TYPICAL QUESTIONS & ANSWERS

TYPICAL QUESTIONS & ANSWERS

PART I

OBJECTIVE TYPES QUESTIONS

Each Question carries 2 marks.

Choose correct or the best alternative in the following:

Q.1 Which of the following is an entire function

(A) (B)

(C) (D)

Ans: C

Q.2 Let . Then which of the following statements is not correct.

(A) f is differentiable at z = 0. (B) f is differentiable at z 0.

(C) f is not analytic at z = 0. (D) f is not analytic at z 0.

Ans: C

Q.3 The image of a square under the transformation is

(A) a square (B) a circle

(C) the upper half plane (D) the right half plane

Ans: A

Q.4 The value of is

(A) (B) 0

(C) (D)

Ans: A

Q.5 The value of line integral , where C is the segment of the parabola from (0, 0) to (1, 1) is

(A) (B) 0

(C) (D)

Ans: C

Q.6 The directional derivative of at the point in the direction of is

(A) (B)

(C) 0 (D) 1

Ans: A

Q.7 The unit normal to the surface at the point is

(A) (B)

(C) (D) None of the above

Ans: C

Q.8 Which of the following probability mass functions can define a probability distribution

(A) (B)

(C) (D)

Ans: B

Q.9 The expected value of a random variable X is 3 and its variance is 2. Then the variance of 2X + 5 is

(A) 8 (B) 9

(C) 10 (D) 11

Ans: A

Q.10 The equation is

(A) Elliptic (B) Parabolic

(C) Hyperbolic (D) None of the above

Ans: B

Q.11 The image of the line under the mapping is a

(A) Circle. (B) Parabola.

(C) Hyperbola. (D) Ellipse.

Ans: B

Q.12 The values of z for which is real is

(A) any multiple of . (B) odd multiples of .

(C) all multiples of . (D) any real number.

Ans: A

Q.13 The value of is

(A) 2 (B) 0

(C) - 4i (D) i

Ans: C

Q.14 Let . Then which of the following is true:

(A) is not continuous at z = 0

(B) is differentiable at z = 0

(C) is not continuous at z = 0 but differentiable at z = 0

(D) is continuous at z = 0 but not differentiable at z = 0

Ans: D

Q.15 The value of integral is

(A) 0 (B) 1

(C) 1+2i (D)

Ans: D

Q.16 The angle between the surfaces and at the point (1, 1, 1)

(A) (B)

(C) (D)

Ans: B

Q.17 A force field is said to be conservative if

(A) Curl = 0 (B) grad = 0

(C) Div = 0 (D) Curl (grad ) = 0

Ans: A

Q.18 The partial differential equation is an example of

(A) Hyperbolic equation (B) Elliptic equation

(C) Parabolic equation (D) None of these

Ans: C

Q.19 The inequality between mean and variance of Binomial distribution which is true is

(A) Mean < Variance (B) Mean = Variance

(C) Mean > Variance (D) Mean Variance = 1

Ans: C

Q.20 Let f (x) be a probability density function defined by , for and f (x) = 0 for x< 0, then the value of cumulative distribution function at x =2 is

(A) (B)

(C) (D)

Ans: B

Q.21 If is an analytic function of , then equals

(A) (B)

(C) (D)

Ans: D

Q.22 The image of the circle under the mapping w = z +(3+2 i) is a

(A) circle. (B) ellipse.

(C) pair of lines. (D) hyperbola.

Ans: A

which is circle in (u,v) plane with centre (3,2).

Q.23 The function at z = 0 has

(A) a removable singularity.

(B) a simple pole.

(C) an essential singularity.

(D) a multiple pole.

Ans: C

Let

Z = 0 is an essential singularity.

Q.24 A unit normal vector to the surface , at the point (1, 1, 1) is

(A) . (B) .

(C) . (D) .

Ans: B

Let f = x³-xyz-1,

The normal to surface at point (1,1,1) is

Q.25 The line integral , where C is the boundary of the region equals

(A) 0. (B) a.

(C) (D)

Ans: A

Using Green’s Lemma.

Q.26 The partial differential equation is elliptic if

(A) (B)

(C) (D)

Ans: B

The given differential equation is

It is elliptic if B²-4AC<0 i.e. x²<y²

Q.27 The expected value of a random variable X is 2 and its variance is 1, then variance of 3X+4 is

(A) 9. (B) 7.

(C) 3. (D) 13.

Ans: A

V(3x+4)=E((3x+4)²) – (E(3x+4))²

= 9E(x²)+24E(x)+16-(3E(x)+4)² = 9E(x²)-9(E(x))²= 9V(x)=9

Q.28 Let X be a random variable having a normal distribution. If P (X < 0) = P(X > 2) = 0.4, then mean value of X equals

(A) 0. (B) 1.

(C) 1.5 (D) 2.

Ans: B

As P(X < 0) = P(X > 2). 0, 2 are symmetrically placed around .

= .

Q.29 The image of the point z = 2+3i under the transformation w = z(z-2i) is

(A) (B)

(C) 0 (D) 1-8i

Ans.: B

By definition, if a point maps into the point through the transformation w = f(z) then is known as the image of . Consequently,

w = f(2+3i) = (2+3i)(2+3i-2i) = ( 2+3i)(2+i)=1+8i

Q.30 Let (i) and (ii) denote the facts

(i) : f is continuous at z = 0

(ii) : f is differentiable at z = 0

Then for function which is correct statement?

(A) both (i) & (ii) are true (B) (i) is true, (ii) is false

(C) (i) is false, (ii) is true (D) both (i) & (ii) are false.

Ans. B

f is continuous at z = 0.

= which depends on slope m.

Therefore, f is not differentiable at z = 0.

Q.31 The order of the pole of the function at is

(A) 2 (B) 1

(C) 0 (D) 4

Ans. A

w=0, is a pole of order 2.

Q.32 The value of the integral where C is circle traversed clockwise, is

(A) (B)

(C) 0 (D) 1

Ans. C

z=3 lies outside the circle. By Cauchy’s integral theorem.

Q.33 The curl of the gradient of a scalar function U is

(A) 1 (B)

Ans. D

Curl girdU =

= 0.

Q.34 The value of the integral where C is the curve from x = 3 to x = 24 is

(A) 156 (B) 153

(C) 150 (D) 158

Ans. A

Q.35 The tangent vector to the curve whose parametric equation is at t=2 is given by

(A) (B)

(C) (D)

Ans. B

The tangent vector to any curve is given by

Q.36. The cumulative distribution function F of a continuous variate X is such that

F(a)=0.5, F(b)=0.7. Then value of P(a≤X≤b) is given as

(A) 0 (B) 0.5

(C) 0.2 (D) 0.7

Ans. C

Q.37 A discrete random variate X has probability mass function f which is positive at x = -1,0,1 and is zero elsewhere. If f(0)=0.5 then the value of E(X²) is

(A) 1 (B) 0

(C) 0.5 (D) -0.5

Ans. C

Q.38 A room has three lamp sockets. From a collection of 10 light bulbs of which only 6 are good,

a person selects 3 at random and puts them in a socket. What is the probability that the room will have light?

