Download AKTU B-Tech 4th Sem 2017-18 RAS401 Mathematics Iii Question Paper

Download AKTU (Dr. A.P.J. Abdul Kalam Technical University (AKTU), formerly Uttar Pradesh Technical University (UPTU)) B-Tech 4th Semester (Fourth Semester) 2017-18 RAS401 Mathematics Iii Question Paper

Printed Pages: 02 Sub Code: RAS401
Paper Id: 1 9 9 4 2 1 Roll No.

B. TECH.
(SEM-IV) THEORY EXAMINATION 2017-18
MATHEMATICS - III
Time: 3 Hours Total Marks: 70
Note: Attempt all Sections. If require any missing data, then choose suitably.

SECTION A
1. Attempt all questions in brief. 2 x 7 = 14
a) Discuss Singularity and its types.
b) Write Cauchy-Riemann equation in polar co-ordinates.
c) The life of army shoes is normally distributed with mean 8 months and standard deviation 2 months. If
5000 pairs are insured, how many pairs would be expected to need replacement after 12 months? Given
that P 0228 . 0 ) 2 ( = ? z .
d) Determine moment generating function of Binomial distribution.
e) Prove that:
1
2
1
2
E ?? =+

f) Write Newton-Cote?s quadrature formula.
g) Find Z-transform of ) ( ) ( k u k f ? = .
SECTION B
2. Attempt any three of the following: 7 x 3 = 21
a) Determine an analytic function f(z) in terms of z if
2
2
sin 2
2 2cos2
y
y
x
u v e x
e
+ = + ? .
b) Find the mean variance of Poisson distribution.
c) Find
6
0
1
x
e
dx
x +
?
using (i) Trapezoidal rule, (ii) Simpson?s 1/3
rd
rule and (iii) Simpson?s 3/8
th
rule.
d) A rod is rotating in a plane. The following table gives the angle ? (in radians) through which the rod has
turned for various values of time t (in seconds).
t: 0 0.2 0.4 0.6 0.8 1.0 1.2
? : 0 0.12 0.49 1.12 2.02 3.20 4.67
Calculate the angular velocity and angular acceleration at t =0.2 and t = 1.2 second.
e) Find Fourier cosine transform of
2
1
1 x +
, hence find Fourier sine transform of
2
1
1 x +




FirstRanker.com - FirstRanker's Choice
Printed Pages: 02 Sub Code: RAS401
Paper Id: 1 9 9 4 2 1 Roll No.

B. TECH.
(SEM-IV) THEORY EXAMINATION 2017-18
MATHEMATICS - III
Time: 3 Hours Total Marks: 70
Note: Attempt all Sections. If require any missing data, then choose suitably.

SECTION A
1. Attempt all questions in brief. 2 x 7 = 14
a) Discuss Singularity and its types.
b) Write Cauchy-Riemann equation in polar co-ordinates.
c) The life of army shoes is normally distributed with mean 8 months and standard deviation 2 months. If
5000 pairs are insured, how many pairs would be expected to need replacement after 12 months? Given
that P 0228 . 0 ) 2 ( = ? z .
d) Determine moment generating function of Binomial distribution.
e) Prove that:
1
2
1
2
E ?? =+

f) Write Newton-Cote?s quadrature formula.
g) Find Z-transform of ) ( ) ( k u k f ? = .
SECTION B
2. Attempt any three of the following: 7 x 3 = 21
a) Determine an analytic function f(z) in terms of z if
2
2
sin 2
2 2cos2
y
y
x
u v e x
e
+ = + ? .
b) Find the mean variance of Poisson distribution.
c) Find
6
0
1
x
e
dx
x +
?
using (i) Trapezoidal rule, (ii) Simpson?s 1/3
rd
rule and (iii) Simpson?s 3/8
th
rule.
d) A rod is rotating in a plane. The following table gives the angle ? (in radians) through which the rod has
turned for various values of time t (in seconds).
t: 0 0.2 0.4 0.6 0.8 1.0 1.2
? : 0 0.12 0.49 1.12 2.02 3.20 4.67
Calculate the angular velocity and angular acceleration at t =0.2 and t = 1.2 second.
e) Find Fourier cosine transform of
2
1
1 x +
, hence find Fourier sine transform of
2
1
1 x +




SECTION C
3. Attempt any one part of the following: 7 x 1 = 7
(a) Verify Cauchy theorem by integrating
iz
e

along the boundary of the triangle with the vertices at
the points i i + ? + 1 , 1 and i ? ?1 .
(b) Evaluate
?
?
?
0
0 ,
sin
m dx
x
mx
..
4. Attempt any one part of the following: 7 x 1 = 7
(a) The following table represents the height of a batch of 100 students. Calculate skewness and
kurtosis:

Height (in cm) 59 61 63 65 67 69 71 73 75
No. of students 0 2 6 20 40 20 8 2 2

(b) Use the method of least squares to fit the curve x c
x
c
y
1
0
+ = to the following table of values:

x 0.1 0.2 0.4 0.5 1 2
y 21 11 7 6 5 6
.
5. Attempt any one part of the following: 7 x 1 = 7
(a) Find the root of the equation x xe
x
cos = using Regula-Falsi method correct to four decimal
places.
(b) Find Newton?s divided difference polynomial for the following data:
x: -3 -1 0 3 5
f(x): -30 -22 -12 330 3458

6. Attempt any one part of the following: 7 x 1 = 7
(a) Solve the initial value problem
2
2tu u ? = ? , 1 ) 0 ( = u with 2 . 0 = h

on the interval ? ? 4 . 0 , 0 . Use
Runge-Kutta fourth order method and compare your result with exact solution.
(b) Solve the following system of linear equations by Matrix decomposition method taking lii =1 for
1? i ? 3.
2 2 2 ; 11 3 2 ; 12 2 3 = ? ? = + + = + ? z y x z y x z y x
7. Attempt any one part of the following: 7 x 1 = 7
(a) Using Z-transform, solve the following difference equation:

k
k k k
y y y 3 3 4
1 2
= + +
+ +
, given that 0
0
= y and 1
1
= y .


(b) The temperature u in the semi-infinite rod ? ? ? x 0 is determined by the differential equation
2
2
x
u
k
t
u
?
?
=
?
?
subject to conditions
(i) 0 = u when 0 , 0 ? = x t (ii) ? ? =
?
?
x
u
(a constant) when 0 = x and 0 ? t , (iii) u (x, t) is bounded.
Determine the temperature u (x, t).

FirstRanker.com - FirstRanker's Choice