If $\alpha$ and $\beta$ are the distinct roots of the equation $x^{2}+(3)^{1/4}x+3^{1/2}=0$,then the value of $\alpha^{96}(\alpha^{12}-1) + \beta^{96}(\beta^{12}-1)$ is equal to:

On each face of a cuboid,the sum of its perimeter and its area is written. Among the six numbers so written,there are three distinct numbers and they are $16, 24$ and $31$. The volume of the cuboid lies between

The remainder when the polynomial $2 x^5-3 x^4+5 x^3-3 x^2+7 x-9$ is divided by $x^2-x-3$ is

Let $\alpha, \beta$ be the roots of the equation $x^2 - 3x + r = 0$, and $\frac{\alpha}{2}, 2\beta$ be the roots of the equation $x^2 + 3x + r = 0$. If the roots of the equation $x^2 + 6x = m$ are $2\alpha + \beta + 2r$ and $\alpha - 2\beta - \frac{r}{2}$, then $m$ is equal to:

If $a, b, c$ are real numbers such that $a+b+c=0$ and $a^2+b^2+c^2=1$,then $(3a+5b-8c)^2+(-8a+3b+5c)^2+(5a-8b+3c)^2$ is equal to

The probability of selecting integers $a \in [-5, 30]$ such that $x^{2}+2(a+4)x-5a+64 > 0$ for all $x \in \mathbb{R}$ is: