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IIT Maths Sample Paper

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:
_¥
å   ^n-r+1C_d . ^r-1C_d-1 = ⁿC_r
^d=1

3.     a, b are 4.     x + y + z = 15 and xy + yz + zx = 72, prove that 3 £ x £ 7.

4.     Let l, a Î 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.

5.     Prove that for each posive integer 'm' the smallest integer which exceeds (Ö3 + 1)^2m is divisible by 2^m+1.

6.     Prove that, for every natural k, the number (k³)! is divisible by (k!)^k2+k+1.

7.     Prove that the inequality: _n=1å^r { _m=1å^r a_ma_n/(m+n)} ³ 0. a_i is any real number.

8.     Prove the following inequality: _k=1åⁿ   Ö[ⁿC_k] £ Ö[n(2ⁿ-1)]

9.     A sequence {U_n, n ³ 0} is defined by U₀=U₁=1 and U_n+2=U_n+1+U_n.Let A and B be natural numbers such that A¹⁹ divides B⁹³and B¹⁹divides A⁹³.Prove by mathematical induction, or otherwise, that the number (A⁴+B⁸)^Un+1 is divisible by (AB)^Un for n ³ 1.

10.     The real numbers a, b satisfy the equations: a³ + 3a² + 5a - 17 = 0, b³ - 3b² + 5b + 11 = 0. Find a+b.

11.   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.

12.   Solve: 4x²/{1-Ö(1+2x²)}² < 2x+9

13.   Find all real roots of: Ö(x²-p) + 2Ö(x²-1) = x

14.   The solutions a, b, g of the equation x³+ax+a=0, where 'a' is real and a¹0, satisfy a²/b + b²/g + g²/a = -8. Find a, b, g.

15.   If a, b, c are real numbers such that a²+b²+c²=1, prove the inequalities: -1/2 £ ab+bc+ca £ 1.

16.   Show that, if the real numbers a, b, c, A, B, C satisfy: aC-2bB+cA=0 and ac-b²>0 then AC-B²£0.

17. 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) + ....

18. Solve for x, y, z:
yz = a(y+z) + r
zx = a(z+x) + s
xy = a(x+y) + t

19. Solve for x, n, r > 1

^xC_r

^n-1C_r

^n-1C_r-1

^x+1C_r

ⁿC_r

ⁿC_r-1

^x+2C_r

ⁿ⁺²C_r

ⁿ⁺²C_r-1

= 0

20. 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 ^mpC_r.

21. 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.

22. 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))

23. Find all real p, q, a, b such that we have (2x-1)²⁰ - (ax+b)²⁰ = (x²+px+q)¹⁰ for all x.

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