The product of the roots of the equation $9 x^{2}-18|x|+5=0,$ is

Let $f(x)=a x^2+b x+c$, where $a, b, c$ are integers, Suppose $f(1)=0,40 < f(6) < 50,60 < f(7) < 70$ and $1000 t < f(50) < 1000(t+1)$ for some integer $t$. Then, the value of $t$ is

The sum of the solutions of the equation $\left| {\sqrt x  - 2} \right| + \sqrt x \left( {\sqrt x  - 4} \right) + 2 = 0\left( {x > 0} \right)$ is equal to

The sum of integral values of $a$ such that the equation $||x\ -2|\ -|3\ -x||\ =\ 2\ -a$ has a solution

The number of roots of the equation $\log ( - 2x)$ $ = 2\log (x + 1)$ are

Exact set of values of $a$ for which ${x^3}(x + 1) = 2(x + a)(x + 2a)$ is having four real solutions is