B.Sc Mathematics
Paper-I

 
OCTOBER 2003 	U/IC 4717/PAN


Time : Three hours 	Maximum: 100 marks

SECTION A - (10 x 2 = 20 marks)

Answer ALL questions.

Each question carries 2 marks.


1.	Obtain the coefficient of x 3 in (1 + X)-n

2.	Define a unitary matrix and give example.

3.	State Fermat's theorem.

4.	If 2 + i is a root of the equation X2 -px+q=0 find

p and q.

5.	State Cayley-Hamilton theorem.

6.	Define the rank of a matrix.

7.	Find the eccentricity of the ellipse 4X2 + 9Y2 = 36.

8.	What is the angle between the asymptotes of the

hyperbola X Y ? a 2

9.	Expand tan n 0 in terms of tan 0 -

10.	Prove that cosh 2 x + sinh 2 x=cosh2x

SECTION B - (5 x 16 = 80 marks)

Answer ALL questions.

Each question carries 16 marks.

11. (a) Show that


1 + 1+2 + 1+2+2 2 +...-= 1 (e _ 1)2
2! 3!	4! 	2

	(b)	Show that n 7 - n is divisible by 42 for any

integer n. 	(10+6)

Or

(c) Show that

.,rX 2		5	21)
	+44a-~+1=~x 1	- -

	2	U2 + 8X4

when x is large.

(d) Show that n(n-1)(n+25)(n +50)=M(24).

(8+8)

12. (a) Find the eigen values and eigen vectors of

8 2 -2 the matrix 3 3 -1

24 8 -6

(b) if A and B are orthogonal matrices of the same order, then show that AB is orthogonal. (12+4)

Or

(c) Show that the equations x + y + z = 6,

x + 2y + 3z = 14, x + 4y + 7z = 30 are consistent and

hence solve them.

		1 2 3

	(d)	Find the rank of the matrix 2 3 4

		-0 2 2-

(12+4)

13. (a) Solve X3 - 7X2 + 36 = 0, given that one root is double the other.

(b) Solve : X4 _ 1OX3 +26X2 _10X+1 =0. (8+8)

Or

(c) Find a positive real root of x 3 - 2x - 5 = 0 by

Horner's method to three places of decimals.	(16)

14. (a) If CP, CD are conjugate semi diameter for an ellipse, prove that Sp. S, p = CD 2.

(b)Show that tangents at the ends of a focal

	chord of a parabola intersect at right angles on the
directrix. 	(8+8)

Or

(c) Find the equation of the chord joining the points 't, ' and 't2' on the rectangular hyperbola XY = C2.

(d) If the diameter CP meets the hyperbola in P and if the conjugate diameter meets the conjugate hyperbola in D, show that Cp2 _CD2 =a 2 -b 2 . (8+8)

15. (a) Separate tan-' (x + iy) into real and imaginary parts.

(b) Prove that

32 sin 6 0 = 10 - 15 cos 20 + 6 cos 4 0 - cos 6 0.

(8+8)