Let $f(x)=x^2+2x+2$,$g(x)=-x^2+2x-1$,and $a, b$ be the extreme values of $f(x)$ and $g(x)$ respectively. If $c$ is the extreme value of $\frac{f}{g}(x)$ (for $x \neq 1$),then $a+2b+5c+4=$

Let $R = \{(1, 3), (2, 2), (3, 2)\}$ and $S = \{(2, 1), (3, 2), (2, 3)\}$ be two relations on set $A = \{1, 2, 3\}$. Then $RoS =$

Let $f(x) = \max(\sin x, \cos x)$ and $g(x) = \min(\cos x, \sin x)$. Define $h(y) = f(x)y^2 + ay + g(x)$. If the equation $h(y) = 0$ has real roots for all $x \in R$,find the complete set of values of $a$.

If $f: R \rightarrow R$ and $g: R \rightarrow R$ are given by $f(x)=|x|$ and $g(x)=[x]$ for each $x \in R$, then $\{x \in R: g(f(x)) \leq f(g(x))\}$ is equal to

If ${e^{f(x)}} = \frac{{10 + x}}{{10 - x}},\;x \in ( - 10,\;10)$ and $f(x) = kf\left( {\frac{{200x}}{{100 + {x^2}}}} \right)$,then $k = $

If $f$ is an even function defined on the interval $(-5, 5),$ then four real values of $x$ satisfying the equation $f(x) = f\left( \frac{x + 1}{x + 2} \right)$ are