If $f(x) = 2\sin x$, $g(x) = {\cos ^2}x$, then $(f + g)\left( {\frac{\pi }{3}} \right) = $

Let for $a \ne {a_1} \ne 0,$ $f\left( x \right) = a{x^2} + bx + c\;,g\left( x \right) = {a_1}{x^2} + {b_1}x + {c_1},p\left( x \right) = f\left( x \right) - g\left( x \right),$ If $p\left( x \right) = 0$ only for  $ x=-1 $ and $p\left( { - 2} \right) = 2$ then value of $p\left( 2 \right)$ is

If $f(x) = \frac{x}{{x - 1}} = \frac{1}{y}$, then $f(y) = $

Let, $f(x)=\left\{\begin{array}{l} x \sin \left(\frac{1}{x}\right) \text { when } x \neq 0 \\ 1 \text { when } x=0 \end{array}\right\}$ and $A=\{x \in R: f(x)=1\} .$ Then, $A$ has

Let $x$ denote the total number of one-one functions from a set $A$ with $3$ elements to a set $B$ with $5$ elements and $y$ denote the total number of one-one functions from the set $A$ to the set $A \times B$. Then ...... .

The total number of functions,$f:\{1,2,3,4\} \cdot\{1,2,3,4,5,6\}$   such that $f (1)+ f (2)= f (3)$, is equal to .