The value of $c$ for which $|{\alpha ^2} - {\beta ^2}| = \frac{7}{4}$,where $\alpha$ and $\beta$ are the roots of $2{x^2} + 7x + c = 0$,is

If the roots of the equation $\frac{x^2 - bx}{ax - c} = \frac{m - 1}{m + 1}$ are equal in magnitude but opposite in sign,then $m = \dots$

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3 + x^2 + x + 1 = 0$,then match the items of List-$I$ with those of List-$II$:
List-$I$:
$(i)$ $\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma}$
(ii) $\alpha^3 + \beta^3 + \gamma^3$
(iii) $\alpha^4 + \beta^4 + \gamma^4$
(iv) $(\alpha - \beta)^2 + (\beta - \gamma)^2 + (\gamma - \alpha)^2$
List-$II$:
$(A)$ $-1$
$(B)$ $-4$
$(C)$ $1$
$(D)$ $3$
$(E)$ $0$

If $-1$ is a twice repeated root of the equation $ax^3+bx^2+cx+1=0$,then

If the ratio of the roots of the equation $ax^2 + bx + c = 0$ is $p:q$,then

If the roots of the equation $ax^2 + ax + c = 0$ are in the ratio $p:q$,then $\sqrt{\frac{p}{q}} + \sqrt{\frac{q}{p}} = $