The product of all real roots of the equation ${x^2} - |x| - \,6 = 0$ is

Let $S=\left\{\sin ^2 2 \theta:\left(\sin ^4 \theta+\cos ^4 \theta\right) x^2+(\sin 2 \theta) x+\right.$ $\left(\sin ^6 \theta+\cos ^6 \theta\right)=0$ has real roots $\}$. If $\alpha$ and $\beta$ be the smallest and largest elements of the set $S$, respectively, then $3\left((\alpha-2)^2+(\beta-1)^2\right)$ equals....................

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 

If $a, b, c \in R$ and $1$ is a root of equation $ax^2 + bx + c = 0$, then the curve y $= 4ax^2 + 3bx+ 2c, a \ne 0$ intersect $x-$ axis at

If $3$ distinct real number $a$,$b$,$c$ satisfy $a^2(a + p) = b^2 (b + p) = c^2 (c + p)$ where $p \in R$, then value of $bc + ca + ab$ is

The number of real roots of the equation, $\mathrm{e}^{4 \mathrm{x}}+\mathrm{e}^{3 \mathrm{x}}-4 \mathrm{e}^{2 \mathrm{x}}+\mathrm{e}^{\mathrm{x}}+1=0$ is