If $a^2+b^2+c^2=1$,where $a, b, c \in \mathbb{R}$,then the set of extreme values of $ab+bc+ca$ is

Let $p, q$ be real numbers. If $\alpha$ is a root of $x^{2}+3 p^{2} x+5 q^{2}=0$, $\beta$ is a root of $x^{2}+9 p^{2} x+15 q^{2}=0$ and $0 < \alpha < \beta$, then the equation $x^{2}+6 p^{2} x+10 q^{2}=0$ has a root $\gamma$ that always satisfies:

If $m$ is a root of the equation $(1 - ab)x^2 - (a^2 + b^2)x - (1 + ab) = 0$ and $m$ harmonic means are inserted between $a$ and $b$,then the difference between the last and the first of the means equals

Let $a, b, c$ and $d$ be any four real numbers. Then $a^{n} + b^{n} = c^{n} + d^{n}$ holds for any natural number $n$ if:

Let $a, b, c$ be the lengths of three sides of a triangle satisfying the condition $(a^2+b^2)x^2-2b(a+c)x+(b^2+c^2)=0$. If the set of all possible values of $x$ is the interval $(\alpha, \beta)$,then $12(\alpha^2+\beta^2)$ is equal to.

Let $\tan A$ and $\tan B$, where $A, B \in (-\frac{\pi}{2}, \frac{\pi}{2})$, be the roots of the quadratic equation $x^2 - 2x - 5 = 0$. Then $20 \sin^2(\frac{A+B}{2})$ is equal to: