Code: A-01/C-01/T-01 Subject: MATHEMATICS

Time: 3 Hours JUNE 2003 Max. Marks: 100

NOTE: There are 11 Questions in all.

· Question 1 is compulsory and carries 16 marks. Answer to Q. 1. must be written in the space provided for it in the answer book supplied and nowhere else.

· Answer any THREE Questions each from Part I and Part II. Each of these questions carries 14 marks.

· Any required data not explicitly given, may be suitably assumed and stated.

Q.1 Choose the correct or best alternative in the following: (2x8)

a. The value of the integral where C is the contour is

(A) . (B) .

(C) 0. (D) .

b. If X has a Poisson distribution such that then the variance of the distribution is

(A) 1. (B) -1.

(C) 2. (D) 0.

c. The vector field function is called solenoidal if

(A) curl =0. (B) div =0.

(C) grad =0. (D) grad div =0.

d. The number of distinct real roots of in the interval is

(A) 0. (B) 2.

(C) 3. (D) 1.

e. The solution of : is

(A) . (B) .

(C) . (D) .

f. If then is equal to

(A) . (B) .

(C) . (D) .

g. The value of Legendre’s Polynomial, is

(A) 1. (B) -1. (C) . (D) 0.

h. The value of integral over the region bounded by the line y = x and the curve is

(A) . (B) . (C) . (D) .

PART I

Answer any THREE questions. Each question carries 14 marks.

Q.2 a. Find the stationary value of subject to the fulfilment of the condition , given a, b, c are not zero. (7)

b. Find the volume enclosed by coordinate planes and portion of the plane lying in the first quadrant. (7)

Q.3 a. If the directional derivative of at (1, 1,1) has maximum magnitude 15 in the direction parallel to line find the value of a, b, c. (7)

b. Verify divergence theorem for the vector field taken over the region bounded by cylinder . (7)

Q.4 a. Show that and hence deduce that . (7)

b. Solve in the interval subject to the boundary conditions :

(i) (ii)

(iii) (iv) . (7)

Q.5 a. Use Cayley - Hamilton theorem to express in terms of A and the identity matrix I, where . (7)

b. Solve . (7)

Q.6 a. Find analytic function whose real part is . (7)

b. Evaluate , using contour integration. (7)

PART II

Answer any THREE questions. Each question carries 14 marks.

Q.7 a. In a normal distribution 31% of the items are under 45 and 8% are over 64. Find the mean and standard deviation of the distribution.

[Given that ,

where is pdf of standard normal distribution.] (7)

b. A can hit a target 3 times in 5 shots, B 2 times in 5 shots and C 3 times in 4 shots. All of them fire one shot each simultaneously at the target. What is the probability that (i) two shots hit (ii) atleast two shots hit? (7)

Q.8 a. Diagonalize the matrix . (7)

b. Investigate the values of and so that equations

have (i) no solution (ii) a unique solution (iii) infinite number of solutions. (7)

Q.9 a. The height h and semi vertical angle of a cone are measured and the total area A of surface of cone including that of base is calculated in terms of h, . If h and are in error by small quantities and respectively, find the corresponding error in the area. Show further that if an error of +1% in h will be approximately compensated by an error of –0.33 degrees in . (7)

b. If what values of n will make (7)

Q.10 a. A vector field is given by . Show that the field is irrotational and find its scalar potential. Hence evaluate line integralfrom (1, 2) to (2, 1). (7)

b. Solve . (7)

Q.11 a. Express in terms of and . (7)

b. Find Taylor’s expansion of about the point z = i. (7)

Code: A-01/C-01/T-01 Subject: MATHEMATICS

Time: 3 Hours December 2003 Max. Marks: 100

NOTE: There are 11 Questions in all.

· Question 1 is compulsory and carries 16 marks. Answer to Q. 1. must be written in the space provided for it in the answer book supplied and nowhere else.

· Answer any THREE Questions each from Part I and Part II. Each of these questions carries 14 marks.

· Any required data not explicitly given, may be suitably assumed and stated.

Q.1 Choose the correct or best alternative in the following: (2x8)

a. The value of the integral where C is the semi-circular arc above the real axis is

(A) . (B) .

(C) . (D) .

b. Residue at z = 0 of the function is

(A) . (B) .

(C) . (D) .

c. In solving any problem, odds against A are 4 to 3 and odds in favour of B in solving the same problem are 7 to 5. The probability that the problem will be solved is

(A) . (B) .

(C) . (D) .

d. The value of the integral over the area in the first quadrant by the curve is

(A) . (B) .

(C) . (D) .

e. The surface will be orthogonal to the surface at the point for values of a and b given by

(A) a = 0.25, b = 1. (B) a = 1, b = 2.5.

