Download VTU BE 2020 Jan [folder1] 4th Sem 17MAT41 Engineering Mathematics IV Question Paper

Download Visvesvaraya Technological University (VTU) BE-B.Tech (Bachelor of Engineering/ Bachelor of Technology) 2020 January [folder1] 4th Sem 17MAT41 Engineering Mathematics IV Previous Question Paper

17MAT41
USN
Fourth Semester B.E. Degree Examination, Dec.2019/Jan.2020
Engineering Mathematics - IV
Time: 3 hrs. Max. Marks: 100
Note: Answer any FIVE full questions, choosing
ONE full question from each module.
Module-1
1 a. From Taylor's series method, find y(0.1), considering upto fourth degree term if y(x)
satisfying the equation dy = x ? y
2
, y(0) = 1. (06 Marks)
dx
b. Using Runge-Kutta method of fourth order ?
dY
- + y = 2x at x = 1.1 given that y = 3 at x = 1
dx
initially. (07 Marks)
C.

If
dx =
2ex ? y, y(0) = 2, y(0.1) = 2.010, y(0.2) = 2.040 and y(0.3) = 2.090, find y(0.4)
correct upto four decimal places by using Milne's predictor-corrector formula. (07 Marks)
OR
2 a.
Using modified Euler's method find y at x = 0.2 given
dx

-I
d
= 3x + ?
1
y with y(0) = 1 taking
2
h = 0.1. (06 Marks)
b. Given ?
dy
+ y + zy
2
=0 and y(0) = 1, y(0.1) = 0.9008, y(0.2) = 0.8066, y(0.3) = 0.722.
dx
Evaluate y(0.4) by Adams-Bashforth method. (07 Marks)
dy y ? x
c. Using Runge-Kutta method of fourth order, find y(0.2) for the equation dx
y+x'
y(0) = 1 taking h = 0.2. (07 Marks)
Module-2
d
2

3 a. Apply Milne's method
dx
2

to compute y(0.8) given that =1? 2y
11
x
- ' and the following table
y
d
of initial values.
x 0 0.2 0.4 0.6
y 0 0.02 0.0795 0.1762
y' 0 0.1996 0.3937 0.5689
(06 Marks)
b. Express f(x) = x
4
+ 3x
3
? x
2
+ 5x ? 2 in terms of Legendre polynomials. (07 Marks)
c. Obtain the series solution of Bessel's differential equation x
2
y" + xy' + (x
2
+ n
2
) y = 0
leading to .1,,(x). (07 Marks)
I of 3
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17MAT41
USN
Fourth Semester B.E. Degree Examination, Dec.2019/Jan.2020
Engineering Mathematics - IV
Time: 3 hrs. Max. Marks: 100
Note: Answer any FIVE full questions, choosing
ONE full question from each module.
Module-1
1 a. From Taylor's series method, find y(0.1), considering upto fourth degree term if y(x)
satisfying the equation dy = x ? y
2
, y(0) = 1. (06 Marks)
dx
b. Using Runge-Kutta method of fourth order ?
dY
- + y = 2x at x = 1.1 given that y = 3 at x = 1
dx
initially. (07 Marks)
C.

If
dx =
2ex ? y, y(0) = 2, y(0.1) = 2.010, y(0.2) = 2.040 and y(0.3) = 2.090, find y(0.4)
correct upto four decimal places by using Milne's predictor-corrector formula. (07 Marks)
OR
2 a.
Using modified Euler's method find y at x = 0.2 given
dx

-I
d
= 3x + ?
1
y with y(0) = 1 taking
2
h = 0.1. (06 Marks)
b. Given ?
dy
+ y + zy
2
=0 and y(0) = 1, y(0.1) = 0.9008, y(0.2) = 0.8066, y(0.3) = 0.722.
dx
Evaluate y(0.4) by Adams-Bashforth method. (07 Marks)
dy y ? x
c. Using Runge-Kutta method of fourth order, find y(0.2) for the equation dx
y+x'
y(0) = 1 taking h = 0.2. (07 Marks)
Module-2
d
2

