(Objective/Multiple type question)

2x
1. The Fourier series for the function f(x) =

a) Sineterm
{
1+
π
2x
;−π ≤ x ≤ 0

1− ; 0 ≤ x ≤ π
π
}
Contains only

b) Cosine term
c) Sine and cosine term
d) ¿

2. Relation between Laplace and Fourier transform is

a) L[f(x)] = 2 F[f(x)]
b) L[f(x)] = (2 π )1/2 F[f(x)]
π
c) L[f(x)] = ( )1/2 F[f(x)].
2
d) L[f(x)] = 2 Fs[f(x)]

3.
f (x)=¿ {0 −π<x<0 ¿¿¿¿ and f( x + 2π) = f(x). Fourier series of f(x) is represented by
a0
+ ∑ (an cos nx+ bn sin nx )
2 then b1 is
a) 0
b) 1
c) 2
d) 3

4.
f (x)=¿ { 1 −1<x<0 ¿¿¿¿ & Fourier series of f(x) is represented by
a0 nπx nπx
+ ∑ (an cos + bn sin )
2 l l then b1 is
2
−
a) π
2
b) π
 3
c) π
2π
d) 3
5. Which one of the following is even function:

 a 1  x3  b x 2 cos  1  x   c sin  1  x 2  d log  1  x 3 

6. Which one of the following is odd function:

 a  b sin  x  x 3   c log  1  x 2  d

3
ex x 3 sin x

h  x , h  x , h  x h  x   h1  2 x   3h2  4 x   5h3  3x 
7. If 1 2 3 are periodic function then is a periodic
function and its period can be given as:

 a  L.C.M.of allfunctions  b  H.C.M.of allfunctions

 c  Linear value according to functions definition  d  None of these

h1  x  , h2  x  , h3  x 
8. If are periodic functions with period 4a,16a,18a respectively, then
h  x   h1  2 x   3h2  4 x   5h3  3x 
is a periodic function with period given by:

 a 48a 3  b 12a  c 4a  3  16a   5  18a  d Noneof these

f  x  l , l  , then which one of the following is true:

9. If is an odd function defined in

 a a1001  0  b b1001  0  c b1000  0 d Noneof these

f  x   sin  1  x 2  ,  0,2  , then which one of the following is true:

10. If be a function defined in

 a  a0 , an  0  b bn  0  c an  0  d  Noneof these

11. Which one of the following is not true:

 2 2

 a  sin 2 x cos 2 x dx  0  b  sin 4 x cos5 x dx  0

0 0
 2

 c   sin 2 x sin 3x dx  0  d   cos 2 x cos 2 x dx  0

0 0

12. Which one of the following is not a part of Dirichlet condion:

a) Function must be piecewise continuous
b) Function must be periodic
c) Function must be continuous
d) Function must be defined in given interval
a 2  x  0
f  x  
13. If a 0  x  2 be a function defined in  2, 2  , then which one of the following is true
a) Function is Odd function
b) Function is Even Function
c) Function is neither even nor odd
d) None of these
z  z  1  2i
14. The solution of the equation is
a) 1  2i
3
2 i
b) 2
3
2i 
c) 2
3
 2i
d) 2
f  z   u  x, y   iv  x, y 
15. The necessary condition for the function to be analytic is
u v u v
 , 
a) x y y x
u v u v
 , 
b) x y y x
u v u v
 , 
c) x y y x
u v u v
 , 
d) x y y x
z2 1
lim
16. The value of z i z  i is
a) i
b) i
c) 2i
 d) 2i
17. Harmonic functions
a) Satisfy Laplace equations
b) Do not Satisfy Laplace equations
18. Cauchy’s theorem requires
a) Function to be analytic
b) Function to be continuous
c) None of these
1
w
19. The transformation z is called
a) Translation
b) Rotation
c) Magnification
d) inversion

