UNIVERSITY OF ENGINEERING & MANAGEMENT, JAIPURQUESTION BANK

SUBJECT NAME: MATHEMATICS-3, SUBJECT CODE: M301

B.TECH, 2nd YEAR, 3TH SEMESTER

GROUP-A

(Objective/Multiple type question)

1. Which of the following is an “even” function of t ? a) t2

b) t2-4tc) sin(2t) + 3t d) t3 +6

2. A “periodic function” is given by a function which. a) has a period T = 2π b) satisfies f (t + T) = f (t) c) satisfies f (t + T) = − f (t)d) has a period T = π

3. Fourier coefficient in the Fourier series of

and is (A)

a)

b)

c)

d)

4. Fourier series representation of periodic function is

, then value of

a)

b)

c)

d)5. The graph of odd function is symmetric about

a) Opposite quadrantb) x-axisc) y-axisd) none of these

6. f ( x )=¿ {π+x −π<x<0 ¿ ¿¿¿

is an even and f(x+2 π) = f(x) then a0 is a) π

b)

π2

c)

π3

d)

π4

7. The Fourier series for the function f(x) = {1+ 2 xπ;−π ≤ x≤0

1−2 xπ;0≤x ≤π } Contains only

a) Sineterm b) Cosine term c) Sine and cosine termd) ¿

8. Relation between Laplace and Fourier transform is

a) L[f(x)] = 2 F[f(x)]b) L[f(x)] = (2π)1/2 F[f(x)]

c) L[f(x)] = (π2

)1/2 F[f(x)].

d) L[f(x)] = 2 Fs[f(x)]

9. f ( x )=¿ {0 −π<x<0 ¿ ¿¿¿

and f( x + 2π) = f(x). Fourier series of f(x) is represented by a0

2+∑ (an cosnx+bn sinnx )

then b1 is a) 0b) 1c) 2d) 3

10.f ( x )=¿ {1 −1<x<0 ¿ ¿¿¿

& Fourier series of f(x) is represented by a0

2+∑ (an cos nπx

l+bnsin nπx

l)

then b1 is

a)− 2π

b)

2π

c)

3π

d)

2π3

11. Which one of the following is even function:

12. Which one of the following is odd function:

13. If are periodic function then is a periodic function and its period can be given as:

14. If are periodic functions with period respectively, then

is a periodic function with period given by:

15. If is an odd function defined in , then which one of the following is true:

16. If be a function defined in , then which one of the following is true:

17. Which one of the following is not true:

18. Which one of the following is not a part of Dirichlet condion:a) Function must be piecewise continuousb) Function must be periodicc) Function must be continuousd) Function must be defined in given interval

19. If be a function defined in , then which one of the following is truea) Function is Odd functionb) Function is Even Functionc) Function is neither even nor oddd) None of these

20. The solution of the equation is

a)

b)

c)

d)

21. The necessary condition for the function to be analytic is

a)

b)

c)

d)

22. The value of isa)b)

c)

d)23. Harmonic functions

a) Satisfy Laplace equationsb) Do not Satisfy Laplace equations

24. Cauchy’s theorem requiresa) Function to be analyticb) Function to be continuousc) None of these

25. The transformation is calleda) Translationb) Rotationc) Magnificationd) inversion

26. If A, B are independent events and , thena) P (A) < P (B)b) P (A) > P (B)c) P (A) = P (B) d) None of these

27. If the mean of a Poisson random variate is 12 then isa) 13/12b) 12/13c) 11/13d) 13/11

28. 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) 70b) 7c) 35d) 21

29. The probability of an event cannot be: a) Equal to zero b) Greater than zero c) Less than zerod) Equal to one

30. 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

31. 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 )

32. Two unbiased dice are thrown, Find the probability that both the dice show the same numbera) 1/6 b) 5/18 c) 5/36 d) 7/36

33. 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

34. A, B and C are three arbitrary events. Find expression for the event only A occur a) A∩B∩Cb) A∪B∪Cc) A∩B∩C d) A∩B∩C

35. A, B and C are three arbitrary events. Find expression for the events at least one occursa) A∩B∩Cb) A∪B∪Cc) A∩B∩Cd) A∩B∩C

36. A, B and C are three arbitrary events. Find expression for the All three events occura) A∩B∩Cb) A∪B∪C

c) A∩B∩C d) A∩B∩C

37. A, B and C are three arbitrary events. Find expression for the events none occura) A∩B∩Cb) A∪B∪C c) A∩B∩C d) A∩B∩C

38. Match the correct expression of probabilities P(A∪B)a) P (B )−P (A∩B )b) P (A )+P (B )−P (A∩B )c) 1−P ( A∪B )d) P (A )−P ( A∩B )

39. Match the correct expression of probabilities P(A∩B)a) P (B )−P (A∩B )b) P (A )+P (B )−P (A∩B )c) 1−P ( A∪B )d) P (A )−P(A∩B)

