Relations and F...

lf $$f:R\rightarrow R$$ such that $$f(x+y)-Kxy = f(x)+2y^2$$ for all $$x,y \in R$$ and $$f(1)=2,f(2)=8$$ then $$f(20)-f(10)$$

Vamsi is "X" years old. His sister''s age is $$\tiny \dpi{150} {\color{DarkBlue} 4\frac{1}{2}}$$ times that of Vamsi. Where as his uncle is 30 years older than him. If the total of their ages is 56 years, what is the age of Vamsi?

If $$f(x)$$ is a polynomial function of degree $$n$$ satisfying eqation $$f(x) f(2x) = xf(3x)$$ then $$f(x)$$ is

If $$f(x) = \cos(  \log x)$$ then $$f(x^2)f(y^2)-\dfrac{1}{2} \left [ f(x^2y^2)+f\left ( \dfrac{x^2}{y^2} \right) \right]=$$

What will be the range, when d $$=$$ 0

If $$A=\left \{ 1,2,3 \right \} $$ and $$B=\left \{ 4,5,6 \right \}$$ then which of the following sets are relation from $$A$$ to $$B$$
(i) $$\displaystyle R_{1}=\left \{ (4,2) (2,6)(5,1)(2,4)\right \}$$
(ii) $$\displaystyle R_{2}=\left \{ (1,4) (1,5)(3,6)(2,6) (3,4)\right \}$$
(iii) $$\displaystyle R_{3}=\left \{ (1,5) (2,4)(3,6)\right \}$$
(iv) $$\displaystyle R_{4}=\left \{ (1,4) (1,5)(1,6)\right \}$$

The number of solution of the equation $$\displaystyle a^{f\left ( x \right )}+g\left ( x \right )=0,$$ where $$\displaystyle a >0, g\left ( x \right )\neq 0$$ and $$\displaystyle g\left ( x \right )$$ has minimum value $$\dfrac14$$, is

Let $$f$$ be a function satisfying $$\displaystyle f(x+y)=f(x)+f(y)$$ for $$x,y\:\epsilon\:R$$. If $$f(1)=k$$ then $$f(n)$$,$$n\:\in\:N$$, is equal to 

If the polynomial satisfies $$f\left( x \right) =\dfrac { 1 }{ 2 } \begin{vmatrix} f\left( x \right)  & f\left( \dfrac { 1 }{ x }  \right) -f\left( x \right)  \\ 1 & f\left( \dfrac { 1 }{ x }  \right)  \end{vmatrix}$$ and $$\displaystyle f \left ( 2 \right ) = 17$$, then the value of $$\displaystyle f\left ( 3 \right )$$ is