(A) 29/120 (B) 39/60

(C) 19/30 (D) 29/30

Ans. Answer is not given. Misprint in the question paper.

Q.39 If the ends x = 0 and x = L are insulated in one dimensional heat flow problems, then the boundary conditions are

(A) at t =0 (B) at t = 0.

(C) for all t (D)for all t

Ans.: C As end points are insulated so by definition.

Q.40 If is a solution of then the value of c is

(A) 1 (B) 2

(C) -2 (D) -1/2

Ans. A Using partial derivative, we get.c= 1, the negative sign is neglected on physical grounds

Q.41 The curves u(x, y) = C and v(x, y) = C' are orthogonal if

(A)u and v are complex functions (B)u+iv is an analytic function

(C)u – v is analytic function (D)u + v is an analytic function

Ans. B By definition of orthogonal function.

Q.42 The value of along the line x =y is

(A) 2/3 (B) -2/3

(C) -2i/3 (D) 2i/3

Ans. Answer is not given. Correct solution is

Q.43 The critical points of transformation are given as

(A) ±1 (B) ±i

(C) ±i/2 (D) ±1/2

Ans. B By definition.

Q.44 If the mean and variance of binomial variate are 12 and 4, then the probabilities of the distribution are given by the terms in the expansion of

(A) (B)

(C) (D)

Ans. C Mean= np=12, variance = npq=4

Q.45 The E(e^tx), t<0.5 of a random variable X is (1-2t)^-4, then E(X) is given as

(A) 5 (B) 6

(C) 7 (D) 8

Ans. D As E'(e^tx) = 8(1-2t)^-5 Therefore E(X) is obtained for t = 0, thus E(X)=8

Q.46 Ifis a conservative force field, then the value of curl is

(A) 0 (B) 1

(C) (D) -1

Ans. A By definition of a conservative field = and curl .

Q.47 If then value of is equal to

(A) 5u (B) 5

(C) 5(u+) (D) 5(u-)

Ans. A

Use the identity div=udiv+ .gradu

Q.48 If then , where S is the surface of unit sphere is

(A) 3π (B) 5π

(C) 4π (D) 6π

Ans. C

= .

Q.49 The partial differential equation of a vibration of a string is

(A) (B)

(C) (D)

Ans.: C

It is not partial differential equation but an ordinary differential equation as y is a function of ‘t’ only. The partial differential equation governing the vibration of a string is a wave equation which is hyperbolic.

Q.50 The partial differential equation is classified as

(A) parabolic (B) elliptic

(C) hyperbolic (D) none of these

Ans. B

For pde to be classified as parabolic, eclipse or hyperbolic

. Here this discriminant is <0. Hence elliptic.

Q.51 The value of is

(A) 3 (B) 2

(C) 1 (D) 0

Ans. D

Singularities lie outside . Therefore by Cauchy’s integral theorem D follows.

Q.52 Residue of tanz at z = π/2 is

(A) -1 (B) 1

(C) 0 (D) 2

Ans. A

Q.53 The Taylor series expansion of is

(A) (B)

(C) (D)

Ans. B

Q.54 Let X be normal with mean 10 and variance 4, then P(X<11) is

(A) (B)

(C) (D)

Ans. A

Q.55 If is a valid probability mass function of x, then the value of K is

(A) (B)

(C) 1/2 (D) 2

Ans. B

Q.56 If X is random variable representing the outcome of the roll of an ideal die, then E(X) is

(A) 3 (B) 2.5

(C) 3.5 (D) 4

Ans. C

Q.57 If S is a closed surface enclosing a volume V and if then is equal to

(A) 3 (B) 3S

(C) 3V (D) 3

Ans. C

Q.58 The unit normal at (2,-2,3) to the surface x²y+2xy=4 is

(A) (B)

(C) (D)

Ans. Answer is not given. Misprint in Question.

At the point (2, -2, 3) we get

Q.59 Eliminating a and b from the , we obtain the partial differential equation

(A) (B)

(C) (D)

Ans.: B

Q.60 Solution of is

(A) (B)

(C) (D)

Ans. A Using Lagrange’s subsidiary equations we get

Q.61 Residue of at z = 0 is

(A) 1 (B) -1

(C) 2 (D) 0

Ans. A Residue is the coefficient of .i.e. 1

Q.62 The function w = log z is analytic everywhere except at the value of z when z is equal to

(A) -1 (B) 1

(C) 2 (D) 0

Ans. D

Q.63 If f(z) is analytic in a simply connected domain D and c is any simple closed curve inside D, then the value of is given by

(A) 1 (B) 2

(C) 0 (D) 3

Ans. C By Cauchy’s integral theorem.

Q.64 If X is a binomial variate with p = 1/5, for the experiment of 50 trials, then the standard

deviation is equal to

(A) 6 (B) -8

(C) 8 (D) 2√2

Ans. D As variance = npq = 50(0.2)(0.8)=8 S.D. = square root of variance = =

Q.65 A unit normal to at (0,1,2) is equal to

(A) (B)

(C) (D)

Ans. C

Q.66 If then value of is equal to

(A) 2u (B) -u

(C) 3u (D) 5u

Ans. D

Using the identity

Q.67 is independent of the path joining any two points if it is

(A) irrotational field (B) solenoidal field

(C) rotational field (D) vector field

Ans. A

Since the integral is independent of path therefore the vector

which further implies that the field is irrotational.

Q.68 The solution of the partial differential equation is

(A) z = -x² sin(xy) + yf(x) + g(x)

(B) z=-x²sm(xy)-xf(x)+g(x)

(C) z = -y² sin(xy) + yf(x) + g(x)

(D) z = x² sin(xy) + yf(x) + g(x)

Ans. A

Q.69 When a vibrating string has an initial velocity, its initial conditions are

PART – II

NUMERICALS

Q.1 Compute the limit . (8)

Ans:

By Taylor series expansion,

Therefore,

Similarly, the Taylor series expansion of 1 – e ^z gives,

Therefore,

Q.2 Show that cosh , where . (8)

Ans:

By the Taylor series expansion,

Let r = 1, z = e ^iθ then

as the second integral is 0

Q.3 Evaluate the integral , by contour integration. (10)

Ans:

Consider the integral

where C is the contour given in figure below

Since z = 0 is a pole of f (z), by Cauchy-Residue theorem,

Since f (z) is analytic in C, we have Therefore

Taking limit R → ∞ and ε → 0 we get

Therefore,

Hence

Q.4 Evaluate , where C is the circle traversed counter clockwise. (6)

Ans:

Now by Taylor series expansion,

Therefore,

Therefore Residue of f (z) at z = 0 is -2.

Hence

Q.5 Show that the function in polar coordinates (r, ) is harmonic. Find the corresponding harmonic conjugate function and construct the analytic function f (z) = u + iv such that f(1) = 1. (8)

Ans:

By Cauchy-Riemann equations

From the definitions v (r, )

Therefore,

Now,

Q.6 Find the Fractional Linear Transformation which maps the unit disc onto the right half plane . (8)

Ans:

Then the Cross Ratio

implies

Q.7 A thin rectangular homogenous thermally conducting plate lies in the xy-plane defined by . The edge y = 0 is held at the temperature x (x – ), while the remaining edges are held at temperature . The other faces are insulated and no internal sources and sinks are present. Find the steady state temperature inside the plate. (10)

Ans:

The given problem can be formulated as

Assume the solution be in the form u (x,y) = X (x)Y (y). Then substituting this in the equation we get,

From this, we get

Case 1: k > 0.

