The equation $\sqrt {3 {x^2} + x + 5} = x - 3$ , where $x$ is real, has

If $\alpha $, $\beta$, $\gamma$  are roots of ${x^3} - 2{x^2} + 3x - 2 = 0$ , then the value of$\left( {\frac{{\alpha \beta }}{{\alpha  + \beta }} + \frac{{\alpha \gamma }}{{\alpha  + \gamma }} + \frac{{\beta \gamma }}{{\beta  + \gamma }}} \right)$ is

The number of distinct real roots of the equation $x ^{7}-7 x -2=0$ is

One root of the following given equation $2{x^5} - 14{x^4} + 31{x^3} - 64{x^2} + 19x + 130 = 0$ is

The number of distinct real roots of the equation $|\mathrm{x}+1||\mathrm{x}+3|-4|\mathrm{x}+2|+5=0$, is ...........

The solution set of the equation $pq{x^2} - {(p + q)^2}x + {(p + q)^2} = 0$ is