- 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(1-a^{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} - Sum the series: 1 + 4x + 9x
^{2}+ ... - The eqns ax
^{2}+ bx + c=0 and x^{3}=k have a common root. Prove thata b c b c ak c ak bk = 0 - If w
is a root of x
^{4}=1 then Show that a + bw + cw^{2}+ dw^{3}is a factor ofa b c d b c d a c d a b d a b c ^{2}+(b-d)^{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^{2n-1}f(r/2n).