
IIT Maths Sample Paper 2
Algebra
 Simplify the expression (a > 0, a ¹
0) : (a^{x}/Ö
5)[2a^{2x}a^{x}(2a^{x}1)]
{1(Ö5a^{x}/2a^{x}1)^{2}}^{1/2}´
Ö[(a^{x}+2)^{2}5](a^{2x}+4)[a^{2x}+4(1a^{x})]^{1/2}+4a^{x}[1+(a^{x}+2)(a^{2x}4a^{x}+4)^{1/2}]{a^{x}+2+(a^{2x}4a^{x}+4)^{1/2}}^{1} and determine for which values of x this expression is equal to 1.
 Prve that log_{4}18 is an irrational number.
 Determine all such integers a and b for which one of the roots of 3x^{3}+ax^{2}+bx+12=0 is equal to 1 +
Ö3.
 Solve in terms of complex numbers: z^{3} +
(w^{7})*=0; z^{5}.w^{11}
= 1. (* indicates conjugate).
 Prove that if a > 0, b > 0 then for any x and y the following inequality holds true: a.2^{x}+b.3^{y}+1 £
Ö(4^{x}+9^{y}+1)Ö(a^{2}+b^{2}+1)
 Prove the inequality n^{n+1} > (n+1)^{n}, n ³ 3, n Î N.
 Prove that
(b+c)^{2}  a^{2}  a^{2} 
b^{2}  (c+a)^{2}  b^{2} 
c^{2}  c^{2}  (a+b)^{2} 
 = 2abc(a+b+c)^{3} 
(Without expanding)
 Sum the series: 1 + 4x + 9x^{2} + ...
 The eqns ax^{2} + bx + c=0 and x^{3}=k have a common root. Prove that
 If w
is a root of x^{4}=1 then Show that a + bw
+ cw^{2} +
dw^{3} is a factor of
Hence Show that the det is equal to (a+b+c+d)(ab+cd){(ac)^{2}+(bd)^{2}}.
 Find the coefficient of x^{4} in (1 + 2x + 3x^{2})^{5}.
 The sum of squares of 3 terms of a GP is S^{2}. If their sum is aS, Prove that a^{2} Î (1/3,1) È (1,3).
 Find the sum to n terms: 0!/5! + 1!/6! + 2!/7! + ...
 If f(x)=a^{x}/(a^{x} + Öa) (a > 0), evaluate _{r=1}å^{2n1} f(r/2n).
 