The minimum value of $\frac{9 \cdot 3^{2x} + 6 \cdot 3^x + 4}{9 \cdot 3^{2x} - 6 \cdot 3^x + 4}$ is

Suppose,$\alpha$ is the minimum value of $x^2+bx+5$ and $\beta$ is the maximum value of $-x^2+ax+5$. If $[\alpha, \beta]$ is the interval of maximum length for $x$ in which $x^2-10x+24 \leq 0$,then $a^2b^2$ is equal to

Let $f(x)=a x^2+b x+c$ and the $GCD$ of $a, b, c$ be $1$. If $\frac{-7+\sqrt{11} i}{6}$ is a root of $f(x)=0$ and $f\left(\frac{x}{k}\right)-L=(x+4)(3 x-5)$,then $k$ and $L$ are respectively:

If $\left|\frac{x^2+kx+1}{x^2+x+1}\right| < 3$ for all real $x$,then $k$ is in the interval

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

If $\alpha$ and $\beta$ are two distinct negative roots of $x^5-5x^3+5x^2-1=0$,then the equation of least degree with integer coefficients having $\sqrt{-\alpha}$ and $\sqrt{-\beta}$ as its roots is