20. If A, B are independent events and

A  B , then
a) P (A) < P (B)
b) P (A) > P (B)
c) P (A) = P (B)
d) None of these
Pr( x  13)
21. If the mean of a Poisson random variate is 12 then Pr( x  11) is
a) 13/12
b) 12/13
c) 11/13
d) 13/11
22. In an experiment, a fair coin was 7 times and the result of each toss was noted. How many of the 128
possible outcomes have exactly 4 heads?
a) 70
b) 7
c) 35
d) 21
23. The probability of an event cannot be:
a) Equal to zero
b) Greater than zero
c) Less than zero
d) Equal to one
24. When the occurrence of one event has no effect on the probability of the occurrence of another event, the
events are called:
a) Independent
b) Dependent
c) Mutually exclusive
d) Equally likely
25. If A1, A2, A3, ..., AK are k mutually exclusive events, then:
a) P(A1⋃A2⋃A3⋃ ...⋃AK ) = P(A1)+P(A2)+P(A3)+...+ P(AK)
b) P(A1⋃A2⋃A3⋃ ...⋃AK ) > 1
c) P(A1⋂A2⋂A3⋂ ...⋂AK) = 1
d) P(A1⋂A2⋂A3⋂ ...⋂AK) = P(A1⋃A2⋃A3⋃ ...⋃AK )
26. Two unbiased dice are thrown, Find the probability that both the dice show the same number
a) 1/6
b) 5/18
c) 5/36
 d) 7/36
27. Two unbiased dice are thrown, Find the probability that the total of the numbers on the dice is greater than
8
a) 1/6
b) 5/18
c) 5/36
d) 7/36
28. A, B and C are three arbitrary events. Find expression for the event only A occur
a) A ∩ B́∩ Ć
b) A ∪ B ∪ C
c) A ∩ B∩ C
d) Á ∩ B́∩ Ć
29. A, B and C are three arbitrary events. Find expression for the events at least one occurs
a) A ∩ B́∩ Ć
b) A ∪ B ∪ C
c) A ∩ B∩ C
d) Á ∩ B́∩ Ć
30. A, B and C are three arbitrary events. Find expression for the All three events occur
a) A ∩ B́∩ Ć
b) A ∪ B ∪ C
c) A ∩ B∩ C
d) Á ∩ B́∩ Ć
31. A, B and C are three arbitrary events. Find expression for the events none occur
a) A ∩ B́∩ Ć
b) A ∪ B ∪ C
c) A ∩ B∩ C
d) Á ∩ B́∩ Ć
32. Match the correct expression of probabilities ´ B)
P( A ∪
a) P ( B )−P ( A ∩ B )
b) P ( Á ) + P ( B )−P ( Á ∩ B )
c) 1−P ( A ∪ B )
d) P ( A )−P ( A ∩B )
33. Match the correct expression of probabilities P( Á ∩ B)
a) P ( B )−P ( A ∩ B )
b) P ( Á ) + P ( B )−P ( Á ∩ B )
c) 1−P ( A ∪ B )
d) P ( A )−P( A ∩ B)
34. Match the correct expression of probabilities P( Á ∪ B)
a) P ( B )−P ( A ∩ B )
b) P ( Á ) + P ( B )−P ( Á ∩ B )
c) 1−P ( A ∪ B )
d) P ( A )−P( A ∩ B)
35. Match the correct expression of probabilities P(B/ Á)
a) A . P ( B )−P ( Á ∩ B )
b) P ( Á ) + P ( B )−P ( Á ∩ B )
 c) 1−P ( Á ∪ B )
d) P ( Á ∩B ) /P ( Á )

36.  A partial differential equation requires

a) exactly one independent variable
b) two or more independent variables
c) more than one dependent variable
d) equal number of dependent and independent variable
37. Using substitution, which of the following equations are solutions to the partial differential equation?

