Marks)
b. Discuss the transformation w = z
-
(05 Marks)
c. Find the bilinear transformation which maps the points z = 1 , i ?1 into w = 2 , i , ?2.
(06 Marks)
OR
6 a. Find the analytic function whose real part is
x
4
- y
4
x
X
2
? y2
(05 Marks)
b. State and prove Cauchy Integral formula. (05 Marks)
c. Evaluate
e
dz where c is the circle : z = 3 using Cauchy's Residue
(z +1)(z ?2)
theorem_
(06 Marks,__-_--
Module-4
7 a_ The probability function of a variate x is
x 0 1 2 3 4 5 6 7
p(x) 0 k 2k 2k 3k k
2
2k
2
7k
2
+ k
(i) Find k (ii) Evaluate p(x < 6), p(x 6) and p(3 < x 6). (05 Marks)
b. Obtain mean and standard deviation of Binomial distribution. (05 Marks)
c. The joint distribution of two discrete variables x and y is f(x, y) = (2x + y) where x and
y are integers such that 0 x 2 y 3.
Find : (i) Marginal distribution of x and y.
(ii) Are x and y independent. (06 Marks)
OR
8 a. The marks of 1000 students in an examination follows a normal distribution with mea.
70 and standard deviation 5. Find the number of students whose marks will be
(i) less than 65 (ii) more than 75 (iii) between 65 and 75 [Given (1(1) = 0.3413]
(05 Marks)
b. If the probability of a bad reaction from a certain injection is 0.001, determine the chance
that out of 2000 individuals, more than two will get a bad reaction. (05 Marks)
c. The joint distribution of the random variables X and Y are given. Find the corresponding
marginal distribution. Also compute the covariance and the correlation of the random
variables X and Y. (06 Marks)
X \ Y 1 3 9
2 1/8 1/24 1/12
4 1/4 1/4 0
_ 6 1/8 1/24 1/12
f.- . th
,
LINCit L
ip
h
C PIIKOM
LI
MART
(r%
?
`waik of En09 lir
t&
2 of 3
FirstRanker.com - FirstRanker's Choice
socao"
USIV5
15MAT41
Fourth Semester B.E. Degree Examination, June/July 2019
Engineering Mathematics - IV
Time: 3 hrs.
Max. Marks: 80
Note: Answer any FIVE full questions, choosing ONE full question front each module,
ti
U
. -
tj
Module-I
P
1 a. Employ Taylor's series method, find y(0.1) considering upto third degree term if y(x) cl
-0
satisfies the equation ?
dy
= x - y` , y(0) = 1.
,.. -
,,,,
(05 Marks)
O dx
P
,--:
t.0 .
?
.c b. Using Runge-Kutta method of fourth order, find y(0.1) for the equation ?
dy
=
y - x
,,, .. ?
dx y + x '
cd ?
q '='
y(0) = 1 taking h = 0.1.
(05 Marks)
,
......,
c
,,
.c
II
dv
C.
Apply Milne's method to compute y(1.4) correct to four decimal places given --=, = x
-
,
+ y ?
dx 2
and following the data : y(1) = 2, y(1.1) = 2.2156, y(1.2) = 2.4649, y(1.3) = 2.7514. ,
, ''''
a
. - (06 Marks)
?.'
OR
. ,-.
.,..
?
2 a.
Use Taylor's series method to find y(4.1) given that (x
2
+ y)y' = 1 and y(4) = 4. (05 Marks)
, cz
..,.
h. Find y at x = 0.8, given y' = x - i and y(0) = 0, y(0.2) = 0.02, y(0.4) = 0.0795,
U
=
1 '6
y(0.6) = 0.1762. Using Adams - Bashforth method. Apply the corrector formula. (05 Marks)
O
7
0
CI CZ c.
Using Modified Euler's method find y at x = 0.1 given y --- 3x + Y with y(0) = 1 taking
.1) c
???-??
2
, -
... ,t,
-ti
,.. _ h = 0.1. (06 Marks)
csi
>
,
.
..
.0
,
7
Module-2
,,
? 3 a.
Obtain the solution of the equation 2y" = 4x + y' with initial conditions y(1) = 2,
O .
O c.
d
y(1.1) - 2.2156, y(1.2) = 2.4649, y(1.3) = 2.7514 and y(1) = 2, y'(1.1) = 2.3178,
, ,
....?