Let k = p ² > 0. Then the solution u (x, y) is

Now,

Case 2: k = 0

In this case we get

This again leads to trivial solution.

Case 3: k < 0

For nontrivial solution

By superposition principle,

The boundary condition

Therefore,

From the boundary condition u (x, 0) = x (x – 1) we get

Which immediately implies

Therefore

where

Q.8 It has been claimed that in 60% of all solar heat installations, the utility bill is reduced by at least one third. Accordingly what are the probabilities that the utility bill will be reduced by at least one-third in

(i) four of five installations,

(ii) at least four of five installations. (6)

Ans:

Given problem can be formulated as

Let u (x, t) = X (x) T (t) be the solution. Then from the equation we get

implies X(0) = X(l) = 0

The acceptable solution of the Sturm-Liouville problem

The solutions of

Hence the solution u (x, t) may be written as

From the initial condition u (x, 0) = x we get

Multiplying on both sides and integrating from 0 to , we get

Q.9 Let the probability density function of a random variable x be given by

Find the corresponding cumulative distribution function and determine the probabilities that the random variable x will take a value

(i) greater than (ii) between 0 and . (8)

Ans:

The probability density of the random variable is given by

So, the density function F (x) is F (x) =

If 0 < x < , then

If x > , then

(i) The probability that the random variable will take a value greater than is

(ii) The probability that the random variable will take a value between 0 and is

Q.10 Consider the heat flow in a thin rod of length l, l. The ends x = 0 and x = l are insulated. The rod was initially at temperature f (x) = x. By the method of Separation of Variables, find the temperature distribution u (x, t) in the rod, where u (x, t) is governed by the partial differential equation . (8)

Ans:

This is a binomial distribution with p = 0.6.

(i) Substituting p = 0.6, n = 5, x = 4 in the formula for the binomial distribution, we get

(ii) Similarly

So, the required probability is

b(4,5,0.6) + b(5,5,0.6) = 0.259 + 0.078 = 0.337.

Q.11 If the amount of a cosmic radiation to which a person is exposed while flying by jet across India is a random variable having the normal distribution with mrem and mrem, find the probability that the amount of cosmic radiation to which a person will be exposed on such a flight is between 4.00 and 5.00 mrem. Given that F(1.10) = 0.8643, F(-0.59) = 0.2776, where F is the distribution function of the standard normal distribution. (4)

Ans:

Let X be the given random variable. Given that

Since X has the Normal distribution, the random variable has the standard normal distribution.

Let F be the density function of Z.

(i) The required probability is

(ii) The required probability is

Q.12 Let the probabilities that there are 0, 1, 2 and 3 power failures in a certain city during the month of July be respectively 0.4, 0.3, 0.2 and 0.1. Find the mean and variance of the number of power failures during the month of July in the city.

(4)

Ans:

x	0	1	2	3
f(x)	0.4	0.3	0.2	0.1

The given data:

Hence

= 0.3 + 0.4 + 0.3 = 1

Q.13 Show that is a conservative force field. Find its scalar potential and the work done in moving an object in this field from (1, -2, 1) to (3, 1, 4). (8)

Ans:

A necessary and sufficient condition that a force will be conservative is that

Now

Thus is a conservative force field.

Therefore,

Integrating,

These agree if we choose f (y, z) = 0, g (x, z) = x z³ ,h (x, y) = x|x|y

so that for some constant c.

Then work done =

Q.14 Find the work done in moving the particle in the force field along the space curve , from x = 0 to x = 2. (8)

Ans: Parametric form of given space curve is

Therefore

And

Q.15 Evaluate , where , S is the surface of the plane x + y + z = 1 in the first octant and is unit outward normal to the surface S. (8)

Ans:

Let f (x, y, z) = x +y + z – 1 = 0 be the surface.

Then and the unit outward normal is

Consider the projection of S on the xy plane. The projection of the portion of the plane in the first octant is the triangle bounded by x = 0, y =0 and x + y = 1.

We have

Therefore,

Q.16 Verify the divergence theorem for on the surface S of the sphere . (8)

Ans:

Gauss-divergence theorem states

Where is the unit outward normal to S.

The normal to the surface and the unit outward normal

and

Therefore ,

Q.17 Verify Stokes theorem for , on the surface S of the sphere above the xy-plane. (8)

Ans:

The boundary C of S is the circle in the xy-plane of radius 3 and center at the origin.

Let

Be a parametric equation of C. Then

Now,

Since the projection R of S on to the xy-plane is

the circle ,

we have

Thus we get

which verifies the Stokes theorem.

Q.18 Show that the function is analytic everywhere except on the half line y =1 , x 0. (8)

Ans:

Where

Therefore, the function is single valued and continuous for all

The point z = I is the branch point and the line y = 1 is the branch cut.

We have

Therefore,

The Cauchy-Riemann equations are satisfied. Since, the partial derivatives are continuous. The given function is analytic everyone except on the half line y =1,

Q.19 If u is a harmonic function of two variables (x,y), then show that is not a harmonic function, unless u is a constant. (8)

Ans:

Since u is an harmonic function,

Now, will be a harmonic function if

we have

Therefore,

Now, that is u is a constant function.

Hence is not a harmonic function unless u is a constant.

Q.20 Evaluate the integral from point 1-2i to point 2-i along the curve C, . (8)

Ans:

The curve C is continuous but not differentiable at z = 2, as

Also for any t . Therefore, the curve C is pieceuise smooth.

On the interval [1, 2], we have z = t – 2i, i – e x = t, y = -2.

On the interval [2, 3], we have z = 2 – I (4 – t) i.e. x = 2, y = t - 4.

Hence,

Q.21 Find the residue of the function at z =. (8)

Ans:

The point z = -2 is an isolated essential singular point of f (z). The residue at z = -2 is the coefficient of in the Laurent series expansion of f (z) about z = -2. We write

We note that first and the third product do not contain (z +2) ^-1 term. From the second and the fourth products, collecting the coefficients of (z +2) ^-1 , we obtain

Q.22 Find all possible Taylor’s and Laurent series expansions of the function about the point z = 1. (10)

Ans:

The given function is not analytic at the points z = -1 and z = -2. The distances between the point z = 1 and the points z = -1, z = -2 and 2 and 3 respectively. Therefore, we consider the regions (i) |z - 1| < 2 (ii) 2 < |z – 1| < 3 (iii) |z – 1| < 3. In the region, |z – 1| < 2, the function is analytic. Therefore, we obtain a Taylor’s series expansion in this region. In the other regions, we obtain Laurent series expansions. We write

(i) In the region |z – 1| < 2, we write

The first series is valid in |z – 1| < 2 and the second and third series are rated in |z – 1| < 3. Hence the sum is valid in |z – 1| < 2.

(ii) In the region 2 < |z – 1| < 3, we have

The first series is valid in |z – 1| > 2, and Second series is valid in |z – 1| < 3. Hence the sum is valid in 2 < |z -1 | < 3.

