Let $f: R -\{0\} \rightarrow(-\infty, 1)$ be a polynomial of degree $2$ , satisfying $f( x ) f\left(\frac{1}{ x }\right)=f( x )+f\left(\frac{1}{ x }\right)$. If $f(K)=-2 K$, then the sum of squares of all possible values of $K$ is :

Let $m$ and $n$ be the numbers of real roots of the quadratic equations $x^2-12 x+[x]+31=0$ and $x ^2-5| x +2|-4=0$ respectively, where $[ x ]$ denotes the greatest integer $\leq x$. Then $m ^2+ mn + n ^2$ is equal to $..............$.

The locus of the point $P=(a, b)$ where $a, b$ are real numbers such that the roots of $x^3+a x^2+b x+a=0$ are in arithmetic progression is

If the equation $\frac{1}{x} + \frac{1}{{x - 1}} + \frac{1}{{x - 2}} = 3{x^3}$ has $k$ real roots, then $k$ is equal to -

The number of solutions of $\frac{{\log 5 + \log ({x^2} + 1)}}{{\log (x - 2)}} = 2$ is

What is the sum of all natural numbers $n$ such that the product of the digits of $n$ (in base $10$ ) is equal to $n^2-10 n-36 ?$