3 a. Apply Milne's method
dx
2

to compute y(0.8) given that =1? 2y
11
x
- ' and the following table
y
d
of initial values.
x 0 0.2 0.4 0.6
y 0 0.02 0.0795 0.1762
y' 0 0.1996 0.3937 0.5689
(06 Marks)
b. Express f(x) = x
4
+ 3x
3
? x
2
+ 5x ? 2 in terms of Legendre polynomials. (07 Marks)
c. Obtain the series solution of Bessel's differential equation x
2
y" + xy' + (x
2
+ n
2
) y = 0
leading to .1,,(x). (07 Marks)
I of 3
OR
4 a.
Given y" - xy' - y = 0 with the initial conditions y(0) = 1, y'(0) = 0, compute y(0.2)
y'(0.2) using fourth order Runge-Kutta method.
b. Prove J_
Iii
(k) =
1
1 ?
2
cos x .
7EX
Prove the Rodfigues formula P (x) =
1 d'y
(X
2 ?
1)
"

2" n! dx"
Module-3
5 a. Derive Cauchy-Riemann equations in Cartesian form.
b. Discuss the transformation w = z
-
.
e
2.2
c. By using Cauchy's residue theorem, evaluate
dz if C is the circle Izi = 3 .
(z +1)(z + 2)
c.
(06 Marks)
(07 Marks)
(07 Marks)
(06 Marks)
(07 Marks)
(07 Marks)
OR
a
, a ,
6 a. Prove that + If (z)
,

l
-
= 41f
1
(z)1"
ex' ay
e

State and prove Cauchy's integral formula.
Find the bilinear transformation which maps z = oo, i, 0 into w = -1, -i, 1.
b.
c.
Module-4
7 a. Find the mean and standard of Poisson distribution. (06 Marks)
b. In an examination 7% of students score less than 35 marks and 89% of the students score
less than 60 marks. Find the mean and standard deviation if the marks are normally
distributed given A(I.2263) = 0.39 and A(1.4757) = 0.43 (07 Marks)
c. The joint probability distributio
Determine:
i) Marginal distribution of X and Y
ii) Covariance of X and Y
iii) Correlation of X and Y (07 Marks)
OR
8 a. A random variable X has the following robability function:
x 0 1 2 3 4 5 6 7
P(x) 0 K 2k 2k 3k K
2
2k
2
7k
2
+k
Find K and evaluate P(x 6) P(3 < x 6). (06 Marks)
b.
The probability that a pen manufactured by a factory be defective is 1/10. If 12 such pens are
manufactured, what is the probability that
i) Exactly 2 are defective
ii) Atleast two are defective
iii) None of them are defective. (07 Marks)
c.
The length of telephone conversation in a booth has been exponential distribution and found
on an average to be 5 minutes. Find the probability that a random call made
i) Ends in less than 5 minutes
ii) Between 5 and 10 minutes.
(07 Marks)
2 of 3
V\
-2 -1 4
1 0.1 0.2 0 0.3
2 0.2 0.1 0.1 0
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17MAT41
USN
Fourth Semester B.E. Degree Examination, Dec.2019/Jan.2020
Engineering Mathematics - IV
Time: 3 hrs. Max. Marks: 100
Note: Answer any FIVE full questions, choosing
ONE full question from each module.
Module-1
1 a. From Taylor's series method, find y(0.1), considering upto fourth degree term if y(x)
satisfying the equation dy = x ? y
2
, y(0) = 1. (06 Marks)
dx
b. Using Runge-Kutta method of fourth order ?
dY
- + y = 2x at x = 1.1 given that y = 3 at x = 1
dx
initially. (07 Marks)
C.

If
dx =
2ex ? y, y(0) = 2, y(0.1) = 2.010, y(0.2) = 2.040 and y(0.3) = 2.090, find y(0.4)
correct upto four decimal places by using Milne's predictor-corrector formula. (07 Marks)
OR
2 a.
Using modified Euler's method find y at x = 0.2 given
dx

-I
d
= 3x + ?
1
y with y(0) = 1 taking
2
h = 0.1. (06 Marks)
b. Given ?
dy
+ y + zy
2
=0 and y(0) = 1, y(0.1) = 0.9008, y(0.2) = 0.8066, y(0.3) = 0.722.
dx
Evaluate y(0.4) by Adams-Bashforth method. (07 Marks)
dy y ? x
c. Using Runge-Kutta method of fourth order, find y(0.2) for the equation dx
y+x'
y(0) = 1 taking h = 0.2. (07 Marks)
Module-2
d
2

