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-(Ö5a^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 log₄18 is an irrational number.
3. Determine all such integers a and b for which one of the roots of 3x³+ax²+bx+12=0 is equal to 1 + Ö3.
4. Solve in terms of complex numbers: z³ + (w⁷)*=0; z⁵.w¹¹ = 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(a²+b²+1)
6. Prove the inequality nⁿ⁺¹ > (n+1)ⁿ, n ³ 3, n Î N.
7. Prove that

   (b+c)2 a2 a2 b2 (c+a)2 b2 c2 c2 (a+b)2

   = 2abc(a+b+c)3

   (Without expanding)
8. Sum the series: 1 + 4x + 9x² + ...
9. The eqns ax² + bx + c=0 and x³=k have a common root. Prove that

   a b c b c ak c ak bk

   = 0

10. If w is a root of x⁴=1 then Show that a + bw + cw² + dw³ 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 x⁴ in (1 + 2x + 3x²)⁵.
12. The sum of squares of 3 terms of a GP is S². If their sum is aS, Prove that a² 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 + Öa) (a > 0), evaluate _r=1å^2n-1 f(r/2n).