(iii) In the region |z – 1| > 3, we have

Q.23 Evaluate the integral (6)

Ans:

We have

Therefore,

Consider the corresponding centre integral

where f (z) = and C is the path , i.e. semicircle from A to B and then from B to A along real axis.

The function f (z) = is analytic in the upper half plane except for the ample pole at z = i.

we find that

we now write

since

Hence as

Therefore

Q.24 Find the directional derivative of the scalar point function at the point (2, 2, 2) in the direction of the normal to the surface at the point . (6)

Ans:

We have

Hence

The given surface can be written as u = 2

Where u =

which is a vector along the normal to the surface at (2, -1, 3). Therefore repunned directional derivative is the component of

Q.25 If and are constant vectors and show that and hence show that , where r = (10)

Ans:

We know that

Using (1) we get

Also we know that if a is a constant vector.

Therefore,

Q.26 Find the value of the surface integral where S is the curved surface of the cylinder bounded by the planes x = 0, x = 2. (8)

Ans:

We know that d = idydz + jdzdx + kdxdy in terms of the projection of d on the coordinate planes. Taking = , the given integral can be written as

To find , Let .

Hence .

Thus the given integral =

Let

Q.27 The vector field is defined over the volume of the cuboid given by , . Evaluate the surface integral , where S is the surface of the cuboid. (8)

Ans:

The surface integral has to be evaluated as the sum of six integrals corresponding to the six faces of the cuboid. Since S is a closed surface, the Gaurs divulgence thrm. is applicable.

Hence

Q.28 A tightly stretched string with end points fixed at x = 0 and x = L, is initially at rest in equilibrium state. If it is set vibrating by giving to each of its points a velocity find the displacement of the string at any point x from one end, at any point of time t. (12)

Ans:

The partial differential equation for vibrating string is

……….(1)

As per the boundary conditions provided, the form of solutions of (1) is

………..(2)

Further, y (0, t) = 0 gives o = (C₁ + 0)

…………(3)

Further, since the string is initially at rest,

Also from the condition y (l, t) = 0, we get

………..(4)

which gives

Thus the most general solution can be written as

…………(5)

From the boundary condition we get

…………(6)

To determine we expand in a half range Fourier sine series in (0, L), we get

………….(7)

where

comparing (6) and (7) yields

Thus

Q.29 Evaluate the integral (4)

Ans:

The integrand is not analytic at the point Z = ½ which lies within C. Using Cauchy integrand formula

Q.30 A continuous type random variable X has probability density f(x) which is proportional to x² and X takes values in the interval [0, 2]. Find the distribution function of the random variable use this to find P (X >1.2) and conditional probability P(X > 1.2/ X>1). (8)

Ans:

Suppose there are 100 bank account holders. So, 20 persons have taken loans among 20, 18 are males and 2 females. Among 80, who are not loan takers, 76 are males and 4 females. So total males are 94 and females are 6 among account holders.

Males who have taken loans = 18.

Totals male accounts holders = 94.

So, the probability of an account holders who is randomly selected turns out a male that he has taken loans with the bank = .

Q.31 Suppose that on an average 1 house in 1000 houses gets fire in a year in a district. If there are 2000 houses in that district find the probability that exactly 5 houses will have fire during the year. Also find approximate probability using Poisson distribution. (8)

Ans:

In Poison distribution,N = 2000, p = 1/1000

Mean = m = np =2

Q.32 Derive the mean and variance of binomial distribution. (8)

Ans:

Mean =

Variance =

Q.33 Determine the analytic function f (z) = u +i v, given that . (8)

Ans:

It is given that 3u+2v=y²-x²+16xy, thus differentiating partially w.r.t.x and y

Solving, we get u_x=2x+4y and v_x=-4x+2y

Thus f ’(z)= u_x+ iv_x=2x+4y+i(2y-4x)

By Milne’s Thomson method, putting x=z and y=0, we get

f’(z)=2(1-2i)z Thus f(z) = (1-2i)z²+c.

Q.34 If w = u + i v is an analytic function, then show that the family of curves u (x, y) = a, cut the family of curves v (x, y) = b orthogonally, a, b being parameters. (6)

Ans:

Let w=u+iv, and

Since w is an analytic function, thus =and

Thus . Hence cut orthogonally.

Q.35 Find the image of infinite strip , under the mapping . (7)

Ans:

Let w = 1/z, then z = 1/w. Thus

Since y ≤ 1/2, thus u²+ v²+2v ≥ 0 or u²+ (v+1)²≥ 1 i.e.

The boundary of this region is the outside of the circle with centre at(0,-1) and

radius 1,The region y ≥ 1/4 is transformed to

The boundary of this region is the outside of the circle with centre at (0,-2i) and

radius 2. Hence, the infinite strip maps into inside of the circle and outside of the circle . See the shaded region in the figure.

Q.36 Find the linear fractional transformation that maps the points i, -1, 1 of z-plane into the points 0, 1, of w-plane respectively. Where in w-plane is the interior of unit disc mapped by the fractional transformation obtained? (7)

Ans:

Since, w₁=0, w₂=1, w₃=∞, and z₁= i, z₂=-1, z₃=1.

The bilinear transformation is given by

Solving for z gives

Thus interior of the circle , in z plane is mapped onto the Imw > 0.

Q.37 Show that is irrotational and hence find its scalar potential. (8)

Ans:

It is given that

\ Given vector is irrotational. Thus it can be expressed as where f is scalar function.

Integrating w.r.t. x, y, z we get

Since these three must be equal

Q.38 Find the directional derivative of the scalar function at the point (2, -1, 1) in the direction of the normal to the surface at the point (-1, 2, 1). (6)

Ans:

A vector normal to the surface xln(z) - y²+ 4 = 0 is given by

which at point (-1,2,1) becomes . The required directional derivative is the component of along ,

Q.39 Find the work done by a force by moving a particle once around the circle . (7)

Ans:

At C: x² + y² = a², z=0, thus.

Work =

where R is the region bounded by circle x² + y² = a². Let x=rcosθ, y=rsinθ, then

dxdy=rdrdθ , where r changes from 0 to a and θ changes from 0 to 2π.

Thus work = .

Q.40 Show that the vector field is conservative. Hence evaluate the line integral along a path joining the points (0, 0, 0) to (1, 1, 1) (7)

Ans:

\ Given vector represents a conservative field. Thus it can be expressed as where f is scalar function.

Integrating w.r.t. x, y, z we get

Since these three must be equal

Also,

Q.41 A rod of length has its lateral surface insulated and is so thin that heat flow in the rod can be regarded as one dimensional. Initially the rod is at the temperature 100 throughout. At t=0 the temperature at the left end of the rod is suddenly reduced to 50 and maintained thereafter at this value, while the right end is maintained at 100. Let u (x, t) be the temperature at point x in the rod at any subsequent time t.

a. Write down the appropriate partial differential equation for u (x, t), with initial and boundary conditions.

b.Solve the differential equation in (i) above using method of separation of variables and show that Where is the constant involved in the partial differential equation. (3+11)

Ans:

(i) Let the equation for conduction of heat be

Prior to temperature change at end B, when t = 0, the heat flow was independent of time (steady state condition), when u depends only on x i.e.