3
, ?,. ..z_ ,;,
0
y(1_2) = 2.6725, y'(1 _3) = 3.0657 by computing y(1.4) applying Milne's method. (05 Marks
;...
t
,
,
1
ca -,
b. If a and r3 are two distinct roots ofJ,,(x) = 0 then prove that i x.1?(ax),I
n
( x)dx = 0 if a # 0.
O
?
--
D
,,,?
to
(05 Marks)
c. J .i4)
E
O cL)
C.
Show that .1
1
,(x) =
1
cos x
-,
(06 Marks)
2
0-
. -
7X
,- _,
ad
- ,-,i
0
4
z
-
a_
OR
Given Y
r
- xy' - y = 0 with the initial conditions y(0) = 1, y'(0) = 0. Compute y(0.2) and
y(0.2) by taking h = 0.2 using Runge - Kutta method of fourth order. (05 Marks)
b. If ,('+.2x
2
- x + 1 = aPo(x) + bPi(x) + cP2(x) + dP3(x) then, find the values of a, b, c, d.
F.
(05 Marks)
c. Derive Rodrigue's formula
1 d" 2
1
y
(06 Marks)
p
n
) =
2'n! dx"
1 of 3
Module-3
5 a. State and prove Cauchy-Reimann equation in polar form. (05 Marks)
b. Discuss the transformation w = z
-
(05 Marks)
c. Find the bilinear transformation which maps the points z = 1 , i ?1 into w = 2 , i , ?2.
(06 Marks)
OR
6 a. Find the analytic function whose real part is
x
4
- y
4
x
X
2
? y2
(05 Marks)
b. State and prove Cauchy Integral formula. (05 Marks)
c. Evaluate
e
dz where c is the circle : z = 3 using Cauchy's Residue
(z +1)(z ?2)
theorem_
(06 Marks,__-_--
Module-4
7 a_ The probability function of a variate x is
x 0 1 2 3 4 5 6 7
p(x) 0 k 2k 2k 3k k
2
2k
2
7k
2
+ k
(i) Find k (ii) Evaluate p(x < 6), p(x 6) and p(3 < x 6). (05 Marks)
b. Obtain mean and standard deviation of Binomial distribution. (05 Marks)
c. The joint distribution of two discrete variables x and y is f(x, y) = (2x + y) where x and
y are integers such that 0 x 2 y 3.
Find : (i) Marginal distribution of x and y.
(ii) Are x and y independent. (06 Marks)
OR
8 a. The marks of 1000 students in an examination follows a normal distribution with mea.
70 and standard deviation 5. Find the number of students whose marks will be
(i) less than 65 (ii) more than 75 (iii) between 65 and 75 [Given (1(1) = 0.3413]
(05 Marks)
b. If the probability of a bad reaction from a certain injection is 0.001, determine the chance
that out of 2000 individuals, more than two will get a bad reaction. (05 Marks)
c. The joint distribution of the random variables X and Y are given. Find the corresponding
marginal distribution. Also compute the covariance and the correlation of the random
variables X and Y. (06 Marks)
X \ Y 1 3 9
2 1/8 1/24 1/12
4 1/4 1/4 0
_ 6 1/8 1/24 1/12
f.- . th
,
LINCit L
ip
h
C PIIKOM
LI
MART
(r%
?
`waik of En09 lir
t&
2 of 3
15MAT41
Module - 5
9 a. Explain the terms: (i) Null hypothesis (ii) type-I and type-II errors (iii) Significance level
(05 Marks)
b. In 324 throws of a six faced 'die', an odd number turned up 1 8 1 times. Is it reasonable to
think that 'die' is an unbiased one? (05 Marks)
c. Three boys A, B, C are throwing ball to each other. A always throws the ball to B and B
always throws the ball to C. C is just as likely to throw the ball to B as to A. If C was the
first person to throw the ball find the probabilities that after three throws (i) A has the ball
(ii) B has the ball (iii) C has the ball. (06 Marks)
OR
10 a. Find the unique fixed probability vector for the matrix
0 2/3 1/3
1/2 0 1/2
1/2 1/2 0
(05 Marks)
b.
A random sample for 1000 workers in company has mean wage of Rs. 50 per day and
standard deviation of Rs. 15. Another sample of 1500 workers from another company has
mean wage of Rs. 45 per day and standard deviation of Rs. 20. Does the mean rate of wages
varies between the two companies? (05 Marks)
c.
A die is thrown 264 times and the number appearing on the face (x) follows the following
frequency distribu
Calculate the value of x
2
. (06 Marks)
x 1 2 3 4 5 6
f 40 32 28 58 54 60
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This post was last modified on 01 January 2020