Let $\phi (x) = (x) + {2^{\log _x^3}} - {3^{\log _x^2}}$ then

If non-zero real numbers $b$ and $c$ are such that $min \,f\left( x \right) > \max \,g\left( x \right)$, where $f\left( x \right) = {x^2} + 2bx + 2{c^2}$  and $g\left( x \right) = {-x^2} - 2cx + {b^2}$$\left( {x \in R} \right)$; then $\left| {\frac{c}{b}} \right|$ lies in the interval

The domain of the function $f(x){ = ^{16 - x}}{\kern 1pt} {C_{2x - 1}}{ + ^{20 - 3x}}{\kern 1pt} {P_{4x - 5}}$, where the symbols have their usual meanings, is the set

Let $A = \{ {x_1},\,{x_2},\,............,{x_7}\} $ and $B = \{ {y_1},\,{y_2},\,{y_3}\} $ be two sets containing seven and three distinct elements respectively. Then the total number of functions $f : A \to B$ that are onto, if there exist exactly three elements $x$ in $A$ such that $f(x)\, = y_2$, is equal to

If $f\left( x \right) + 2f\left( {\frac{1}{x}} \right) = 3x,x \ne 0$ and $S = \left\{ {x \in R:f\left( x \right) = f\left( { - x} \right)} \right\}$;then $S :$

Domain of $f (x)$ = $\sqrt {{{\log }_2}\left( {\frac{{10x - 4}}{{4 - {x^2}}}} \right) - 1} $ , is