3 a. Apply Milne's method
dx
2

to compute y(0.8) given that =1? 2y
11
x
- ' and the following table
y
d
of initial values.
x 0 0.2 0.4 0.6
y 0 0.02 0.0795 0.1762
y' 0 0.1996 0.3937 0.5689
(06 Marks)
b. Express f(x) = x
4
+ 3x
3
? x
2
+ 5x ? 2 in terms of Legendre polynomials. (07 Marks)
c. Obtain the series solution of Bessel's differential equation x
2
y" + xy' + (x
2
+ n
2
) y = 0
leading to .1,,(x). (07 Marks)
I of 3
OR
4 a.
Given y" - xy' - y = 0 with the initial conditions y(0) = 1, y'(0) = 0, compute y(0.2)
y'(0.2) using fourth order Runge-Kutta method.
b. Prove J_
Iii
(k) =
1
1 ?
2
cos x .
7EX
Prove the Rodfigues formula P (x) =
1 d'y
(X
2 ?
1)
"

2" n! dx"
Module-3
5 a. Derive Cauchy-Riemann equations in Cartesian form.
b. Discuss the transformation w = z
-
.
e
2.2
c. By using Cauchy's residue theorem, evaluate
dz if C is the circle Izi = 3 .
(z +1)(z + 2)
c.
(06 Marks)
(07 Marks)
(07 Marks)
(06 Marks)
(07 Marks)
(07 Marks)
OR
a
, a ,
6 a. Prove that + If (z)
,

l
-
= 41f
1
(z)1"
ex' ay
e

State and prove Cauchy's integral formula.
Find the bilinear transformation which maps z = oo, i, 0 into w = -1, -i, 1.
b.
c.
Module-4
7 a. Find the mean and standard of Poisson distribution. (06 Marks)
b. In an examination 7% of students score less than 35 marks and 89% of the students score
less than 60 marks. Find the mean and standard deviation if the marks are normally
distributed given A(I.2263) = 0.39 and A(1.4757) = 0.43 (07 Marks)
c. The joint probability distributio
Determine:
i) Marginal distribution of X and Y
ii) Covariance of X and Y
iii) Correlation of X and Y (07 Marks)
OR
8 a. A random variable X has the following robability function:
x 0 1 2 3 4 5 6 7
P(x) 0 K 2k 2k 3k K
2
2k
2
7k
2
+k
Find K and evaluate P(x 6) P(3 < x 6). (06 Marks)
b.
The probability that a pen manufactured by a factory be defective is 1/10. If 12 such pens are
manufactured, what is the probability that
i) Exactly 2 are defective
ii) Atleast two are defective
iii) None of them are defective. (07 Marks)
c.
The length of telephone conversation in a booth has been exponential distribution and found
on an average to be 5 minutes. Find the probability that a random call made
i) Ends in less than 5 minutes
ii) Between 5 and 10 minutes.
(07 Marks)
2 of 3
V\
-2 -1 4
1 0.1 0.2 0 0.3
2 0.2 0.1 0.1 0
17MAT41
Module-5
9 a. A die is thrown 9000 times and a throw of 3 or 4 was observed 3240 times. Show that the
dia cannot be regarded as an unbiased die.
(06 Marks)
b. A group of 10 boys fed on diet A and another group of 8 boys fed on a different disk B for a
period of 6 months recorded the following increase in weight (lbs):
Diet A: 5 6 8 1 12 4 3 9 6 10
Diet B: 2 3 6 8 10 1 2 8
Test whether diets A aid B differ significantly t.05 = 2.12 at 16d1 (07 Marks)
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 terms:
i) Null hypothesis
Type-I and Type-II error
iii) Confidence limits
1/2 0 1/2
(06 Marks)
h. The t.p.m. of a Markov chain is given by P = 1 0 0 . Find the fined probabilities
vector.
1/4 1/2 1/4
(07 Marks)
c.
Two boys B1 and B2 and two girls G1 and G2 are throwing ball from one to another. Each
boy throws the ball to the Other boy with probability 1/2 and to each girl with probability
1/4. On the other hand each girl throws the ball to each boy with probability 1/2 and never to
the other girl. In the long run how often does each receive the ball? (07 Marks)
3 of 3
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