(C) a = 1.5, b = 2. (D) .

f. If and and if z = u + v then equals

(A) 4 v. (B) 4 u.

(C) 2 u. (D) 4 u + v.

g. The series equals

(A) . (B) . (C) . (D) .

h. The value of integral , where is a Legendre polynomial of degree 3, equals

(A) . (B) 0. (C) . (D) .

PART I

Answer any THREE Questions. Each question carries 14 marks.

Q.2 a. Examine the following system of equations for consistency :

Reduce the augmented matrix of the above system of equations to Echelon form and find the solution of the above system, if it exists. (7)

b. Find the eigen values and the corresponding eigen vectors of the matrix A defined by

Obtain the modal matrix and reduce the given matrix to the diagonal matrix. (7)

Q.3 a. If and is an analytic function of , find f(z) subject to the condition that at . (7)

b. Prove that the relation transforms the real axis in the z-plane into a circle in the w-plane. Find the centre and the radius of the circle and the point in the z-plane which is mapped on the centre of the circle. (7)

Q.4 a. Solve in series the differential equation :

(8)

b. Prove that

where c is an arbitrary constant. (6)

Q.5 a. Show that the vector field represented by

is irrotational but not solenoidal. Also obtain a scalar function such that grad = . (5)

b. If and , show that . Also show that is a possible solution of where A and B are arbitrary constants. (5)

c. Evaluate where and s is the surface of the cylinder included in the first octant between z = 0 and z = 5. (4)

Q.6 a. If , then show that . (7)

b. The Luminosity L of a star is connected with its mass M by the relation where and a, b being given constants. If p is the percentage error made in the estimate of M, express the resulting percentage error in the calculated luminosity in terms of p and and show that it lies between p and 3p. (7)

PART II

Answer any THREE Questions. Each question carries 14 marks.

Q.7 a. Solve the following differential equations :

(i) . (5)

(ii) . (5)

b. Change the order of integration in the following integral :

(4)

Q.8 a. A string is stretched and fastened to two points at distance l apart. Motion is ensued by displacing the string into the form from which it is released at time t = 0. Find the displacement at any point x and any time t. (5)

b. The ends A and B of an insulated rod of length , have their temperatures at and until steady state conditions prevail. The temperatures of these ends are changed suddenly to and respectively. Find the temperature distribution in the rod at any time t. (9)

Q.9 a. Represent the function in Laurent’s series

(i) within

(ii) in the annular region within and

(iii) Exterior to . (8)

b. Apply the calculus of residue to evaluate . (6)

Q.10 a. Show that the stationary values of , where and are the roots of the equation . (7)

b. Expand in powers of and upto

3^rd degree terms. (7)

Q.11 a. A man takes a step forward with probability 0.4 and backward with probability 0.6. Find the probability that at the end of 11 steps, he is just one step away from the starting point. (7)

b. In a certain factory turning out razor blades, there is a small chance of for any blade to be defective. The blades are supplied in packets of 10. Using Poisson’s distribution calculate the approximate number of packets containing

(i) no defective blade (ii) one defective blade (iii) two defectives blades

in a consignment of 10,000 packets . (7)

Code: A-01/C-01/T-01 Subject: MATHEMATICS

Time: 3 Hours JUNE 2004 Max. Marks: 100

NOTE: There are 11 Questions in all.

· Question 1 is compulsory and carries 16 marks. Answer to Q. 1. must be written in the space provided for it in the answer book supplied and nowhere else.

· Answer any THREE Questions each from Part I and Part II. Each of these questions carries 14 marks.

· Any required data not explicitly given, may be suitably assumed and stated.

Q.1 Choose the correct or best alternative in the following: (2x8)

a. For what values of x, the matrix is singular?

(A) 0, 3 (B) 3, 1

(C) 1, 0 (D) 1, 4

b. If then

(A) 3 ab (B) 2 abz

(C) abz (D) 3 abz

c. The value of the integral is

(A) . (B) 2.

(C) -2. (D) 0.

d. If and then div

(A) 5 (B) 5u

(C) (D) 0

e. The solution of the differential equation is given as

(A) (B)

(C) (D)

f. The value of the integral where C is the circle is given as

(A) (B)

(C) 0 (D)

g. The value of the Legendre’s polynomial if

(A) (B) (C) (D)

h. Two persons A and B toss an unbiased coin alternately on the understanding that the first who gets the head wins. If A starts the game, then his chances of winning is

(A) (B) (C) (D)

PART I

Answer any THREE questions. Each question carries 14 marks.