Since u = 100 for x = 0 and x = L

\ b = 100 and a = 0.

Thus initial condition is expressed as

The boundary conditions are

(ii) Assuming product solution u(x,t) = X(x). T(t) and substituting in equation (1), we get

Case I : If (4) gives

Solving we have the solution

This solution is rejected as exponential term makes temperature u(x,t) increases without bounds as t → ∞.

Case II : If = 0. (4) gives

Integrating, we obtain X = (A X + B) and T = C.

Thus we can write u(x,t) = (A₁ X + B₁), where A₁ = AC and B₁ =BC are arbitrary constant. Using boundary conditions (3), we get is a solution of heat equation.

Case III : If then from (4) (as in case (I)) we conclude that

Since we already have a solution (5) satisfying boundary conditions (3) we can find A, B in (6) by satisfying the condition u(0,t) = 0 =u(l,t) which gives A = 0,

B sinλl = 0 .

As B = 0 leads to trivial solution we must have sinλl = 0 or , n =1,2,….. Combining (5) and (6), we have

as a general solution of (1).

Applying initial condition (2) to the general solution we must have

implies that are the coefficient in half range sine series expansion of

Putting in (7) we get required solution.

Q.42 Evaluate the complex integral .

Also find . (6)

Ans:

If then f(z) = is analytic within and on C. Thus =0,

by Cauchy Theorem if .

If the singularity of f(z) = lies within C and by Cauchy

integral formula

Since ,

Since is constant, then .

Q.43 Find the residues of at its isolated singularities, using Laurent’s series expansions. (8)

Ans:

, z=1, is apole of order 1 and z=-2 is a pole of order 2.

Expanding about z=1,

let z-1=t, i.e. z=t+1,

Since there is only one term in negative powers of (z-1), therefore z = 1,

is a pole of order 1. Residue at z = 1 is the coefficient of 1/t, which is 1/9.

Expanding about z=-2,

let z+2=t, i.e. z=t-2,

Since there are only two terms in negative powers of (z+2), therefore z = -2, is a pole of order 2. Residue is the coefficient of 1/t, which is 8/9.

Q.44 Let u (x, y) be continuous with continuous first and second partial derivatives on a simple closed path C and throughout the interior D of C. Show that where is the directional derivative of u along the outer normal to the curve C. (6)

Ans: Let the position vector of a point on C, in terms of arc length s be

. Then the tangent vector to C is given by

and a normal vector is given by

. Thus

since is a directional derivative of u in the direction of . Now, using Green’s theorem, we obtain

Q.45 Verify Gauss divergence theorem for on the surface

S of the cuboid formed by the planes x = 0, x = a, y = 0, y = b, z = 0 and

z = c. (8)

Ans:

Also, ,

where S₁, S₂, S₃, S₄, S₅and S₆ are the six faces of the cuboid.

On S₁,

On S₂ ,

On S₃,

On S₄,

On S₅,

On S₆,

Thus

Hence Gauss Divergence theorem is verified.

Q.46 The probability of an airplane engine failure (independent of other engines) when the aircraft is in flight is (1-P). For a successful flight at least 50% of the airplane engines should remain operational. For which values of P would you prefer a four engine airplane to a two engine one? (7)

Ans:

Let X be the number of engines that do not fail and let S_k denote the successful flight

with k engine plane. Let 1-p=q,

P(S₂) = P(X ≥ 1) = 1-P(X=0)= 1-q²,

P(S₄) = P(X≥2) = 1-P(X=0)-P(X=1)= 1-4q³+3q⁴.

For P(S₄) > P(S₂), we have

1-4q³+3q⁴> 1- q² or q²(1-q)(1-3q) > 0.

If 0<q<1/3, i.e. 2/3<p<1, the four engine plane is preferred.

Q.47 If the resistance X of certain wires in an electrical networks have a normal distribution with mean of 0.01 ohm and a standard deviation of 0.001 ohm, and specification requires that the wires should have resistance between 0.009 ohm and 0.011 ohms, then find the expected number of wires in a sample of 1000 that are within the specification. Also find the expected number among 1000 wires that cross the upper specification.

(You may use normal table values ). (7)

Ans:

Given that Expected number of wires in a sample of 1000 with this specification

= 1000(0.6826)=682.6=683 approximately.

Hence, expected number among 1000 wires that cross the upper specification = 1000(0.1587)=158.7=159 approximately.

Q.48 Suppose that certain bolts have length where X is a random

variable with probability distribution function.

(i) Determine C so that with probability , a bolt will have length between 400 – C and 400 + C.

(ii) Find the mean and variance of bolt length L. Also find mean and variance of (2 L+5). (4+10)

Ans:

(i) P(400-C≤ L≤ 400+C)=11/16

Since C≠ as it is either >1 or <1. Thus C=1/2.

(ii) E(L) = E(400+X) = 400+E(X)

Thus E(L) = 400.

V(L) = V(400+X) = V(X) = E(X²), as E(X) = 0

Thus E(L) = 400, V(L) = 0.2. Therefore,

E(2L+5) = 2E(L) + 5=805

V(2L+5) = 4V(L) = 0.8.

Q.49 Evaluate the integral dx, using contour integration. (7)

Ans:

Since integrand is an even function, thus

Consider the contour integral and C is the

path from –R to R along the real axis and from R to –R along C_R . Now f(z) is

analytic in upper half of the plane except at z=ai, which is pole of order 1.

Residue of f(z) at

by Residue theorem.

Now, . Therefore, by Jordan’s Lemma

Equating imaginary part, we get

Q.50 Prove that . (7)

Ans:

Q.51 The two equal sides of an isosceles triangle are of length a each and the angle θ between them has a probability density function proportional to in the range and zero otherwise. Find the mean value and the variance of the area of the triangle. (8)

Ans. The area of triangle is

Q.52 Using complex integration, compute (8)

Ans:

The integrand can be written as

Now z =0 is an essential singularity of the integrand.

The Laurent series expansion of f(z) is given by

Q.53 Show that if X has Poisson distribution with mean 1 then its mean deviation about mean is 2/e. (8)

Ans.

Poisson distribution is given by P(r) = where m is the mean of the distribution.

Here mean is 1 = m, S.D. =

We require

= with

Mean deviation about mean =

(since m =1)

Q.54 A person plays an independent games. The probability of his winning any game is (a,b are positive numbers). Show that the probability that the person wins an odd number of games is (8)

Ans.

The probability that a thing is received by a man is p = a/(a+b), a thing is received by women is q = b/(a+b), hence the probability that (2r+1) things are received by men is , the chance that the number of things received by men is odd is

Subtracting we get

Q.55 An infinitely long uniform plane plate of breadth π is bounded by two parallel edges and an end right angles to them. This end is maintained at temperature u₀ for all points and the other edge at zero temperature. Determine the temperature at any point of the plate in the steady state. (8)

Ans.

In the steady state the temperature u(x, y) at any point P(x, y) satisfies Laplace equation.

Thus, we have to solve the following boundary-value problem:

; ; for .

