The roots of the equation $a(x^2 + 1) - (a^2 + 1)x = 0$ are

Which of the following six statements are true about the cubic polynomial $P(x) = 2x^3 + x^2 + 3x - 2$?
$(i)$ It has exactly one positive real root.
$(ii)$ It has either one or three negative roots.
$(iii)$ It has a root between $0$ and $1$.
$(iv)$ It must have exactly two real roots.
$(v)$ It has a negative root between $-2$ and $-1$.
$(vi)$ It has no complex roots.

The sum of the roots of the equation $x^2 + |2x - 3| - 4 = 0$ is:

If $\sin A, \sin B, \cos A$ are in $G.P.$,then the roots of the equation $x^2 + 2x \cot B + 1 = 0$ are always

The quadratic equation $p(x) = 0$ with real coefficients has purely imaginary roots. Then the equation $p(p(x)) = 0$ has

The number of solutions of the equation $e^{\sin x} - 2e^{-\sin x} = 2$ is