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

The number of ordered pairs $(x, y)$ of real numbers that satisfy the simultaneous equations $x+y^2=x^2+y=12$ is

The sum of all the rational roots of the equation $6x^6-25x^5+31x^4-31x^2+25x-6=0$ is

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 both roots of the quadratic equation $x^2 + (\sin \theta + \cos \theta)x + \frac{3}{8} = 0$ are positive and distinct,then the complete set of values of $\theta$ in $[0, 2\pi]$ is:

Let $f(x)=(x-a)(x-b)-\left(\frac{a+b}{2}\right)$. If $f(x)=0$ has both non-negative roots,then the minimum value of $f(x)$ is: