In a triangle $PQR$, $\angle R = \pi / 2$. If $\tan(P/2)$ and $\tan(Q/2)$ are the roots of the quadratic equation $ax^2 + bx + c = 0$, where $a \neq 0$, then which of the following is true?

If $\alpha$ and $\beta$ are the roots of the equation $2x^2 + 2(a + b)x + a^2 + b^2 = 0$,then the equation whose roots are $(\alpha + \beta)^2$ and $(\alpha - \beta)^2$ is

If the product of the roots of the equation $2x^2 + 6x + \alpha^2 + 1 = 0$ is $-\alpha$,then the value of $\alpha$ will be

The number of integers $k$ for which the equation $x^3-27x+k=0$ has at least two distinct integer roots is

Let $\alpha$ and $\beta$ be the roots of the equation $px^2+qx-r=0$,where $p \neq 0$. If $p, q,$ and $r$ are the consecutive terms of a non-constant $G$.$P$. and $\frac{1}{\alpha}+\frac{1}{\beta}=\frac{3}{4}$,then the value of $(\alpha-\beta)^2$ is:

If $\alpha, \beta, \gamma$ are the roots of $x^3+p x^2+q x+r=0$,then the value of $(1+\alpha^2)(1+\beta^2)(1+\gamma^2)$ is