West Bengal Institute of Technology (WBIT) 2009-3rd Sem B.Tech Electronics and Communications Engineering Electronics & Comm ( - ) Mathematics - Question Paper

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).

vii)    The value of

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* ⁺ b_n sin jvc ),

n = 1

then the value of a_Q 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 | = ~

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.

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.

/( x) = cx² 0 < x< I = 0, otherwise.

1

Find (i) c (ii) PI 0 < X < ₂

8. a)

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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.

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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.

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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