                     

cos (3 x− y)
a)
b) x 2− y 2
c) sin ⁡(3 x −3 y)
d) e−3 x sinπy
38. Equation ptany+qtanx =sec 2 z is of order
a) 1
b) 2
c) 0
d) none of these
39. The equation (2 x+3 y ) p+ 4 xq−8 pq=x+ y
a) Linear
b) quasi-linear
c) semi –linear
d) non-linear
40. The general solution of (y-z) p+ (z-x) q = x-y is
a) f ( x + y + z , x2 + y2 + z2 ) = 0
b) f ( xyz , x+ y+ z ) = 0
c) f ( xyz , x 2+ y 2+ z 2 ) = 0
f ( x − y−z , x 2− y 2− z2 ) = 0
d)
y2 z 2
41. Subsidiary equations for equation ( ) p + zxyq= y are
x
dx dy dz
a) = =
y 2 z zx y 2
dx dy dz
b) = =
x 2 y 2 zx
dx dy dz
c) = =
x2 y2 z2
 dx dy dz
= =
d) 1 1 1
x 2
y 2
zx
42. The general solution of linear partial differential equation Pp+Qq=R is
a) f (u , v )=1
b) f (u , v )=−1
c) f (u , v )=0
d) none of these
∂2 z ∂2 z ∂z 2
43. Equation −2 +( ) = 0 is of order
∂ x2 ∂x ∂ y ∂ y
a) 1
b) 2
c) 3
d) none of these
44. The equation Pp + Qq = R is known as
a) Lagrange’s equation
b) Bernoulli’s equation
c) Charpit’s equation
d) Clairaut’s equation
∂z ∂z
45. Q.10 The integral surface satisfying equation y ( ) ( )
∂x
−x
∂y
=x 2 + y 2 and passing through the curve

x=1−t , y=1+t , z=1+t 2

( x 2− y 2 )
a) z=xy +
2
(x − y 2 )2
2
b) z=xy +
8
2 2 2
(x − y )
c) xy +
4
(x 2− y 2 )2
d) z=xy +
16

GROUP-B

(Short answer type questions)

1. Obtain the a0 for f(x)=|sinx| for –π<x<π

Obtain a1 for f(x)= x in (0,1).

2. Express f(x)= {10forfor0<x>x <π π as a Fourier sine integral or evaluate

 ∞

∫ 1−cosπλ
λ
sinx𝞴 d𝞴.
0
3. Find the relationship between Fourier and Laplace Transform.
−d
4. Show that FS [xF(x)] = F́ C(S)
ds
5. Check wither the following functions are even or odd:
 x   x  0   a l  x  0
f  x   f  x  
(a) 2  x 0  x   (b) a 0 xl
0 0  x  l
f  x  
6. Find the value of a0 in the Fourier Series expansion of the periodic function  a l  x  2l
7. Write the Dirichlet’s condition for the convergence of Fourier series of a function.
f  x   e  x , x  0.
8. Find the Fourier sine transform of
1
F  f  x  cos ax   F  s  a   F  s  a   , F  s   F  f  x  ; s .
9. Show that 2 where
10. Write the convolution of two functions and convolution theorem for fourier transform.
z  axe y   1/ 2  a 2 e 2 y  b.
11. Find a PDE by eliminating a and b from the equation
f  x   x2 ,    x  
12. Obtain the Fourier series for the function and deduce the following
2 1 1 1
 1  2  2  2  .....
6 2 3 4
l
f  x   x, 0  x  l .
13. Find the half range sine series for the function 2
f  x   e x .
2

14. Find the Fourier transform of

 2 z z
 ,
x 2 y
15. Find the solution of which satisfied the following conditions

u  0, t   u  3, t   0 and u  x, 0   5sin 4 x.
16. Evaluate the following integral by using Cauchy’s integral formula
sin  z 2  cos  z 2
C  z  1  z  2  dz C is the circle z  3 .
i. , where
u  e 2 x  x cos 2 y  y sin 2 y 
17. Determine the analytic function w  u  iv , if .
f  z
18. If complex function is analytic, then prove that its real and imaginary part satisfies Laplace
equation; also prove that the family of curves formed by its real and imaginary parts is orthogonal to
each other.
e x  cos y  i sin y 
19. Prove that the function is analytic and find its derivative.