40. Match the correct expression of probabilities P(A∪B)a) P (B )−P (A∩B )b) P (A )+P (B )−P (A∩B )c) 1−P ( A∪B )d) P (A )−P(A∩B)

41. Match the correct expression of probabilities P(B/ A)a) A .P (B )−P ( A∩B )b) P (A )+P (B )−P (A∩B )c) 1−P ( A∪B )d) P (A ∩B )/P ( A )

42.  A partial differential equation requiresa) exactly one independent variableb) two or more independent variablesc) more than one dependent variabled) equal number of dependent and independent variable

43. Using substitution, which of the following equations are solutions to the partial differential equation?

                     

a) cos (3 x− y )b) x2− y2

c) sin (3 x−3 y)d) e−3 xsinπy

44. Equation ptany+qtanx=sec2 z is of order

a) 1 b) 2 c) 0 d) none of these

45. The equation (2 x+3 y ) p+4 xq−8 pq=x+ ya) Linear b) quasi-linearc) semi –lineard) non-linear

46. The general solution of (y-z) p+ (z-x) q = x-y isa) f (x+ y+z , x2+ y2+z2 ) = 0

b) f ( xyz , x+ y+z ) = 0

c) f (xyz , x2+ y2+z2 ) = 0

d) f (x− y−z , x2− y2−z2 ) = 0

47. Subsidiary equations for equation ( y2 zx

) p+zxyq= y2 are

a)dxy2 z

=dyzx

=dzy2

b)dxx2 =dy

y2 =dzzx

c)dxx2 =dy

y2 =dzz2

d)dx1x2

= dy1y2

= dz1zx

48. The general solution of linear partial differential equation Pp+Qq=R isa) f (u , v )=1

b) f (u , v )=−1

c) f (u , v )=0 d) none of these

49. Equation ∂2 z∂x2 −2 ∂2 z

∂x ∂ y+( ∂z∂ y

)2

= 0 is of order

a) 1 b) 2 c) 3 d) none of these

50. The equation Pp + Qq = R is known asa) Lagrange’s equationb) Bernoulli’s equation c) Charpit’s equation d) Clairaut’s equation

51. Q.10 The integral surface satisfying equation y ( ∂ z∂ x )−x ( ∂ z∂ y )=x2+ y2 and passing through the curve

x=1−t , y=1+t , z=1+t 2

a) z=xy+(x2− y2)2

b) z=xy+(x2− y2)2

8

c) xy+(x2− y2)2

4

d) z=xy+(x2− y2)2

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)={1 for 0<x<π0 for x>π

as a Fourier sine integral or evaluate

∫0

∞ 1−cosπλλ

sinx𝞴 d𝞴.

3. Find the relationship between Fourier and Laplace Transform.

4. Show that FS [xF(x)] =−dds

FC(S)

5. Check wither the following functions are even or odd:

(a) (b)

6. Find the value of in the Fourier Series expansion of the periodic function 7. Write the Dirichlet’s condition for the convergence of Fourier series of a function.

8. Find the Fourier sine transform of

9. Show that where 10. Write the convolution of two functions and convolution theorem for fourier transform.

11. Find a PDE by eliminating a and b from the equation

12. Obtain the Fourier series for the function and deduce the following

13. Find the half range sine series for the function

14. Find the Fourier transform of

15. Find the solution of which satisfied the following conditions

16. Evaluate the following integral by using Cauchy’s integral formula

i. , where is the circle .

17. Determine the analytic function , if .

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.

19. Prove that the function is analytic and find its derivative.

20. Determine the analytic function, whose real part is . Also find its conjugate.

21. Evaluate the following integral by using Cauchy’s integral formula

ii. .

22. If find its corresponding analytic function (by Milne’s Method).

23. Evaluate along the curve

24. (a) Examine that the function is harmonic or not.

25. (b) Examine that the function is harmonic or not.

26. (a) Examine that the function is harmonic or not.

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.

32. In a shooting game the probability of A, B and C to hit the target are 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 , then prove that (i) (ii) .

37. If A and B are independent events, then prove that 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

tanθ=1−r2

r.σ xσ yσx2+σ y

2

.45. Find the differential equation of all spheres of radiusr , having centre in the xy-plane.46. Form a partial differential equation fromz ¿ax3+b y3 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− y2)¿.

49. Form a partial differential equation by eliminating the arbitrary function f from the equation x+ y+z=f (x2+ y2+z2)

50. Form partial differential equation by eliminating arbitrary function f and g from

z=f (x2− y )+g (x2+ y)51. Find the bounded solution u(x , t) ,0< x<1 , t>0 of the boundary value problem

∂u∂x

−∂u∂ t

=1−e−tSubject to u(x ,0)=x

52. Solve ∂u∂ t

=5 ∂2u∂ x2 , u ( x ,0 )=cos5 x ,ux (0 , t )=0 and u(

π2, t )=0.

