West Bengal Institute of Technology (WBIT) 2009-3rd Sem B.Tech Electronics and Communications Engineering Electronics & Comm ( - ) Mathematics - Question Paper
Name :.................................................................*/ \
i?oll No. :...........................................................
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Invigilator's Signature:.........................................
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CS/B.TechCECE,EE,EIE,EEE,PWE,BBE,ICE)/SEM-3/M-302726d9-10
Time Allotted : 3 Hours Full Marks : 70
The figures in the margin indicate JiUl marks.
Candidates are required to give their answers in their own words
as far as practicable.
GROUP - A ( Multiple Choice Type Questions )
1. Choose the correct alternatives for any ten of the following :
10x1*10
i) The probability that a leap-year selected at random will contain 53 sundays is
1 3 w 2
a) b) rj
C) 7 d) g.
ii) If a coin is tossed 6 times in succession, the probability of getting at least one head is
.63 w 3_
64 } 64 7
c) gTj d) None of these.
ill) The probability that the 4 children of a family have different birthdays is
a) 0-9836 b) 0-4735
c) 0-9 d) 0-757.
iv) A tree has n vertices. The number of its edges is a) n + 1 b) n - 1
c) 2n d) none of these.
v) The value of m such that 3y - 5x 2 + my 2 is a harmonic function is
a) 5 b) - 5
c) 0 d) 3.
vi) Let X and Y be two random variables such tfriat
Y = a + bx where a and b are constants. Then, Var (y) is
a) b2 Var (X) b) Var(X)
c) a2 Var (X) d) ( b/a) Var (X).
d z
where C is a circle Izl = 1 is
vii) The value of
z+3
a) 0 b) 1
c) 2 d) - 1.
viii) If/( z ) = 4 j22 3 . then z = 0 is a pole of order
a) 3 b) 2
c) 1 d) 4.
a) 1 b) 2
c) 3 d) 4.
x) The period of the function/( x) = sin 2kxis
1
2
c) 0 d)
xi) If/( x) = x sin x, - n < x < n, be presented in Fourier series as + S ( a n cos n* + bn sin jvc ),
n = 1
then the value of aQ will be
a) 2 b) 0
c) 4 d) 1.
xii) If two variables x and y are uncorrelated, then r, is
a) 1 b) 2
c) 3 d) 0.
xiii) If x = 4y + 5 be a regression line of x on y then bxy is
a) | b) 4
c) 0 d) 1.
GROUP -B ( Short Answer Type Questions )
Answer any three of the following. 3x5= 15
2. Show that/( x) given by
fix) = x; 0 < x < 1
= Jc - x ; 1 < x <2
= 0 ; elsewhere,
is a probability density function for a suitable value of k. Calculate the probability that the random variable lies between and
e _ax
3. Find the Fourier sine transform of - .
3 z 2 2 1
-z dz, where c is the circle I z | = ~
4. Evaluate
c
z l
5. An urn contains 3 white and 5 black balls. One ball is drawn and its colour is unnoted, kept aside and then another ball is drawn. What is the probability that it is (i) black
(ii) white ?
6. Find the mean and standard deviation of a bionomial distribution.
GROUP -C ( Long Answer Type Questions )
Answer any three of the following. 3 x 15 = 45 If A and B are mutually independent events, prove that A c and B c are also mutually independent events.
7. a)
b)
There are three identical urns containing white and black balls. The first urn contains 3 wKite and 4 black balls, the 2nd urn contains 4 white and 5 black balls and the 3rd urn contains 2 white and 3 black balls. An urn is chosen at random and a ball is drawn from it. If the drawn ball is white, what is the probability that
the 2nd um chosen ?
A random variable X has the following p.d.f.
c)
/( x) = cx2 0 < x< I = 0, otherwise.
1
5 + 5 + 5
Find (i) c (ii) PI 0 < X < 2
8. a)
Find the Fourier series expansion of the periodic function of period 2rc,
-ti<x<7c. Hence deduce
1 1 J_ +. + ......
f 2 "2 + 3 2 42 .........
The following marks have been obtained by students in
1LZ
12
Compute me uu-ciuu -------
data. Find also the equations of the lines of regrssion.
b) | ||||||||||||||||||||||||
| ||||||||||||||||||||||||
7 + 8 |
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9. a) Solve
dt = k dx* * x> 1
If u ( 0. t) = 0. u ( x, 0 ) = e x > 0. u I x. t ) is unbounded.
b) If /( z ) is a regular function of z, then prove that
(&&) |/'z,|2 = 412,i2- 8 + 7
10. a) Apply Di/kstra's algorithm to determine a shorterst path
between s to z in the following graph :
b) Define isomorphism of two graphs. Examine whether the following graph G and C1 are isomorphic. Give reasons.
6
33803
3 2 2 + z - 1
dz
( z2- 1 )(2-3)
11, a) Use residue theorem to evaluate
around the circle |z| =2.
b) Expand the function / { z ) = [z2+\)(z+ 2T in the region | z | < 1.
2 2 for z * 0 x * + y z J
c) Show that the function / ( z ) = > is continuous at z = 0.
for z = 0 5 + 7 + 3
12. a) Show that a simple graph with n vertices and . + f t tn-fc)( n - fc + 1 )
'Components can have at most 2
edges.
b) Find the incidence matrix of the following graph.
c) Find the Fourier sine transform of the functon [ 1 for 0 < x <, n
/(*) =
I 0 for x > k and hence evaluate the integral
1 - cos UK , , -sin pxap.
5 + 5 + 5
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Attachment: |
Earning: Approval pending. |