20. Determine the analytic function, whose real part is x  3 xy  3 x

3 2 2
 3 y 2  2 x  1 . Also find its
conjugate.
21. Evaluate the following integral by using Cauchy’s integral formula
 e2 z
  z  1
C
4
dz , C : z  3
ii. .

22. If u  x  y find its corresponding analytic function (by Milne’s Method).

2 2

2  4i

 zdz ,
2
23. Evaluate 0 along the curve y  x .
f  x, y   y 3  3 x 2 y
24. (a) Examine that the function is harmonic or not.
1
f  x, y   log  x 2  y 2 
25. (b) Examine that the function 2 is harmonic or not.

x y
f  x, y  
26. (a) Examine that the function x 2  y 2 is harmonic or not.
f  x, y   2 x  1  y 
27. (b) Examine that the function is harmonic or not.
28. Tickets numbered 1 to 20 are mixed up and then a ticket is drawn at random. What is the probability
that the ticket drawn has a number which is a multiple of 3 or 5?
29. Two dice are thrown simultaneously. What is the probability of getting two numbers whose product is
even?
30. Define Exhaustive, Mutually Exclusive and Independent events.
31. An urn contains 2 white, 3 red and 4 black balls. Three balls are drawn from the urn. Find the chance
that (i) all are of the same colour (ii) all are of different colours.
1 2 3
, ,
32. In a shooting game the probability of A, B and C to hit the target are 2 3 4 respectively. If all of
them fire at a target, find the probability that at least one of them hits the target.
33. A and B take turns in throwing of two dice, the first to throw 9 will be awarded a prize. If A has the
first turn, show that their chances of winning are in the ratio 9:8.
34. Three groups of children contain respectively 3 girls 1 boy, 2 girls 2 boys, 1 girl 3 boys. One child is
selected at random from each group. Find the probability of selecting 1 girl and 2 boys.
35. There are two bags, one of which contains 3 black and 4 white balls, while the other contains 4 black
and 3 white balls. A fair dice is cast, if the face 1 or 3 turns up a ball is taken from the first bag and if
any other face turns up a ball is chosen from the second bag. Find the probability of choosing a black
ball.

36. If B  A , then prove that (i)

 
P A  B  P  A  P  B 
(ii)
P  B   P  A
.

37. If A and B are independent events, then prove that A, B are also independent.
38. A card is drawn from pack of 52 cards; find the probability of getting a king or a heart or a red card?
39. A card is drawn from a pack of 52 cards, if the value of faces cards 10, aces cards 1 and other
according to denomination, find the expected value of the no. of point on the card.
40. A bag contains 10 red and 15 white balls. Two balls are drawn in succession. What is the probability
that one of them is white and other red?
41. A manufacturer supplies quarter horsepower motors in lots of 25. A buyer, before taking a lot, tests at
random a sample of 5 motors and accepts the lot if they are all good; otherwise he rejects the lot. Find
the probability that: (i) he will accept a lot containing 5defective motors; (ii) he will reject a lot
containing only one defective motors.
42. In an examination with multiple-choice questions, each question has four, out of which one is correct.
A candidate ticked the answer either by his skill or by copying from his neighbors, the probability of
guess is 1/3, copying is 1/6. The probability of correct answer by copying is 1/8. If a candidate answers
a question correctly find the probability that he know the answer.
43. In a partially destroyed laboratory record of an analysis of correlation data, the following results only
are legible: Variance of x =9, Regression equation: 8x - 10y + 66=0; 40x - 18y - 214=0. Find (i) the
mean values of x and y (ii) the S.D. of y (iii) coefficient of correlation between x and y
44. Show that , the acute angle between the two lines of regression, is given by
1−r 2 σ x σ y
tan θ= .
r σ 2 + σ 2y
x .
45. Find the differential equation of all spheres of radiusr , having centre in the xy -plane.
3 3
46. Form a partial differential equation from z ¿ ax +b y by eliminating arbitrary constants a and b.
47. Find the partial differential equation of all spheres whose centre lie on z-axis.
48. Eliminate arbitrary function f from f =(x ¿ ¿ 2− y 2) ¿.
49. Form a partial differential equation by eliminating the arbitrary function f from the equation
x + y + z=f ( x 2+ y 2 + z 2)
50. Form partial differential equation by eliminating arbitrary function f and g from
z=f ( x 2− y )+ g (x 2+ y )
51. Find the bounded solution u(x , t) ,0< x<1 , t> 0 of the boundary value problem
∂u ∂u
− =1−e−t Subject to u(x , 0)=x
∂x ∂t
∂u ∂2 u π
52. Solve =5 2 , u ( x , 0 ) =cos 5 x , u x ( 0 , t )=0 and u( , t )=0.
∂t ∂x 2
∂u ∂2 u
53. Obtain the solution of = −4 u ,u ( 0 , t ) =0 ,u ( π , t ) =0 ,u ( x , 0 )=6 sinx−4 sin 2 x .
∂ t ∂ x2
∂u ∂2 u
54. Find the bounded solution of = , where u ( 0 , t ) =1, u ( x , 0 ) =0.
∂ t ∂ x2
∂u ∂2 u π ∂u
55. Solve =3 2 where u( , t )=0, ¿¿ =0 ,u ( x ,0 )=30 cos 5 x .
∂t ∂x 2 ∂ x x=0
∂u ∂u
56. Find the solution of =2 +u , u ( x , 0 ) =6 e−3 x which is bounded for x >0 , t>0.
∂x ∂t

