Number of rational roots of equation $x^{2016} -x^{2015} + x^{1008} + x^{1003} + 1 = 0,$ is equal to

If $\log _{(3 x-1)}(x-2)=\log _{\left(9 x^2-6 x+1\right)}\left(2 x^2-10 x-2\right)$, then $x$ equals

Let $a, b, c, d$ be numbers in the set $\{1,2,3,4,5,6\}$ such that the curves $y=2 x^3+a x+b$ and $y=2 x^3+c x+d$ have no point in common. The maximum possible value of $(a-c)^2+b-d$ 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 $n^2-10 n-36 ?$

Product of real roots of the equation ${t^2}{x^2} + |x| + \,9 = 0$

Let $f(x)={{x}^{2}}-x+k-2,k\in R$ then the complete set of values of $k$ for which $y=\left| f\left( \left| x \right| \right) \right|$ is non-derivable at $5$ distinict points is