If for the function $f(x) = \frac{1}{4}{x^2} + bx + 10$ ; $f\left( {12 – x} \right) = f\left( x \right)\,\forall \,x\, \in \,R$ , then the value of $'b'$ is 

If $f$ is a function satisfying $f(x+y)=f(x) f(y)$ for all $x, y \in N$ such that $f(1)=3$ and $\sum\limits_{x = 1}^n {f\left( x \right) = 120,} $ find the value of $n$

Let $f : R \rightarrow R$ be a function defined by $f ( x )=$ $\log _{\sqrt{m}}\{\sqrt{2}(\sin x-\cos x)+m-2\}$, for some $m$, such that the range of $f$ is $[0,2]$. Then the value of $m$ is $…………$

Let $\sum\limits_{k = 1}^{10} {f\,(a\, + \,k)} \, = \,16\,({2^{10}}\, – \,1),$ where the function $f$ satisfies $f(x + y) = f(x) f(y)$ for all natural numbers $x, y$ and $f(1) = 2.$ Then the natural number $‘ a '$ is

Numerical value of the expression $\left| {\;\frac{{3{x^3} + 1}}{{2{x^2} + 2}}\;} \right|$ for $x = – 3$ is