If $f(x)$ satisfies the relation $f\left( \frac{5x - 3y}{2} \right) = \frac{5f(x) - 3f(y)}{2}$ for all $x, y \in R$,with $f(0) = 1$ and $f'(0) = 2$,then the period of $\sin(f(x))$ is:

Find $\sum_{t=1}^{39} f(t)$ if $f: R \rightarrow R$ is defined as $f(x+y)=f(x)+f(y)$ for all $x, y \in R$ and $f(1)=7$.

Let a function $f : R \rightarrow R$ be defined such that $3f(2x^2 - 3x + 5) + 2f(3x^2 - 2x + 4) = x^2 - 7x + 9$ for all $x \in R$. Then the value of $f(5)$ is:

Let $f: R \rightarrow R$ be a twice differentiable function such that $(\sin x \cos y)(f(2x+2y) - f(2x-2y)) = (\cos x \sin y)(f(2x+2y) + f(2x-2y))$ for all $x, y \in R$. If $f'(0) = \frac{1}{2}$,then the value of $24f''\left(\frac{5\pi}{3}\right)$ is:

Let $f: \mathbb{R} - \{0\} \rightarrow (-\infty, 1)$ be a function satisfying $f(x)f(\frac{1}{x}) = f(x) + f(\frac{1}{x})$. If $f(x)$ is a polynomial of degree $2$ and $f(K) = -2K$,then the sum of squares of all possible values of $K$ is:

Let $f: R \rightarrow R$ be a function defined by $f(x)=(2+3a)x^2 + \left(\frac{a+2}{a-1}\right)x + b$,where $a \neq 1$. If $f(x+y) = f(x) + f(y) + 1 - \frac{2}{7}xy$,then the value of $28 \sum_{i=1}^3 |f(i)|$ is: