The value of $a$ such that the sum of the squares of the roots of the equation $x^2 - (a - 2)x - a + 1 = 0$ assumes the least value is:

The roots of the quadratic equation $3x^2 - px + q = 0$ are the $10^{\text{th}}$ and $11^{\text{th}}$ terms of an arithmetic progression with a common difference $d = \frac{3}{2}$. If the sum of the first $11$ terms of this arithmetic progression is $88$,then $q - 2p$ is equal to:

If $\operatorname{cosec} \theta$ and $\cot \theta$ are the roots of $cx^2+bx+a=0$ $(bc \neq 0)$,then $b^2(b^2-4ac)=$

If the difference between the roots of the equation $x^2 + px + q = 0$ is $1$,then:

If $p$ and $q$ are the roots of the equation $x^{2}+px+q=0$, then:

The condition that $x^3 - p x^2 + q x - r = 0$ may have two of its roots equal to each other but of opposite sign is