West Bengal Institute of Technology (WBIT) 2008-3rd Sem B.Tech Electronics and Communications Engineering Electronics
C8/B.Tech(ICS/M/EM/*I*/rWI/Btt/IC*)/8m-3/M-30a/0S/(09) <->
ENGINEERING & MANAGEMENT EXAMINATIONS, DECEMBER - 2008
Time : 3 Hours ) I Full Marks : 70
GROUP - A ( Multiple Choice Type Questions)
1. Choose the correct alternatives for any ten of the following : 10x1 = 10
1) If/( z) = z, then/'(0) is
a) 1 b) - 1
. c) 0 d) does not exist.
11) If/( z ) = u ( x, y ) + iv ( x, y ), then/' ( z) is
, dv . du K1 du dv
, dii.dv dv
ill) The probability P ( a < X < b )
[ where F ( x ) is distribution function of a continuous random variable X 1 is defined by
a) F ( a) - F ( b) b) F(b) + F(a)
c) F( b) - F( a) d) F(a)F(b). S ~j
iv) The number of vertices of odd degree in an undirected graph is even.
a) True b) False. 1
CS/B.Teck(CCB/n/nB/KIK/rK/BMK/ICB)/HM-a/l(-302/M/(0
Let G be a graph with n vertices and e edges. G has a vertex of degree m s.t. 2e
V)
2e m<-
b)
d)
a)
c)
m =
n
2e
n
2e
n
m >
m >
vi) Given, P ( A ) = |, P ( B ) = |, P ( AB ) = |. The value of P ( AB ) is
, _5
12
i 3
c) 4
b)
d)
12
7_ 12
vii) Two variables x auid y are related by x = 2y + 5. The median of x is 25. The median of y is
b) 10 d) 20.
a) 9 c) 8
viii) Evaluate
d?, where C is the circle whose equation is | z - | = p
Z - oe
b) 2ni d) n i.
a)
c)
471 i
n i 2
A purse contains 4 copper coins, 3 silver coins and an another purse contains 6 copper coins and 2 silver coins. A purse is chosen at random and a coin is taken out of it. The probability that it is a copper coin is
3
4
37 56
4
7
3
7
b)
d)
a)
c)
x} A vertex of a tree is a cut vertex iff its degree is
b) greater than 2 d) equals to 2.
a) greater than 1 c) equals to 1
C8/B.Tech(KCE/KE/KE/EIE/PW*/Bltt/ICE)/8EM-'3/M~302/08/(09)
xi) If x = 4y + 5 and y = kx + 4 be two regression equations of x on y and y on x respectively, then the value of k lies in the interval
b) [0,4]
a) [ 4, 5 ]
c) [ 0, 5 ]
d) none of these.
xii) A function /( x) = x2 , - rc x < 7t is represented by a Fourier series
oo
- + X ( a n C0S + b n sin T*1611 value f b n ls
n = 1
2n 2
4(- 1 )" 3
b)
a)
d) none of these.
c)
xiii) The period of the function/( x) = sin 2 nx is
b)
d)
a) 2
_1
3
c)
sin z
xiv) The order of the pole z = 0 of function ~z 3~ is
b) 2 d) 4.
a) 1
c) 3
C
xv) Let G be a connected graph with n vertices and e edges and T be a spanning tree of G. Then T has
a) n + 1 branches, e - n - 1 edges
b) n - 1 branches, e + n - 1 edges
c) n - 1 branches, e - n + 1 edges
d) n + 1 branches, e + n + 1 edges.
GROUP-B .
( Short Answer Type Questions )
Answer any three of the following. 3 x 5 = 15
2. Expand f(z) = sin z in a Taylor series about z =
3. Define component of a graph. Prove that a simple graph with n vertices and K
- components can have at most ( n - K) [ n - K + 1 ) edges.
4. a) Suppose G is a non-directed graph with 12 edges. If G has 6 vertices each of
degree 3 and the rest have degree less than 3, find the minimum number of vertices G can have.
% '
b) Determine the poles and residue at each pole of the function/! z ) = cot z.
3 + 2
5. a) Compute Spearman's rank correlation coefficient r for the following data :
|
Person |
A |
B |
C |
D |
E |
F |
G |
H |
I |
J |
|
Rank in statistics |
9 |
10 |
6 |
5 |
7 |
2 |
4 |
8 |
1 |
3 |
|
Rank in income |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
b) Find the Fourier sine transform of/( x) = ~ . 3 + 2
6. State and prove Bayes theorem.
7. Find the Fourier series to represent x - x 2 from x = -ntox = n and hence find the
GROUP -C ( Long Answer Type guestions )
Answer any three of the following. 3 x 15 = 45
i. a) For two variables x and y the equations of two regression lines are x + 4y + 3 = 0 and 4x + 9y + 5 = 0. Identify which one is of y on x. Find the means of x and y. Find the correlation coefficient between x and y. Estimate the value of x when y = 1-5.
CS/B.TcMSCE/K/*M/B/PW*/N/IC*}/Ml<-S/M-a03/OS/(OB]
b) The mean and s.d. of marks of 70 students were found to be 65 and 5-2 respectively. Later it was detected that the value 85 was recorded wrongly and therefore it was removed from the data set. Then find the mean and s.d. for the remaining 69 students.
|
c) Find the median from the following data : | ||||||||||||||||
|
5 + 5+5
9, a) If a random variable X follows normal distribution such that P ( 9*6 < X < 13-8 ) = 0-7008 and P( X > 9-6 ) = 0-8159. where
12
09
e~ 2 dt = 0-8159 and
V2rc
find the mean and variance of X.
b) In answering a question on a multiple choice test, a student either knows the
answer or he guesses. Let P be the probability that he knows the answer and
1-p be the probability that he guesses. Assume that a student who guesses the answer will be correct with probability g. What is the conditional probability that
a student knew the answer to a question given that he answered it correctly ?
c) The overall percentage of failures in a certain examination is 40. What is the probability that out of a group of 6 candidates at least 4 passed the
. o 5 + 5 + 5
examination ?
C*/BT*cl{*t/K/WC/X[*/FVY/lllC*/ICS)/0EN'3/V-3O3/60/(e0]
fXi
i32s
8
10. a) A define Isomorphism of two graphs. Examine whether the following two graphs G and G1 are isomorphic. Give reasons.
|
A |
 |
G
b) Applying Dijkstra's method, find the shortest path between the two vertices a *
. and / in the following graph.
8 + 7
2ji
f _c. J 5 +
cos 20
11. a) Evaluate | K . 4'cos~8 uslng Cauchy's Residue theorem.
|
b) Use the method of least-squares to fit a linear curve ysdg + cjctofit the data: | ||||||||||
|
c) Expand f{ z) = -rn-5"7 in a Laurent's series valid for
c J (z+1)(2+3)
0 < I 2 + 1 I < 2. 7 + 3 + 5
CS/B.Ttch(BCE/EE/EEE/EIE/PWE/BiIE/ICK)/Sni-3/M-302/08/(08) . 9
12. a) Prove that the number of vertices of odd degree in a graph is always even.
b) Show that u(x, y ) = x3 - 3xy 2 is harmonic in C and find a function v ( x, y )
'
such that/{ z) = u + tv is analytic.
c) Find the Fourier Transform of the function f(x) = 1, | x j < a
= 0, | x | > a.
Hence evaluate
I
sin sa cos sx , . _ _ _ --- ds. 3 + 6 + 6
END
j 33803 (12/12)
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Attachment: |
| Earning: Approval pending. |
