If $\alpha$ and $\beta$ are the roots of the equation $2^{6x} - 3(2^{3x+2}) + 32 = 0$ with $\beta < 1$,then $2\alpha + 3\beta =$

If a polynomial $x^4+x^2+1$ is divisible by $x^2+mx+1$ and $x^2+nx+1$,then $m+n$ is equal to:
$(1)$ $2$
$(2)$ $0$
$(3)$ $3$
$(4)$ $4$

The $x$-coordinates of the vertices of a square of unit area are the roots of the equation $x^2 - 3|x| + 2 = 0$ and the $y$-coordinates of the vertices are the roots of the equation $y^2 - 3y + 2 = 0$. Then the possible vertices of the square are:

The roots of the given equation $2(a^2 + b^2)x^2 + 2(a + b)x + 1 = 0$ are

The number of integral values of $m$ for which the equation $(1 + m^2) x^2 - 2(1 + 3m) x + (1 + 8m) = 0$ has no real root is

If $x = \sqrt{7} + \sqrt{3}$ and $xy = 4$,then $x^4 + y^4 =$