A real valued function $f(x)$ satisfies the function equation $f(x - y) = f(x)f(y) - f(a - x)f(a + y)$ where a is a given constant and $f(0) = 1$, $f(2a - x)$ is equal to

Let $A=\{1,3,7,9,11\}$ and $B=\{2,4,5,7,8,10,12\}$. Then the total number of one-one maps $\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}$, such that $\mathrm{f}(1)+\mathrm{f}(3)=14$, is :

For a real number $x,\;[x]$ denotes the integral part of $x$. The value of $\left[ {\frac{1}{2}} \right] + \left[ {\frac{1}{2} + \frac{1}{{100}}} \right] + \left[ {\frac{1}{2} + \frac{2}{{100}}} \right] + .... + \left[ {\frac{1}{2} + \frac{{99}}{{100}}} \right]$ is

$f(x,\;y) = \frac{1}{{x + y}}$ is a homogeneous function of degree

Product of all the solution of the equation  ${x^{1 + {{\log }_{10}}x}} = 100000x$ is

The set of values of $'a'$ for which the inequality ${x^2} - (a + 2)x - (a + 3) < 0$ is satisfied by atleast one positive real $x$ , is