For solving this we make use of the product solution:

The boundary conditions are

The solution is given as

Q.56 Using the method of separation of variables solve (8)

Ans.

Given

Let u(x, y) = X(x)Y(y) be the solution of (1), then

Three possibilities arise

(i) k = 0 (ii) k > 0 (iii) k < 0

For k = 0, X(x) = Ax + B, Y = C; u(x, y) =

(ii) k > 0; ,

Hence .

(iii) ;

and

Thus .

Q.57 Verify Stoke’s theorem for the function where C is the curve of intersection of cone by the plane z = 4 and S is surface of cone below

z = 4. (8)

Ans:

As per Stoke’s theorem we have to prove that

Here

CurlF = ;

;

, ,

In obtaining . nds we transform it to polar coordinates by using , . Thus we get

Hence Stoke’s theorem is verified.

Q.58 Verify Green’s theorem for the function and C is the square with vertices . (8)

Ans:

Along ,

Using Green’s theorem we get

Hence Green’s theorem is satisfied.

Q.59 Show that the vector field is conservative. Find its scalar potential and work done by it in moving a particle from (-1,2,1) to (2,3,4). (8)

Ans:

Therefore the vector field is conservative.

Thus

Q.60 Find a normal vector and the equation of tangent plane to surface at point (3,4,5). (6)

Ans:

; By definition is a vector normal to the surface

, ,

;

Equation of the plane through is

Here a, b, c are the direction ratios of the normal to the plane and are given by , 1 .

Using these values we get equation of the tangent plane as .

Q.61 If is a constant vector and Show that (5)

Ans:

Let

Q.62 Find the values of constants and so that the surfaces intersect orthogonally at the point (1,-1,2). (5)

Ans.

Let the given point (1,-1,2) must lie on both the surfaces. Thus we have . The two surfaces will intersect orthogonally if normals to them at (1,-1,2) are perpendicular to each other. Therefore at (1,-1,2)

Q.63 Show that the function is not analytic at the origin even though CR equations are satisfied at this point. (8)

Ans.

If at the origin, we have

thus CR equations are satisfied at the origin. which depends on m, thus f(z) is not analytic at the origin.

Q.64 If f(z) = u+iv is an analytic function of z and find f(z) subject to the condition (8)

Ans.

Putting x = z, y =0, we get

Q.65 Discuss the transformation w = z + 1/z and show that it maps the circle |z|=a onto an ellipse. In particular discuss the case when a=1 (8)

Ans:

,it is conformal everywhere except at z=1, -1 which corresponds to w =2,-2 of w plane. Let transform to polar coordinate ,

Eliminating we get …(2)

Eliminating r we get…..(3) From (2) it follows that the circle r = a of z plane are mapped into a family of ellipses in the w plane. The ellipses are confocal since a constant. In particular, the unit circle r=1 in the z plane gives from (1) i.e. the unit circle flattens out to become the segment u=-2 to u=2 of real axis in w plane.

Q.66 Obtain the first three terms of the Laurent series expansion of the function about the point z = 0 valid in the region (8)

Ans:

The given function is not analytic when e^z =1, at z = 0 and z= The requires Laurent series expansion is about the point z = 0 . Its region of convergence is , we have

Q.67 Evaluate the integral by contour integration. (10)

Ans:

Since the integrand is an even function we write

Consider the contour integral The function f(z) is analytic in the upper half plane except for the pole of order 2 at z = ai

Now,

Q.68 Evaluate where C is unit circle described in the positive direction. (6)

Ans:

Poles are given by z=0, which is a pole of order 2, sinz =0, z = nπ. Thus only z=0 lies within C.

Q.69 Solve the differential equation for the conduction of heat along a rod without radiations, subject to the following conditions:

(i) u is not infinite for t → ∞

(ii) for x = 0 and x = L.

(iii) u = Lx – x², for t = 0 between x = 0 and x = L. (10)

Ans:

Let u = X(x)T(t) then

From condition (ii) we get

c₂=0, kl=nπ.,

Q.70 Solve subject to the boundary condition

(6)

Ans:

Let u = X(x)Y(y) using the boundary conditions we get

Q.71 From a bag containing a black and b white balls, n ball are drawn at random without replacement. Let X denote the number of black balls drawn. Find the probability mass function of random variable X and compute expectation of Y = 2+3X. (5)

Ans:

Since .

Q.72 If the probability of a bad reaction from a certain injection is 0.001, determine the chance that out of 2000 individuals more than two will get a bad reaction. (5)

Ans:

Mean= np= 2, P(more than two bad reaction)=1-(P(0)+P(1)+P(2))=1-5/e²

Q.73 If X is a continuous random variable with p.d.f. given by

Find the value of k and mean value of X (6)

Ans:

By definition

Mean =

= = 3.

Q.74 Using the method of separation of variables, solve where (8)

Ans:

Let u=X(x)T(t), then thus

Q.75 If the directional derivative of at the point (1,1,1) has maximum magnitude 15 in the direction parallel to the line. Find the value of a,b,c. (8)

Ans.

Thus we get

But directional derivative is maximum parallel to the line

Thus we get

Q.76 If where show that (8)

Ans.

Q.77 Show that the integral is independent of the path joining the points (1,2) and (3,4). Hence evaluate the integral. (8)

Ans:

For integral to be independent of path obviously

and

curl=0

+ f(y)

= 254.

Q.78 Use Stroke’s theorem, to evaluate where and C is the bounding curve of the hemisphere oriented in the +ve direction. (8)

Ans.

Q.79 The vector field is defined over the volume of the cuboid given by Evaluate the surface integralwhere S is the surface of the cuboid. (8)

Ans:

Q.80 Find the points where CR equations are satisfied for the function . Where does exist? Where f(z) analytic? (8)

Ans:

Let

Now

Thus at origin C-R equations are satisfied. exists at the origin only and f(z) is analytic at the origin only.

Q.81 Find the analytic function where (8)

Ans:

We have

thus

du = d(-r²sin2θ+rsinθ)

Thus

after some simplifications turns out to be

Q.82 Find the image in w-plane of

(i) the circle with centre (2.5, 0) and radius 0.5

(ii)The interior of the circle in (i) in z plane under the mapping (8)

Ans:

Equating real and imaginary parts we get

ux – vy -2u = 3 – x, vx – 2v + uy = -y

or (u + 1)x – vy = 2u + 3, vx + (u + 1)y = 2v (1, 2)

or solving eqns(1) & (2) we get

, (3, 4)

Equation of the circle with centre (2.5, 0) and radius 0.5 is (5, 6)

On combining eqns (3,4,5,6) we get

After some simplifications we get

Or which is an equation of imaginary axis.

Equation of the interior of the circle is

When transformed 2, u, v coordinates we get

As as u > 0 which is an equation of the right half plane.

Q.83 Expand in Laurent Series valid for (8)

Ans:

The function

Q.84 Evaluate (10)

Ans:

The integrand can be written as Poles are z = ± 3i, ±i

Q.85 Use Cauchy Integral formula to evaluate where C is the circle |z| = 3 traversed counter clock wise. (6)

Ans:

Poles are at z = 1, 2. Thus

Q.86 An elastic string of length l which is fastened at its ends x = 0 and x = L is released from its horizontal position (zero initial displacement) with initial velocity g(x) given as Find the displacement of the string at any instant of time.

