Consider two numbers whose arithmetic mean is $9$ and geometric mean is $4$. These numbers are the roots of which equation?

The number of real values of $m$ such that the equation $x^2+(2m+1)x+m=0$ has equal roots is

If $a, b, c$ are real numbers, then both the roots of the equation $(x-b)(x-c)+(x-c)(x-a)+(x-a)(x-b)=0$ are always

Let $[x]$ be the greatest integer less than or equal to $x$ for a real number $x$. Then the equation $[x^2] = x + 1$ has:

If the roots of the given equation $(\cos p-1) x^2+(\cos p) x+\sin p=0$ are real,then

Let $\alpha, \beta$ be two roots of the quadratic equation $x^2+ax-b=0, b \neq 0$. If the straight line $x \cos \theta + y \sin \theta = c$ touches the curve $(\frac{x}{\alpha})^n + (\frac{y}{\beta})^n = 2$ at the point $(\alpha, \beta)$, then $(\frac{a}{b})^2 + \frac{2}{b} =$