Madras University (UnOM) 2006 B.Sc Mathematics - I - Question Paper

Time : 3 hours
Maximum: 100 marks

part A - ( [10 x two = Marks 20]
ans ALL ques..

every ques. carries two marks.


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

2. Define a unitary matrix and provide example.

3. State Fermat's theorem.

4. If two + i is a root of the formula X2 -px+q=0 obtain 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 ranging from the asymptotes of the hyperbola X Y ? a 2

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

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

part B - [5 x 16 = Marks 80]

ans ALL ques..

every ques. carries 16 marks.

11. (a) Show that


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

(b) Show that n seven - 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). (Marks 8+8)

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

8 two -2 the matrix three 3 -1

24 eight -6

(b) if A and B are orthogonal matrices of the identical 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 two 3

(d) Find the rank of the matrix two three 4

-0 two 2-

(12+4)

13. (a) Solve X3 - 7X2 + 36 = 0, provided that 1 root is double the other.

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

Or

(c) obtain a positive real root of x three - 2x - five = 0 by

Horner's method to 3 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) obtain the formula 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 two -b two . (8+8)

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

(b) Prove that

32 sin six 0 = 10 - 15 cos 20 + six cos four 0 - cos six 0.

(8+8)

Earning:   Approval pending.