IIT Maths Sample Paper 1

Algebra

  1. Let x be a real number with 0<x<p. Prove that, for all natural numbers n, the sum sinx + sin3x/3 + sin5x/5 + ... + sin(2n-1)x/(2n-1) is positive.
  2. Use combinatorial argument to prove the identity:

      n-r+1Cd . r-1Cd-1 = nCr
    d=1
  3. a, b are roots of x2+ax+b=0, g, d are roots of x2-ax+b-2=0. Given 1/a + 1/b + 1/d + 1/g =5/12 and abdg = 24, find the value of the coefficient ‘a’.
  4. x + y + z = 15 and xy + yz + zx = 72, prove that 3 x 7.
  5. Let l, a R, find all the set of all values of l for which the set of linear equations has a non-trivial solution.
    lx + (sin a) y + (cos a) z = 0
    x + (cos a) y + (sin a) z = 0
    -x + (sin a) y - (cos a) z = 0
    If l = 1, find all values of a.
  6. Prove that for each posive integer 'm' the smallest integer which exceeds (3 + 1)2m is divisible by 2m+1.
  7. Prove that, for every natural k, the number (k3)! is divisible by (k!)k2+k+1.
  8. Prove that the inequality: n=1r  { m=1r  aman/(m+n)} 0. ai is any real number.
  9. Prove the following inequality: k=1n   [nCk] [n(2n-1)]
  10. A sequence {Un, n 0} is defined by U0=U1=1 and Un+2=Un+1+Un.Let A and B be natural numbers such that A19 divides B93 and B19 divides A93.Prove by mathematical induction, or otherwise, that the number (A4+B8)Un+1 is divisible by (AB)Un for n 1.
  11. The real numbers a, b satisfy the equations: a3 + 3a2 + 5a - 17 = 0, b3 - 3b2 + 5b + 11 = 0. Find a+b.
  12. Given 6 numbers which satisfy the relations:
    y2 + yz + z2 = a2
    z2 + zx + x2 = b2
    x2 + xy + y2 = c2
    Determine the sum x+y+z in terms of a, b, c. Give geometrical interpretation if the numbers are all positive.
  13. Solve: 4x2/{1-(1+2x2)}2 < 2x+9
  14. Find all real roots of: (x2-p) + 2(x2-1) = x
  15. The solutions a, b, g of the equation x3+ax+a=0, where 'a' is real and a0, satisfy a2/b + b2/g + g2/a = -8. Find a, b, g.
  16. If a, b, c are real numbers such that a2+b2+c2=1, prove the inequalities: -1/2 ab+bc+ca 1.
  17. Show that, if the real numbers a, b, c, A, B, C satisfy: aC-2bB+cA=0 and ac-b2>0 then AC-B20.
  18. When 0<x<1, find the sum of the infinite series: 1/(1-x)(1-x3) + x2/(1-x3)(1-x5) + x4/(1-x5)(1-x7) + ....
  19. Solve for x, y, z:
    yz = a(y+z) + r
    zx = a(z+x) + s
    xy = a(x+y) + t
  20. Solve for x, n, r > 1
    xCrn-1Crn-1Cr-1
    x+1CrnCrnCr-1
    x+2Crn+2Crn+2Cr-1
    = 0
  21. Let p be a prime and m a positive integer. By mathematical induction on m, or otherwise, prove that whenever r is an integer such that p does not divide r, p divides mpCr.
  22. Let a and b be real numbers for which the equation x4 + ax3 + bx2 + ax + 1 = 0 has at least 1 real solution. For all such pairs (a,b), find the minimum value of a2+b2.
  23. Prove that:
    2/(x2 - 1) + 4/(x2 - 4) + 6/(x2 - 9) + ... + 20/(x2 - 100) =
      11/((x - 1)(x + 10)) + 11/((x - 2)(x + 9)) + ... + 11/((x - 10)(x + 1))
  24. Find all real p, q, a, b such that we have (2x-1)20 - (ax+b)20 = (x2+px+q)10 for all x.