Download AKTU B-Tech 3rd Sem 2015-2016 AS 303 Mathematics Iii Question Paper

Download AKTU (Dr. A.P.J. Abdul Kalam Technical University (AKTU), formerly Uttar Pradesh Technical University (UPTU) B-Tech 3rd Semester (Third Semester) 2015-2016 AS 303 Mathematics Iii Question Paper

Printed Pages: 6 4 ' AS-303
(Following Paper ID?and Roll N o. to be ?lled in your
Answer Book)
Roll No.
B.Tech.
(SEM. III) THEORY EXAMINATION, 2015-16
MATHEMATICS?lll
[Time:3 hours] JMaximumMarksleO]
Note: Attempt, all questions from each Section as indicated.
The symbols have their usual meaning.
Section-A
1. Attempt all parts of this section. Each part carry 2 marks.
(2 x 1 0=20)
(a) Show that w=iz is the rotation of the z-plane
through an angle 7r/2 in the counterclockwise
direction.
(b) Determine and Classify all the Singularity of
1 1
+
z(z?2)5 (z?2)? ' '
3800 . (1) P.T.O

(C)
(d)
(f)
(g)
(h)
0)
De?ne F ourier Transform of a function f (x).
Find the z- Transform of {(-1)-}.
De?ne Probability density function.
What is Karl Pearson?s coefficient of skewness.
Show that V _ A = ?VA..
De?ne Bisection method.
What is cubic spline?
Find missing value in following table:
\
x 45 50 55 60~64
Y 3 - 2 - -2.4
Section-B
Attempt any five questions from this section; (5 ><10=50)
2.
3800
(a)
(b)
Show that the function de?ned by f (x) = [.131] is
not regular at origin, although Cauchy-Riemann
equations are satis?ed. '
Determine the analytic ?inction f(z) =n+iv, in terms
ofz, whose u ? v = e? (cos y ? sin y).
(2) / As-303
(a) F ind inverse Z-Transform of W , when 2 > 5
(b) Solve the following difference equation using Z-
transfonn um+2u?+l +u,, = n,uo = ul = 0.
(a) In a normal distribution, 31% of the items are
under 45 and 8% are over 64. Find the mean and
standard deviation of the distribution. It is given
1 . 1 1 2
that if f(t)=J27t?Ioe??ch, when of
(0.5)=0.19, and f(1.4)=0.42.
' (b) In a bombing action, there is a 50% chance that
any bomb will strike the target. Two direct hits are
needed to destory the targey completely. How
many bombs are required to be dropped to give a
99% chance of better of completely destroying the
target.
5. (a) F ind to four places of decimal, the smallest root
the equation 9* = sin x,
(b) a From the? following table ?nd the value of e024 .
x 0.1 0.2 0.3 0.4 0.5
v 1.10517 1.2214 1.34986 1.49182 1.64872
3800 - (3) - _ P.T.O.

6. (a) The distance covered an athlete for the 50 meter
race is given as:
Time(sec) 0 1 2 3 4 5 6
Distance(meter) 0 2.5 8.5 15.5 24.5 36.5 50
Determine speed of the athlete at t=5 sec correet
to two decimal.
ab?
1+):2
taking h=1/6.?
. 1 , I
(b) Evaluate I0 using Simpson?s 3/8? rule, by
7. (a) Evaluateusing Cauchy intergral formula.
?m h. C, m . 1 :4
C (Z-lXZ-Z) dZ,W etc 15 ecueeizl >.
(b) ' Find the Fourier Sine transform of :
f(x)=e--, forx 2 Oanda>0.
hence show that,
3800 (4) A_S'-303
10.?
3800?
(a) Six coins are tossed 6400 times: Using the Poisson
distn'bution, determine the probability of getting ?
six heads x times.
(b) Using Newton?s divided difference formula ?nd a
polynomial which takeszthe values 3, 12, 15, -21
when x has the values 3, 2, 1 and -1 reSpectively.
(a) - using Milne?s-methbd, SOIVe % =1+y2 with initial
I conditions? J ,
y(0)=?o, y(0.2)=0.2027, y(0.4)= 0.4228,y
(0.6)=0.6841, ?nd y(0.8). ' "
(b) Find the value of y (0.6) by Ranga Kutta' fourth
order method taking h=0.2 for the initial value
problem; .
Section-C
Attempt any two parts of this Section. (15x2_=30)
(a) Apply calculus of residues to evaluate.
???dx,a>0. ?
Inc xsinx
0 x2 +a2

2
= 6 u ? .
" '. ?=?-?, >0,t>0
(b) . Solve the equetlon . a: ? 6:2 x
Subject to the conditions:
(i) y = 0 when x = 0, (ii) f(x) = {3331" (iii) 11 (x,t)
i?bounded. ?
(c) The ?rst four moments about yvorking mean 28.5
of a distribution are 0.294; 7.144, 42.409, and
454.98. Calculate the moments about mean. Also
calculate 5' and ?z ahd comment upon the
',skewness and kurtosis?of the distribution.
((1). use Gauss-Seidal method to solve the following
* equations, 7 .
- 2x+10y+z=51
10x+y+22;44
x+2y+102=6l
.?.I?.X~?v
3800 , (-6) ' . AS-3 03