# IIT Maths Sample Paper 1

### Algebra

- 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.
- Use combinatorial argument to prove the identity:

_{¥}

å^{n-r+1}C_{d}.^{r-1}C_{d-1}=^{n}C_{r}

^{d=1} - a, b are roots of x
^{2}+ax+b=0, g, d are roots of x^{2}-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’. - x + y + z = 15 and xy + yz + zx = 72, prove that 3 £ x £ 7.
- 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. - Prove that for each posive integer 'm' the smallest integer which exceeds (Ö3 + 1)
^{2m}is divisible by 2^{m+1}. - Prove that, for every natural k, the number (k
^{3})! is divisible by (k!)^{k2+k+1}. - Prove that the inequality:
_{n=1}å^{r}{_{m=1}å^{r}a_{m}a_{n}/(m+n)} ³ 0. a_{i}is any real number. - Prove the following inequality:
_{k=1}å^{n}Ö[^{n}C_{k}] £ Ö[n(2^{n}-1)] - A sequence {U
_{n}, n ³ 0} is defined by U_{0}=U_{1}=1 and U_{n+2}=U_{n+1}+U_{n}.Let A and B be natural numbers such that A^{19}divides B^{93 }and B^{19 }divides A^{93}.Prove by mathematical induction, or otherwise, that the number (A^{4}+B^{8})^{Un+1}is divisible by (AB)^{Un}for n ³ 1. - The real numbers a, b satisfy the equations: a
^{3}+ 3a^{2}+ 5a - 17 = 0, b^{3}- 3b^{2}+ 5b + 11 = 0. Find a+b. - Given 6 numbers which satisfy the relations:

y^{2}+ yz + z^{2}= a^{2}

z^{2}+ zx + x^{2}= b^{2}

x^{2}+ xy + y^{2}= c^{2}

Determine the sum x+y+z in terms of a, b, c. Give geometrical interpretation if the numbers are all positive. - Solve: 4x
^{2}/{1-Ö(1+2x^{2})}^{2}< 2x+9 - Find all real roots of: Ö(x
^{2}-p) + 2Ö(x^{2}-1) = x - The solutions a, b, g of the equation x
^{3}+ax+a=0, where 'a' is real and a¹0, satisfy a^{2}/b + b^{2}/g + g^{2}/a = -8. Find a, b, g. - If a, b, c are real numbers such that a
^{2}+b^{2}+c^{2}=1, prove the inequalities: -1/2 £ ab+bc+ca £ 1. - Show that, if the real numbers a, b, c, A, B, C satisfy: aC-2bB+cA=0 and ac-b
^{2}>0 then AC-B^{2}£0. - When 0<x<1, find the sum of the infinite series: 1/(1-x)(1-x
^{3}) + x^{2}/(1-x^{3})(1-x^{5}) + x^{4}/(1-x^{5})(1-x^{7}) + .... - Solve for x, y, z:

yz = a(y+z) + r

zx = a(z+x) + s

xy = a(x+y) + t - Solve for x, n, r > 1

^{x}C_{r}^{n-1}C_{r}^{n-1}C_{r-1}^{x+1}C_{r}^{n}C_{r}^{n}C_{r-1}^{x+2}C_{r}^{n+2}C_{r}^{n+2}C_{r-1}= 0 - 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
^{mp}C_{r}. - Let a and b be real numbers for which the equation x
^{4}+ ax^{3}+ bx^{2}+ ax + 1 = 0 has at least 1 real solution. For all such pairs (a,b), find the minimum value of a^{2}+b^{2}. - Prove that:

2/(x^{2}- 1) + 4/(x^{2}- 4) + 6/(x^{2}- 9) + ... + 20/(x^{2}- 100) =

11/((x - 1)(x + 10)) + 11/((x - 2)(x + 9)) + ... + 11/((x - 10)(x + 1)) - Find all real p, q, a, b such that we have (2x-1)
^{20}- (ax+b)^{20}= (x^{2}+px+q)^{10}for all x.