# 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:
¥
å   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 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.
6. Prove that for each posive integer 'm' the smallest integer which exceeds (Ö3 + 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=1år  { m=1år  aman/(m+n)} ³ 0. ai is any real number.
9. Prove the following inequality: k=1ån   Ö[nCk] £ Ö[n(2n-1)]
10. A sequence {Un, n ³ 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 ³ 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-Ö(1+2x2)}2 < 2x+9
14. Find all real roots of: Ö(x2-p) + 2Ö(x2-1) = x
15. 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.
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-B2£0.
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
 xCr n-1Cr n-1Cr-1 x+1Cr nCr nCr-1 x+2Cr n+2Cr n+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. 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