GROUP-C

(Long type questions)

1. Using Fourier transform, show that the solution of the PDE

 2u  2u
 c 2 2 ,    x  , t  0
a. t 2
x
2. satisfying the conditions
u
 i u  f  x ,  0at t  0
t
u
 ii  u  x, t  , both tends to zero as x  
x
 
1
u  x, t   e
 isx
F  s  cos cst ds,
3. Can be written in the form 2 

F  s f  x .
4. Where is the Fourier Transform of
5. Find the steady temperature distribution in a thin plate bounded by the line
x  0, x  l , y  0, y   assuming that heat cannot escape from either surface; the sides
x  0, x  l are being kept at temperature zero. The lower edge y  0 is kept at temperature
f  x
and the edge y   at temperature zero. (Laplace Equation)
0
6. A uniform rod of 20m length is insulated over its sides. Its ends are kept at 0 C . Its initial
sin   x / 20  u  x, t 
temperature is at a distance x from an end. Find temperature at any time t.
7. Solve the given BVP using Fourier Cosine Transform
 2 u u

x 2 t
u x 0  x  1
given that  0at x  0 and u  x,0    .
x 0 x 1
8. (a) Prove the sufficient condition of analytic function with C-R theorem statement.

(b)Show that the function u(x,y)=4xy-3x+2 is a harmonic function and construct the corresponding
analytic function.

9. (a)If v=4xy-3y-4, then find u such that f(z)= u+iv is an analytic function with f(1+i)=-3i.

(b)Write the statement of Fourier integral Theorem.

 
a0   an cos nx   bn sin nx
f  x n 1 n 1 an is
©If the Fourier series of function is given by , then
given by?
 
a0   an cos nx   bn sin nx
f  x n 1 n 1 bn is
10. (a)If the Fourier series of function is given by , then
given by?
F  p f  x , f  ax 
(b)If is the Fourier transform of then the Fourier transform of is given by?
F  p f  x , f  x  a
©If is the Fourier transform of then the Fourier transform of is given by?
11. (a)Define periodic function
(b)Define even function
©Write the relation between two orthogonal functions.
 1 

F  f  u  g  x  u  du; p  
(d)IF convolution of two functions exists then the value of 
2  
f  z  xy
12. Show that the function is not regular at the origin although C- R equations are
satisfied at that point.
1 i

  x  y  ix  dz
2

13. Find the value of the integral 0 , along

.
f  z
14. Prove that the function defined by
 x  1  i   y3  1  i 
3

 ,z0
f  z   x2  y2
0, z0

f   0
15. Function is continuous and that Cauchy- Riemann equation are satisfied at the origin, yet
does not exist.
 x 3 y ( y  ix )
 ,z0
f ( z )   x6  y 2  f ( z )  f (0) 
0,
 z0  z  0   0 as z  0
16. If prove that along any radius

vector but not as z  0 along the curve

y  ax 3
. is this function differentiable at z = 0.
  2
2  2 2
 2  2 
f (z)  4 f ' (z) .
 x y 
17. If f(z) a analytic function of z, prove that
1 i

 z dz
2

18. (a) Evaluate 0 , where z is complex number.

