Download VTU B-Tech/B.E 2019 June-July 1st And 2nd Semester 10 Scheme 10MAT41 Engineering Mathematics IV Question Paper

Download VTU ((Visvesvaraya Technological University) B.E/B-Tech 2019 July ( Bachelor of Engineering) First & Second Semester (1st Semester & 2nd Semester) 10 Scheme 10MAT41 Engineering Mathematics IV Question Paper

10MAT41
cncia
Fourth Semester B.E. Degree Examination, June/July 2019
Engineering Mathematics - IV
Time: 3 hrs.
Max. Marks:100
Note: Answer FIVE full questions, selecting
atleast TWO questions _from each part.
PART - A
a. Find by Taylor's series method the value of y at x = 0.1 and x = 0.2 correct to four decimal
places from = x
2
y -1, y(0)=1. (07 Marks)
dx
b. Using modified Euler's method find y(20.2) and y(20.4) given that ?
dy
= lo2
10
(I-.) with
dx Y
y(20) = 5 taking h = 0.2 (use modified formula twice). (07 Marks)
Given
dY
= x
2
(1 + y) and y(1)=1, y(1.1)=1.233, y(1.2)=1.548, y(1.3)=1.979.
dx
Evaluate y(1.4) by Adam's Bashforth method. (06 Marks)
2 a. Obtain the solution using fourth order Runge-Kutta method of the system of equations
?
dx
= 2x + y,
d
= x -3y ; t = 0, x = 0, y = 0.5. Take h = 0.2. (07 Marks)
dt dx
Obtain second approximation values of y and z to x = 0.1 using Picard's method, given that
y(0) = 2, z(0) = 1 and ?
dy
= x + z, ?
dz
=
dx
x - y` . (07 Marks)
,
dx
Given y" + xy' + y - 0. Calculate y(0.4) using Milne's method by the following data :
x 0 0.1 0.2 0.3
y 1 0.995 0.9801 0.956
y' 0 -0.995 -0.196 -0.2867
(06 Marks)
3 a. Derive Cauchy-Riemann equations in polar form. (06 Marks)
b. If (I) + ill' represents the complex potential of an electrostatic field where
kv = x
2
? y
2
+ , find the complex potential as a function of z and hence determine 4).
x + y
-
(07 Marks)
c. If f(z) is analytic, show that
')
2 \if(z)1
2
=41r(z)1
2

a-
a
(07 Marks)
c.
b.
c.
4 a. Find the bilinear transformation which maps the points z = 1, i, -1 into w = 0, 1, 00.
(07 Marks)
b.
Find the image of the circles Izi = 1 and IzI = 2 under the mapping w = z + (06 Marks)
State and prove Cauchy's integral formula for the analytic function f(z) inside and on a
simple closed curve. (07 Marks)
1 o 2
C.
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10MAT41
cncia
Fourth Semester B.E. Degree Examination, June/July 2019
Engineering Mathematics - IV
Time: 3 hrs.
Max. Marks:100
Note: Answer FIVE full questions, selecting
atleast TWO questions _from each part.
PART - A
a. Find by Taylor's series method the value of y at x = 0.1 and x = 0.2 correct to four decimal
places from = x
2
y -1, y(0)=1. (07 Marks)
dx
b. Using modified Euler's method find y(20.2) and y(20.4) given that ?
dy
= lo2
10
(I-.) with
dx Y
y(20) = 5 taking h = 0.2 (use modified formula twice). (07 Marks)
Given
dY
= x
2
(1 + y) and y(1)=1, y(1.1)=1.233, y(1.2)=1.548, y(1.3)=1.979.
dx
Evaluate y(1.4) by Adam's Bashforth method. (06 Marks)
2 a. Obtain the solution using fourth order Runge-Kutta method of the system of equations
?
dx
= 2x + y,
d
= x -3y ; t = 0, x = 0, y = 0.5. Take h = 0.2. (07 Marks)
dt dx
Obtain second approximation values of y and z to x = 0.1 using Picard's method, given that
y(0) = 2, z(0) = 1 and ?
dy
= x + z, ?
dz
=
dx
x - y` . (07 Marks)
,
dx
Given y" + xy' + y - 0. Calculate y(0.4) using Milne's method by the following data :
x 0 0.1 0.2 0.3
y 1 0.995 0.9801 0.956
y' 0 -0.995 -0.196 -0.2867
(06 Marks)
3 a. Derive Cauchy-Riemann equations in polar form. (06 Marks)
b. If (I) + ill' represents the complex potential of an electrostatic field where
kv = x
2
? y
2
+ , find the complex potential as a function of z and hence determine 4).
x + y
-
(07 Marks)
c. If f(z) is analytic, show that
')
2 \if(z)1
2
=41r(z)1
2

a-
a
(07 Marks)
c.
b.
c.
4 a. Find the bilinear transformation which maps the points z = 1, i, -1 into w = 0, 1, 00.
(07 Marks)
b.
Find the image of the circles Izi = 1 and IzI = 2 under the mapping w = z + (06 Marks)
State and prove Cauchy's integral formula for the analytic function f(z) inside and on a
simple closed curve. (07 Marks)
1 o 2
C.
10MAT41
PART ? B
5 a. Find the series solution of Bessel's differential equation leading to Bessel function.
(07 Marks)
b. If a and 13 are two roots of .1,
i
(x) = 0, then show that xJ
n
(ax).1,(13x)dx = 0 if a 13.
0
(07 Marks)
c. Show that x
4
? 3x
2
+ x
:5
P 4(x)
?10
P
2
(
x
)
Pl(x)=-
4
110(x)-
7 5
(06 Marks)
6 a. A problem in mathematics is given to three students A, B and C whose changes of solving it
are 3,
3
and
4
respectively. What is the probability that the problem will be solved?
(07 Marks)
b. Give P(A) =
3
, P(B)= 1 ? and P(A n B) = 1 ?.
4 5 20
Find : i) P(A / B) ii) P(A / B) iii) P(A / B) . (06 Marks)
c. State and proved Baye's theorem o conditional probability. (07 Marks)
7 a. A die is tossed thrice. A success is 'getting 1 or 6' on a toss. Find the mean and variance of
the number of successes. (07 Marks)
b. A die is thrown 8 times. Find the probability that '3' falls,
i) Exactly 2 times
ii) Atleast once
iii) At the most 7 times. (06 Marks)
c. Define exponential distribution and obtain the mean and standard deviation of exponential
distribution. (07 Marks)
8 a. Certain tubes manufactured by a company have mean life time of 800 hours and standard
deviation of 60 hours. find the probability that a random sample of 16 tubes taken from the
group will have a mean life time :
i) Between 770 hours and 830 hours
ii) Less than 785 hours
iii) More than 820 hours
(Given (I)(2) = 0.4772 ; 4)(1) = 0.3413; (1(1.33) = 0.4082). (07 Marks)
b. A sample of 900 days was taken in a coastal town and it was found that on 100 days the
weather was very hot. Obtain the probable limits of the percentage of very hot weather.
(06 Marks)
c. A sample analysis of examination results of 500 students was made. It was found that 220
students had failed, 170 had secured third class, 90 had secured second class and 20 had
secured first class. Do these figures support the general examination result which is in the
ratio 4 : 3 : 2 : 1 for the respective categories (x6
.05
= 7.81 for 3 d.f) (07 Marks)
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