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IIT Maths Sample Paper 2

Algebra

1. Simplify the expression (a > 0, a ¹ 0) :  (a -x / Ö 5)[2a 2x -a x (2a x -1)] {1-( Ö 5a x /2a x -1) -2 } -1/2 ´ Ö [(a x +2) 2 -5]-(a 2x +4)[a 2x +4(1-a x )] -1/2 +4a x [1+(a x +2)(a 2x -4a x +4) -1/2 ]{a x +2+(a 2x -4a x +4) 1/2 } -1 and determine for which values of x this expression is equal to 1.
2. Prve that log 4 18 is an irrational number.
3. Determine all such integers a and b for which one of the roots of 3x 3 +ax 2 +bx+12=0 is equal to 1 + Ö 3.
4. Solve in terms of complex numbers: z 3 + ( w 7 )*=0; z 5 . w 11 = 1. (* indicates conjugate).
5. Prove that if a > 0, b > 0 then for any x and y the following inequality holds true: a.2 x +b.3 y +1 £ Ö (4 x +9 y +1) Ö (a 2 +b 2 +1)
6. Prove the inequality n n+1 > (n+1) n , n ³ 3, n Î N.
7. Prove that

   (b+c) 2 a 2 a 2 b 2 (c+a) 2 b 2 c 2 c 2 (a+b) 2

   = 2abc(a+b+c) 3

   (Without expanding)
8. Sum the series: 1 + 4x + 9x 2 + ...
9. The eqns ax 2 + bx + c=0 and x 3 =k have a common root. Prove that

   a b c b c ak c ak bk

   = 0

10. If w is a root of x 4 =1 then Show that a + b w + c w 2 + d w 3 is a factor of

    a	b	c	d
    b	c	d	a
    c	d	a	b
    d	a	b	c

    Hence Show that the det is equal to -(a+b+c+d)(a-b+c-d){(a-c) 2 +(b-d) 2 }.
11. Find the coefficient of x 4 in (1 + 2x + 3x 2 ) 5 .
12. The sum of squares of 3 terms of a GP is S 2 . If their sum is a S, Prove that a 2 Î (1/3,1) È (1,3).
13. Find the sum to n terms: 0!/5! + 1!/6! + 2!/7! + ...
14. If f(x)=a x /(a x + Ö a) (a > 0), evaluate r=1 å 2n-1 f(r/2n).