1 2 i

  1  z  dz
2

(b) Evaluate 0 , where z is complex number.

2i

 e dz
z

19. (a) Evaluate 0 , where z is complex number.

1 i

z  3z  2  dz
2

(c) Evaluate 0 , where z is complex number.

f  z
20. (a) Find the analytic function of which the imaginary part is given by
v  x, y   e sin y.
x

ez
z2  z  1  z  3 dz.
(b) Evaluate
(c) Statement of the Cauchy’s Residue Theorem.

21. . In an engineering examination, a student is considered to have failed, secured second class, first class
and distinction according as he scores less than 45%, between 45% and 60%, between 60% and 75%
and above 75% respectively. In a particular year 10% of the students failed in the examination and 5%
of the students got distinction. Find the percentage of students who have got first class and second
t z2
1 
f  t  e 2
dz
2 0
f  1.28   0.40 f  1.65  0.45
class. Given that if , then and .
 22. (a) A speaks the truth in 75% cases and B in 80% of the cases. In what percentage of cases are they
likely to contradict to each other in stating the same fact.
(b) Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the
mean and variance of number of aces.
23. If 10% of the pens manufactured by the company are defective, find the probability that a box 12 pens
contain
a) Exactly two defective pens
b) At least two defective pens
c) No defective pen
d) At most two defective pens
24. A letter is known to come either from Calcutta or from Tatanagar. In the half printed postal stamp of
the coming states only two consecutive letter TA are readable. Find the chances of the letter coming
from (i) Calcutta (ii) Tatanagar.
25. Three factories A, B, C do 30%, 50%, and 20% production of certain item. Out of their production
8%, 5%, and 10% of the items produced are defective respectively. An item is purchased and is found
to be defective. Find the probability that it was a product of (i) factory A (ii) factory B .
26. A random variable X has following probability distribution
X 0 1 2 3 4 5 6 7

P X  0 K 2K 2K 3K K2 2K2 7K2+K

a) Find K
P  X  6  , P  X  6  , P  0  X  5
b) Evaluate
27. From a lot of 10 items containing 3 defectives, a sample of 4 items is drawn at random. If the sample is
drawn without replacement and the random variable X denotes the number of defective items in the
sample. Find
a) The probability distribution of X
P  X  1
b)
P  X  1
c)
P  0  X  2
d)
28. In a Normal distribution, 31% of the items are under 45 and 8% are over 64. Find the parameters of the
distribution.
29. Define the correlation by graphically and mathematically. Also prove that the correlation is
independent of change of origin and scale.
30. Obtain the rank correlation coefficient for the following data:
X 85 74 85 50 65 78 74 60 74 90

