The number of real roots of the equation $x|x|-5|x+2|+6=0$, is

The sum of all the roots of the equation is :$\left|x^2-8 x+15\right|-2 x+7=0$ is :

The number of integral solutions $x$ of $\log _{\left(x+\frac{7}{2}\right)}\left(\frac{x-7}{2 x-3}\right)^2 \geq 0$ is

The number of elements in the set $\left\{n \in \mathbb{Z}:\left|n^2-10 n+19\right|<6\right\}$ is $\_\_\_\_$ .

Let $\mathrm{a}, \mathrm{b}, \mathrm{c}$ be three distinct positive real numbers such that $(2 \mathrm{a})^{\log _{\mathrm{a}} \mathrm{a}}=(\mathrm{bc})^{\log _{\mathrm{a}} \mathrm{b}}$ and $\mathrm{b}^{\log _{\mathrm{a}} 2}=\mathrm{a}^{\log _{\mathrm{a}} \mathrm{c}}$. Then...

Let $\alpha, \beta, \gamma$ be the three roots of the equation $x^3+b x+c=0$. If $\beta \gamma=1=-\alpha$, then $b^3+2 c^3-3 \alpha^3-6 \beta^3-8 \gamma^3$ is equal to-

The sum of all the roots of the equation $\left|x^2-8 x+15\right|-2 x+7=0$ is :

The number of distinct real roots of the equation $|x||x+2|-5|x+1|-1=0$ is

The sum of all the solutions of the equation $(8)^{2 x}-16 \cdot(8)^x+48=0$ is is:

Let $S$ be the set of positive integral values of a for which $\frac{\mathrm{ax}^2+2(\mathrm{a}+1) \mathrm{x}+9 \mathrm{a}+4}{\mathrm{x}^2-8 \mathrm{x}+32}<0, \forall \mathrm{x} \in \mathbb{R}$. Then, the number of...