After the roots of the equation $6x^3 + 7x^2 - 4x - 2 = 0$ are diminished by $h$,if the transformed equation does not contain the $x^2$ term,then the product of all the possible values of $h$ is

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:

If $p$ and $q$ are positive real numbers such that $p^2 + q^2 = 1$,what is the maximum value of $(p + q)$?

Let the transformed equation of $2x^4-8x^3+3x^2-1=0$ such that the term containing the cubic power of $x$ is absent be $2x^4+bx^2+cx+d=0$. Then $b=$

Let $m$ and $n$ be the numbers of real roots of the quadratic equations $x^2-12x+[x]+31=0$ and $x^2-5|x+2|-4=0$ respectively,where $[x]$ denotes the greatest integer $\leq x$. Then $m^2+mn+n^2$ is equal to $..............$.

The number of integers $a$ in the interval $[1, 2014]$ for which the system of equations $x+y=a$ and $\frac{x^2}{x-1}+\frac{y^2}{y-1}=4$ has finitely many solutions is