(10)

Ans:

The equation governing the motion of stretched string is given by

--------- (1)

(1) has to be solved under the following initial and b.conditions

b.conditions: u(0, t) = 0 = u(l, t) --------- (2),(3)

initial conditions: u(x, 0) = 0; --------- (4),(5)

For solving (1) we assume solution of the form

y(x, t) = X(x)T(t) --------- (6)

Using (6) in (1) we get --------- (7)

Or (on physical ground)

Hence ---- (8)

Using b.condn(2) we get and (8) reduces to

At x = l, u = 0 yields for all

Either which gives trivial solution y = 0

Or or

Hence we get

Where ,

At t = 0,

Hence

The equation is

With conditions u(0,t)=0, u(L,t)=0, t >0, u(x,0)=0,

Thus

Q.87 Solve by the method of separation of variables

Ans:

Let z = X(x)Y(y) . Using this in the given d.eqn. we get

or ,

Aux. eqn. is ,

Hence

Q.88 The frequency distribution is given as

Calculate Standard deviation and mean deviation about mean. (5)

Ans:

Total Frequency.

Q.89 Suppose the life in hours of a certain kind of radio tube has p.d.f.

Find the distribution function. What is the probability that none of the 3 tubes in a given radio set will have to be replaced during the first 150 hours of operation? What is the probability that all three of the original tubes will be replaced during the first 150 hours? (6)

Ans:

Distribution function = Probability that a tube will fast for first 150 hours is given by Thus the probability that none of the three tubes will have to be replaced during the first 150 hours is The probability that a tube will not last for the first 150 hours is Hence the probability that all three of the original tubes will have to be replaced during the first 150 hours is

Q.90 A variate X has p.d.f.

X	-3	6	9
P(x)	1/6	1/2	1/3

Find E(X), E(X²) and E(2X+1)². (5)

Ans:

Q.91 Fit a Poisson distribution to the following data which gives the number of calls per square for 400 squares.

No. of calls per square (x)	0	1	2	3	4	5	6	7	8	9	10
No. of squares (f)	103	143	98	42	8	4	2	0	0	0	0

It is given that e^-1.32 = 0.2674 (8)

Ans:

Mean =

No. of calls	0	1	2	3	4	5	6	7	8	9	10
Probability	.2674	.353	.233	.103	.034	.009	.002	.0004	.00006	.000009	.000001
Frequency	107	141	93.2=93	41	13.52=14	3.57=4	.78=1	.15=0	.24=0	0	0

Q.92 Find the directional derivative of where at the point (2,0,3) in the direction of the outward normal to the sphere at (3,2,1). (8)

Ans:

;

At the pt (2, 0, 3)

Thus directional derivative along the normal

= .

Q.93 A Fluid motion is given by is the motion is irrotational? If so, find the velocity potential. (8)

Ans:

Thus V is a conservative field. Now

Integrating partially w.r.t. x, we get

Integrating partially w.r.t. y, we get

Integrating partially w.r.t. z, we get

Thus

Q.94 A vector field is given by Evaluate the line integral where C is a circular path given by . (8)

Ans:

----------- (1)

From Green’s theorem we know that

Using this theorem the line integral (1) is transformed to

Using polar coordinates we get .

Q.95 Find where and S is the surface of the sphere having centre (3,-1,2) and radius 3. (8)

Ans:

Q.96 Evaluatewhere and C is boundary of triangle with vertices (0,0,0), (1,0,0) and (1,1,0). (8)

Ans

Since z coordinate is zero thus triangle is in xy plane. Thus .

Q.97 Show that the function is not analytic anywhere. (8)

Ans:

Let now

thus C-R equations are not satisfied anywhere.

Q.98 Show that the function u(x,y)=4xy-3x+2 is harmonic. Construct the corresponding analytic function w = f(z) in terms of complex variables z. (8)

Ans

We have

Thus

Q.99 Find the Taylor series expansion of the function of the complex variable about the point z = 4. Find the region of convergence. (8)

Ans

If centre of a circle is z=4, then the distance of the singularities z =1 and z = 3 from the centre are 3 and 1. Hence, if a circle is drawn with centre at z = 4, and radius 1, then within the circle , then given function f(z) is analytic hence it can be expanded in Taylor’s series within the circle , which is therefore the circle of convergence.

Q.100 Evaluate where . (8)

Ans:

Poles are

These are two circles with centre at (0,1) and (0,-1) with radius .

Thus

Q.101 Using complex variable techniques evaluate the real integral

(10)

Ans:

Poles are z = ½, 2. So inside the contour C there is a simple pole at z = ½.

Q.102 Determine the poles and residue at each pole of the function f(z)= cotz (6)

Ans:

Poles are given by sinz =0, z = nπ. Thus

Q.103 A string is stretched and fastened to two points l apart. Motion is started by displacing the string in the form from which it is released at time t = 0. Show that the displacement of any point at a distance x from one end at time t is given by

(8)

Ans:

Vibrations of the stretched string are governed by the wave equation (under usual notations)

----------------- (1)

Since the end points of the string are fixed for all time, therefore the displacement y(x, t) satisfies the following conditions

y(0, t) = y(l, t) = 0 . ----------------- (2), (3)

Further, as the initial transverse velocity of any point of the string is zero one can write

----------------- (4)

Also, ----------------- (5)

For obtaining solution of (1) under the two boundry conditions (2, 3) and two initial conditions (4, 5) we use the method of product solution and write

Y(x, t) = X(x) T(t) ----------------- (6)

Combining (1) and (6) we get

(on physical ground)

Thus, ----------------- (7)

On using b.c.(2) we get , and on using the b.c.(3) we get

----------------- (8)

Since (8) is valid for all time therefore

cannot be zero as it shall lead to trivial solution. Therefore the only possibility is

----------------- (9)

Consequently, solution (8) assumes the following form:

Where ,

At t = 0,

Hence the solution assumes the following form

------------------ (10)

At t = 0,

Hence n = 1,

Q.104 An infinitely long plate uniform plate is bounded by two parallel edge and an end at right angles to them. The breadth is; this end is maintained at a temperature at all points and other edge at zero temperature. Determine the temperature at any point of the plate in the steady-state. (8)

Ans:

In the steady state the temperature u(x,y) at any point P(x,y) satisfies the equation the boundary conditions are

Or (p is a separation constant)

Thus, X = (Acos px + Bsin px); Y(y) =

Or u(x, y) =

Where A, B, C, D are replaced respectively by .

The solution is given as

Q.105 Show that under the mapping w = 1/z, all circles and straight lines in the z-plane are transformed to circles and straight lines in the w-plane. (8)

Ans:

The equation represents a circle if and a straight line if

a = 0, in the z-plane. Substituting z = x + i y, w = u + i v, in w = 1/z and comparing the real and imaginary parts, we get

is the equation of a circle. If d = 0, we get a + bu – cv =0. We observe the following:

(i) A circle () not passing through the origin () in the z-plane, is transformed into a circle not passing through the origin in the w-plane.

(ii) A circle () passing through the origin (d=0) in the z-plane, is transformed into a straight line not passing through the origin in the w-plane.