53. Obtain the solution of ∂u∂ t

=∂2u∂x2 −4u ,u (0 , t )=0 , u (π , t )=0 , u (x ,0 )=6 sinx−4 sin 2x.

54. Find the bounded solution of ∂u∂ t

=∂2u∂x2 , where u (0 , t )=1, u ( x ,0 )=0.

55. Solve ∂u∂ t

=3 ∂2u∂ x2 where u(

π2, t )=0,

∂u∂x

¿¿x=0=0 ,u ( x ,0 )=30 cos5 x.

56. Find the solution of ∂u∂x

=2 ∂u∂ t

+u , u ( x ,0 )=6e−3x which is bounded for x>0 , t>0.

GROUP-C

(Long type questions)

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

a.2. satisfying the conditions

3. Can be written in the form

4. Where is the Fourier Transform of 5. Find the steady temperature distribution in a thin plate bounded by the line

assuming that heat cannot escape from either surface; the sides

are being kept at temperature zero. The lower edge is kept at temperature

and the edge at temperature zero. (Laplace Equation)6. A uniform rod of 20m length is insulated over its sides. Its ends are kept at . Its initial

temperature is at a distance from an end. Find temperature at any time 7. Solve the given BVP using Fourier Cosine Transform

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.

©If the Fourier series of function is given by , then is given by?

10. (a)If the Fourier series of function is given by , then is given by?

(b)If is the Fourier transform of then the Fourier transform of is given by?

©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.

(d)IF convolution of two functions exists then the value of

12. Show that the function is not regular at the origin although C- R equations are satisfied at that point.

13. Find the value of the integral , along.

14. Prove that the function defined by

15. Function is continuous and that Cauchy- Riemann equation are satisfied at the origin, yet

does not exist.

16. If prove that along any radius

vector but not as along the curve is this function differentiable at z = 0.

17. If f(z) a analytic function of z, prove that

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

(b) Evaluate , where is complex number.

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

(c) Evaluate , where is complex number.

20. (a) Find the analytic function of which the imaginary part is given by

(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

class. Given that if , then and .22. (a) speaks the truth in 75% cases and 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

containa) Exactly two defective pensb) At least two defective pensc) No defective pend) 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 are readable. Find the chances of the letter coming from (i) Calcutta (ii) Tatanagar.

25. Three factories 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 (ii) factory .

26. A random variable has following probability distribution0 1 2 3 4 5 6 7

0 K 2K 2K 3K K2 2K2 7K2+K

a) Find K

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 denotes the number of defective items in the sample. Find

a) The probability distribution of

b)

c)

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 andF 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 , x2+ y2+z2 ) = 0 .what is the order of this partial differential equation?

38. Solve: p+3q=z+tan( y−3 x)39. Solve: xyp+ y2q+2x2−x yz=0.40. Find the integral surface of x2 p+ y2q+z2=0 which passes through the hyperbola

xy=x+ y , z=1.

41. Solve :∂2 z∂x2 +2 ∂2 z

∂x ∂ y +∂2 z∂ y2=2 x+3 y

42. Solve: ∂2 z∂x2 +

∂2 z∂ y2=cosmx . cosmy

43. find the surface passing through the parabola z=0 , y2=4ax and and z=1 , y2=−4ax and

satisfying the equation xr+2 p=0.44. Show that a surface passing through the circle z=0 , x2+ y2=1 and satisfying the differential

equations s=8xy is z=(x2+ y2)2−1.

45. find the bounded solution of ∂2u∂ t2

=9 ∂2u∂ x2 ,u (0 , t )=0, u (2 ,t )=0,

u ( x ,0 )=20 sin 2πx−10 sin 5πxand ut (0 , t )=0.

46. (a)Data on the readership of a certain magazine show that the proportion of ‘male readers

under is ’ and ‘over is ’. if the proportion of readers under is 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 and respectively. If the probability of their making a common

error is and they obtain the same answer, find the probability that their answer is correct.

©The probability distribution of a r.v. is: Determine the

constant and obtain the median.47. (a)The kms in thousands of kms which car owners get with a certain kind of trial is a r.v. having

pdf .

(b)Find the probability that one of these trial will at least kms.©Two unbiased dice are thrown. find the expected values of the sum of numbers of points on them.

48. (a)Express f(x)={1 for 0<x<π0 for x>π

as a Fourier sine integral or evaluate

∫0

∞ 1−cosπλλ

sinx𝞴 d𝞴.

(b) Find the Fourier Series of the periodic function

© Show that FS [xF(x)] =−dds

FC(S)

49. (a)Find the solution of given BVP with the help of Fourier Transform

suchthat

(b)Prove that is harmonic function. Determine its harmonic conjugate and find the

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

(a) P (0<X<5) (b) P