If $\alpha$ and $\beta$ are the roots of the quadratic equation $x^2 + x \sin \theta - 2 \sin \theta = 0$,where $\theta \in (0, \frac{\pi}{2})$,then the value of $\frac{\alpha^{12} + \beta^{12}}{(\alpha^{-12} + \beta^{-12})(\alpha - \beta)^{24}}$ is equal to

Let $2 \sin^2 x + 3 \sin x - 2 > 0$ and $x^2 - x - 2 < 0$ ($x$ is measured in radians). Then $x$ lies in the interval

$A$ 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 $\frac{\Delta}{3p+s}$.

If $x$ is real, then the value of $\frac{x^2-3x+4}{x^2+3x+4}$ lies in the interval

For any real value of $x$,if $\frac{11 x^2+12 x+6}{x^2+4 x+2} \notin(a, b]$,then the value of $x$ for which $\frac{11 x^2+12 x+6}{x^2+4 x+2}=b-a+3$ is

Let $f(x)$ be a quadratic polynomial with leading coefficient $1$ such that $f(0)=p, p \neq 0$ and $f(1)=\frac{1}{3}$. If the equations $f(x)=0$ and $f(f(f(f(x))))=0$ have a common real root,then $f(-3)$ is equal to $........$