Y 78 91 78 58 60 72 80 55 68 70

31. Prove that Poisson distribution is a limiting case of Binomial distribution. Also evaluate mode of
Poisson distribution.
32. Define the properties of Normal distribution. Also prove that mean and mode are equal.
33. State Bye’s theorem. A man is equally likely to choose any one of the three routes A, B, C from his
house to the railway station and his choice of route is not influenced by weather. If the weather is dry,
the probabilities of missing the train by route A, B, C are respectively 1/20, 1/10, 1/5. He sets out on a
dry day and misses the train. What is the probability that the routes chosen were C?
34. The probability that a doctor A will diagnose a disease X correctly is 0.6 .The probability that a patient
will die by his treatment after correct diagnosis is 0.4 and probability of death by wrong diagnosis is
 0.7. A patient of Doctor A who had disease X, died. What is the probability that his disease was
diagnosed correctly?
35. A communication system consists of n components each of which will independently function with
probability p. The total system will be able to operate effectively if at least one-half of its components
function. For what value of p is a 5 component system more likely to operate effectively than a 3
component system?
36. form a partial differential equation by eliminating the arbitrary functions f and F from
z=f ( x +iy )+ F ( x−iy).
37. form a partial differential equation by eliminating the arbitrary function f from
f ( x + y + z , x 2 + y 2 + z 2 ) = 0 .what is the order of this partial differential equation?
38. Solve: p+3 q=z +tan( y−3 x)
39. Solve: xyp+ y 2 q+ 2 x 2−xyz=0.
40. Find the integral surface of x 2 p+ y 2 q+ z 2=0 which passes through the hyperbola xy=x + y , z=1.
∂2 z 2
41. Solve : 2 +2
∂2 z + ∂ z =2 x +3 y
∂x ∂ x ∂ y ∂ y2
∂2 z ∂2 z
42. Solve: + =cosmx . cosmy
∂ x2 ∂ y2
43. find the surface passing through the parabola z=0 , y 2=4 ax and and z=1 , y 2=−4 ax and
satisfying the equation xr +2 p =0.
44. Show that a surface passing through the circle z=0 , x 2+ y 2=1 and satisfying the differential
equations s=8 xy is z=(x2 + y 2 )2−1.
∂2 u ∂2 u
45. find the bounded solution of =9 ,u ( 0 , t ) =0, u ( 2 ,t )=0 ,
∂ t2 ∂ x2
u ( x , 0 ) =20 sin 2 πx−10 sin 5 πx
and ut ( 0 , t )=0 .
46. (a)Data on the readership of a certain magazine show that the proportion of ‘male readers
under 35 is 0.40 ’ and ‘over 35 is 0.02 ’. if the proportion of readers under 35 is 0.70. find the
probability that a randomly selected male subscriber is under 35 year of age .
(b)A and B are two weak students of mathematics and their chances of solving a problem in
mathematics correctly are 1 / 6 and 1 / 8 respectively. If the probability of their making a common
error is 1 / 525 and they obtain the same answer, find the probability that their answer is correct.
f  x   k sin  1 / 5   x, 0  x  5.
©The probability distribution of a r.v. X is: Determine the
constant k and obtain the median.
47. (a)The kms X in thousands of kms which car owners get with a certain kind of trial is a r.v. having

1 / 20e  ( x / 20) , x0

F x   
 0, otherwise
pdf .
(b)Find the probability that one of these trial will at least 30,000 kms.
©Two unbiased dice are thrown. find the expected values of the sum of numbers of points
on them.
48. (a)Express f(x)=
∞
{10forfor0<x>x <π π as a Fourier sine integral or evaluate

∫ 1−cosπλ
λ
sinx𝞴 d𝞴.
0

f  x   x3 ,   x   .
(b) Find the Fourier Series of the periodic function

−d
© Show that FS [xF(x)] = F́ C(S)
ds
49. (a)Find the solution of given BVP with the help of Fourier Transform
 2u 2  u
2
 c ,    x  , t  0.
t 2 x 2 suchthat

u  x, t 
 a u  x, t   f  x  ,  0, at t  0,
t
u  x, t 
(b ) u  x, t   0,  0, at x  0.
x

(b)Prove that u  2 x  2 xy is harmonic function. Determine its harmonic conjugate and find the
f  z .
corresponding analytic function

50. Define the random variable, Explain the types of random variable with example.
An experiment consists of three independent tosses of a fair coin. Let : X= the number of heads, Y =
the number of head runs, Z = the length of head runs, a head run being defined as consecutive
occurrence of at least two heads, it’s length then being the number of heads occurring together in three
tosses of the coin.
Find the probability function of (i) X (ii) Y (iii) Z (iv) X+Y (v) construct probability tables and draw
their probability charts and Evaluate

 1.5  ( X  Y )  4.5 
 
(a) P (0<X<5) (b) P 
(X Y )  2 