Q.2 a. Show that at the point on the surface where x = y = z, we have . (7)

b. Find the volume of greatest rectangular parallelopiped that can be inscribed inside the ellipsoid . (7)

Q.3 a. Change the order of integration and hence evaluate . (7)

b. Find the volume common to cylinders and . (7)

Q.4 Solve the differential equations

(i) . (8)

(ii) . (6)

Q.5 a. Discuss the consistency of the following system of equations for various values of

(7)

and, if consistent solve it.

b. Find the characteristic equation of the matrix and hence, find the matrix represented by

(7)

Q.6 a. Show that the vector field is irrotational as well as solenoidal. Find the scalar potential. (6)

b. Evaluate where and S is the region of the plane 2x + 2y + z = 6 in the first octant. (8)

PART II

Answer any THREE questions. Each question carries 14 marks.

Q.7 a. The odds that a Ph.D. thesis will be favourably reviewed by three independent examiners are 5 to 2, 4 to 3 and 3 to 4. What is the probability that a majority approve the thesis? (7)

b. If the probabilities of committing an error of magnitude x is given by compute the probable error from the following data :

. (7)

Q.8 a. Solve by method of separation of variables . (5)

b. Solve for conduction of heat along a rod without

radiation subject to

(i) u is not infinite for (ii) for x = 0, x =

(iii) for t = 0 between x = 0 and x = . (9)

Q.9 a. Obtain the series solution of equation . (8)

b. Express in terms of and . (6)

Q.10 a. Express as = where . (7)

b. Find analytic function whose real part is . (7)

Q.11 a. Show that under the transformation , real axis in the z-plane is mapped into the circle Which portion of the z plane corresponds to the interior of the circle? (7)

b. Let where C is ellipse . Find value of F(3.5) and . (7)

Code: A-01/C-01/T-01 Subject: MATHEMATICS-I

Time: 3 Hours December 2004 Max. Marks: 100

NOTE: There are 11 Questions in all.

· Question 1 is compulsory and carries 16 marks. Answer to Q. 1. must be written in the space provided for it in the answer book supplied.

· Answer any THREE Questions each from Part I and Part II. Each of these questions carries 14 marks.

· Any required data not explicitly given, may be suitably assumed and stated.

Q.1 Choose the correct or best alternative in the following: (2x8)

a. The value of limit

(A) equals 0. (B) equals .

(C) equals 1. (D) does not exist.

b. If then equals

(A) . (B) .

(C) . (D) .

c. The function has

(A) a minimum at (0, 0).

(B) neither minimum nor maximum at (0, 0).

(C) a minimum at (1, 1).

(D) a maximum at (1, 1).

d. The family of orthogonal trajectories to the family , where k is an arbitrary constant, is

(A) . (B) .

(C) . (D) .

e. Let be two linearly independent solutions of the differential equation . Then , where are constants is a solution of this differential equation for

(A) . (B) .

(C) no value of . (D) all real .

f. If A, B are two square matrices of order n such that AB=0, then rank of

(A) at least one of A, B is less than n.

(B) both A and B is less than n.

(C) none of A, B is less than n.

(D) at least one of A, B is zero.

g. A real matrix has an eigenvalue i, then its other two eigenvalues can be

(A) 0, 1. (B) -1, i. (C) 2i, -2i. (D) 0, -i.

h. The integral , n>1, where is the Legendre’s polynomial of degree n, equals

(A) 1. (B) . (C) 0. (D) 2.

PART I

Answer any THREE questions. Each question carries 14 marks.

Q.2 a. Compute and for the function

(6)

b. Let v be a function of (x, y) and x, y are functions of defined by

where Show that . (8)

Q.3 a. Expand near (1, 1) upto 3^rd degree terms by Taylor’s series. (7)

b. Find the extreme value of subject to the conditions and . (7)

Q.4 a. Find the rank of the matrix

(6)

b. Let

be a linear transformation from to

and

be a linear transformation from to .

Find the linear transformation from to by inverting appropriate matrix and matrix multiplication. (8)

Q.5 a. Prove that the eigenvalues of a real matrix are real or complex conjugates in pairs and further if the matrix is orthogonal, then eigenvalues have absolute value 1. (6)

b. Find eigenvalues and eigenvectors of the matrix . (8)

Q.6 a. Find a matrix X such that is a diagonal matrix, where . Hence compute . (8)

b. Prove that a general solution of the system

can be written as

+ + where are arbitrary. (6)

PART II

Answer any THREE questions. Each question carries 14 marks.

Q.7 a. Let Recognise the region R of integration on the r.h.s. and then evaluate the integral on the right in the order indicated. (7)

b. Compute the volume of the solid bounded by the surfaces and . (7)

Q.8 a. Let be an integrating factor for differential equation Mdx+Ndy=0 and is a solution of this equation, then show that is also an integrating factor of this equation, G being a non-zero differentiable function of . (6)