# IIT Maths Sample Paper 2

### Algebra

1. Simplify the expression (a > 0, a ¹ 0) :  (a-x/Ö 5)[2a2x-ax(2ax-1)] {1-(Ö5ax/2ax-1)-2}-1/2´ Ö[(ax+2)2-5]-(a2x+4)[a2x+4(1-ax)]-1/2+4ax[1+(ax+2)(a2x-4ax+4)-1/2]{ax+2+(a2x-4ax+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.2x+b.3y+1 £ Ö(4x+9y+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)=ax/(ax + Öa) (a > 0), evaluate r=1å2n-1 f(r/2n). IIT JEE - Home Websites of IIT JEE Syllabus IIT Question Papers Books For IIT IIT JEE 2008 IIT JEE 2007 IIT JEE 2006 IIT JEE 2005 IIT Courses Seats Reservation Eligibility Criteria Analysis of IIT JEE IIT Alumni IIT JEE Tips IIT Articles