Code: D-23 / DC-23 Subject: MATHEMATICS - II

Time: 3 Hours 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 points 2i – j + k, i – 3j – 5k, 3i – 4j – 4k are the vertices of a triangle which is

(A) equilateral. (B) isosceles.

(C) right angled. (D) None of these.

b. If then ordered pair (x, y) is

(A) (0, 2). (B) (0, 1).

(C) (1, 0). (D) (1, 1).

c. If then is

(A) . (B) .

(C) . (D) .

d. A vector of magnitude 2 along a bisector of the angle between the two vectors 2i - 2j + and i + 2j - 2 is

(A) . (B) .

(C) . (D) None of these.

e. Let A and B be two matrices such that and AB =0. Then we must have

(A) B = 0. (B) B to be identity matrix.

(C) . (D) None of these.

f. If then is

(A) 0. (B) 1.

(C) 2. (D) 3.

g. exists only when n is

(A) zero. (B) –ve integer.

(C) +ve integer. (D) –ve rational.

h. The differential equation of the curve , where a and b are constants, is

(A) . (B) .

(C) . (D) .

PART I

Answer any THREE Questions. Each question carries 14 marks.

Q.2 a. If the complex numbers be the vertices of an equilateral triangle, prove that . (7)

b. If the roots of represent vertices of a triangle in the Argand plane, then find area of the triangle. (7)

Q.3 a. Reduce to the modulus amplitude form. (7)

b. Prove that . (7)

Q.4 a. If a square matrix A satisfies a relation Prove that exists and that being an identity matrix. (7)

b. Show that any square matrix can be written as the sum of two matrices, one symmetric and the other anti-symmetric. (7)

Q.5 a. Show that x = 2 is one root of the determinant and find other two roots. (6)

b. Show that . (8)

Q.6 a. If and be any two vectors, then show that

(i) .

(ii) . (7)

b. Forces of magnitudes 5, 3, 1 units respectively, act in the directions respectively on a particle. If the particle is displaced from the point to the point , find the work done by the resultant force. (7)