The equation $\left(x^4+1\right)=\frac{1}{a}(x+1)^4$ is a reciprocal equation:

If the lengths of two sides of a triangle are the roots of the equation $x^2-2 \sqrt{3} x+2=0$ and the angle between these sides is $\frac{\pi}{3}$,then the perimeter of the triangle is

Given the equation $4x^2 + 4(a - 1)x + (1 - 2a) = 0$ has roots $\sin \theta$ and $\cos \theta$ $(0 < \theta < \frac{\pi}{2})$,then the maximum value of $(a + \sin \theta)$ is-

The maximum value of the expression $\frac{x^2+x+1}{2x^2-x+1}$,for $x \in R$,is

If $x=2+2^{\frac{2}{3}}+2^{\frac{1}{3}}$,then $x^3-6x^2+6x=$

$A$ student,while solving a quadratic equation in $x$,copied its constant term incorrectly and obtained the roots as $5$ and $9$. Another student copied the constant term and the coefficient of $x^2$ of the same equation correctly as $12$ and $4$ respectively. If $s$,$p$,and $\Delta$ denote the sum of the roots,the product of the roots,and the discriminant of the correct equation respectively,then find the value of $\frac{\Delta}{3p+s}$.