The number of pairs of real numbers $(x, y)$ such that $x = x^2 + y^2$ and $y = 2xy$ is:

If $\alpha, \beta$ are the real roots of $x^2+p x+q=0$ and $\alpha^4, \beta^4$ are the roots of $x^2-r x+s=0$,then the equation $x^2-4 q x+2 q^2-r=0$ has always

If the equation whose roots are $p$ times the roots of the equation $x^4-2ax^3+4bx^2+8ax+16=0$ is a reciprocal equation,then $|p|=$ :

If $x$ is real,then the value of the expression $\frac{x^2 + 14x + 9}{x^2 + 2x + 3}$ lies between

Consider the quadratic equation $nx^2 + 7\sqrt{n}x + n = 0$ where $n$ is a positive integer. Which of the following statements are necessarily correct?
$I$. For any $n$,the roots are distinct.
$II$. There are infinitely many values of $n$ for which both roots are real.
$III$. The product of the roots is necessarily an integer.

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