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)2a2a2
    b2(c+a)2b2
    c2c2(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
    abc
    bcak
    cakbk
    = 0
  10. If w is a root of x4=1 then Show that a + bw + cw2 + dw3 is a factor of
    abcd
    bcda
    cdab
    dabc
    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).