(iii) A straight line ( a=0) not passing through the origin () in the z-plane, is transformed into a circle passing through the origin in the w-plane.

(iv) A straight line ( a=0) passing through the origin (d = 0) in the z-plane, is

transformed into a straight line passing through the origin in the w-plane.

Q.106 The probability that a man aged 60 will live to be 70 is 0.65. What is the probability that out of 10 men, now 60, at least 7 will leave to be 70? (8)

Ans:

The probability that a man aged 60 will live to be 70 = p = 0.65, q = 0.35, n = 10,

Probability that at least 7 will live to 70 = P(7 or 8 or 9 or 10)

Q.107 Solve the telephone equation when assuming that is large compared with unity. (8)

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Let be the solution of the given equation. Substituting in equation, we get

The boundary condition is satisfied when we take –ve sign. Since q can be both –ve as well as +ve, thus the general solution is Using boundary conditions, we get where .

Q.108 Show that the vector field defined by the vector function is conservative. (8)

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If the given vector field is conservative, then it can be expressed as the gradient of a scalar function f(x,y,z), therefore, Comparing, we get integrating the first equation, we obtain substituting in the second and third equation we get that g = k = constant. Hence

Q.109 Evaluate (8)

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The parametric equation for C is therefore

Q.110 If

show that vector E and H satisfy the wave equation (8)

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Consider

Thus vector E and H satisfy the wave equation .

Q.111 Using the Green’s theorem, show that where n is the unit vector outward normal to C. (8)

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Let the position vector of a point on C, be then the tangent vector to C is given by , is the unit normal vector . Thus since is the directional derivative of u in the direction of Using Green’s theorem, we get

In obtaining the double integral from line integral,

We have used the following form of the Green’s theorem

Q.112 Use the Divergence theorem to evaluate and S is the boundary of the region bounded by the paraboloid z = and the plane

z = 4y. (8)

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We have

===

Put ,;,,

y=0 y = 4

= =

= = =

Q.113 Show that the function is continuous at the point z = 0, but not differentiable at z = 0. (8)

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Let now

Thus the function is continuous at the point z = 0. Now at z = 0, choosing now the path y = mx, we have as thus which depends on the value of m, thus function is not differentiable at z =0.

Q.114 Show that the function is harmonic. Find its conjugate harmonic function u(x,y) and the corresponding analytic function f(z). (8)

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We have . Thus the function v(x,y) is harmonic. From Cauchy-Riemann equation we get Integrating w.r.t x, we get where g(y) is an arbitrary function of y. Using Cauchy Riemann equation we getThus

Thus .

Q.115 Evaluate the integral (8)

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The curve C is continuous but not differentiable at z = 2, as

also for any t. Therefore the curve C is piecewise smooth. On the interval [1,2], we have z = t – 2i, x=t, y = -2, and f(z) = (1+2i)t +4, On the interval [2,3], we have

z = 2-i(4-t), x=2, y =t-4, and f(z) = 2 + (t-4)² -2i(t -4),

Hence

Q.116 Show that the function is analytic in the region , obtain the Laurent series expansion about z = 0 valid in the region. (8)

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The function is not analytic when These conditions are satisfied when y = 0, 0≤x≤1, The given function is analytic in the region . In the region , consider the function , .

Integrating term by term, we obtain the Laurent series expansion as

where k is a constant of integration, letting thus we get

Q.117. Prove that is harmonic. Find a function v that is conjugate harmonic to u and hence the analytic function with . (7)

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u_x=3x²-3y²+6x, u_xx=6x+6

u_y=-6xy-6y,u_yy= -6x-6 u_xx+u_yy=0 or u is harmonic

If u is conjugate then u+iu is unalytic and hence CR-equations are satisfied

u_x=v_y=3x²-3y²+6x v=3x²y-y³+6xy+g(x)

v_x=6xy+6y+g¹(x)=-u_g=-(-6xy-6y) g₁(x)=0 g(x)=C

=3x²y-y³+6xy+C, v(1,0)=C,u(1,0) =5, f(1)=5+ic=5+I, c=1

f(z)= z³+3y²+(1+i)

Q.118 If f(z) is a regular function of z, then prove that . (7)

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Q.119. Find the image of the strip under the mapping . (4)

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Q.120 Find the image of the circle under the mapping . (4)

Ans: followed by transtation by 3 unit to the right. The circle is mapped into a circle of same radius with center shifting by 1 to the right.

Q.121 Find the linear fractional mapping that maps the points i, 1, 2 + i to 4i, 3-i, respectively. (6)

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Q.122. Show that where and is irrotational. Find f (r) if it is also solenoidal. (8)

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Q.123 The temperature at a point in a space is given by A fly located at the point (4, 4, 2) desires to fly in a direction that gets cooler fastest. Find the direction in which it should fly. Also find the rate decrease of temperature in the direction of flight. (6)

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The direction of maximum decrease is

Q.124 Evaluate the line integral where C is a simple closed path enclosing origin in its interior. (7)

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Q.125 Show that the following line integral is independent of path C from points to and hence evaluate the integral (7)

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Q.126 Obtain d’Alembert’s solution of the wave equation with initial conditions u(x, 0) = f (x), . (11)

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Let

Q.127 A string stretching to infinity in both direction is given the initial displacement and released from rest. Determine the subsequent motion using d’Alembert’s solution obtained in part (a) of the question. (3)

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Q.128 State Cauchy integral formula for derivatives of an analytic function. If , where C is the circle find using Cauchy integral formula. (7)

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Q.129 Identify the singularities of the function . Classify the singularities and find the residues for each of them. (7)

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Q.130 State Green’s theorem and use this theorem to show that for a solution w (x, y) of Laplace’s equation in a region R with boundary curve C and outer unit normal vector ,

(7)

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Q.131 Verify Stoke’s theorem for , where S is the surface of upper half of the sphere and C is the circular boundary on XOY-plane. (7)

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Q.132 Suppose that two teams A and B are playing a series of games. Team A has a probability p of winning a game against team B. The first team to win three games is declared winner of the series. Find the probability distribution of number of games played in the series for declaring a winner. (7)

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Q.133 The probability density function of a random variable X equals and = 0, otherwise. Find c. Also find the probability that X takes a value greater than its expected value. (7)

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Q.134 A ticket office can serve 4 customers per minute. The average number of customers arriving to the ticket office for purchase of tickets is 120 per hour. Assuming number of customers arriving to the ticket office follow Poisson distribution, find the probability that ticket office is continuously busy during first 30 minutes of opening. (6)

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Q.135 Sick leaves time X used by employees of a company in one month is roughly normal with mean 1000 hrs and standard deviation 100 hrs. How much time t should be budgeted for sick leaves during next month if t is to be exceeded with a probability of 16%?

Also find the probability that in the next year no more than one month will have sick leave time more than 1200 hrs. (You may use the following values of distribution function of standard normal distribution ) (8)

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Q.136. Evaluate the integral where a > 0, by using contour integration. (9)

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Q.137 Let S be a closed surface of volume V, containing the point P in its interior and let N be the outer unit normal to the surface S at a general point. Show that div. (5)

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