For any real value of $x$,if $\frac{11 x^2+12 x+6}{x^2+4 x+2} \notin(a, b]$,then the value of $x$ for which $\frac{11 x^2+12 x+6}{x^2+4 x+2}=b-a+3$ is

If $\tan A$ and $\tan B$ are the roots of the quadratic equation $3x^2 - 10x - 25 = 0$,then the value of $3 \sin^2 (A + B) - 10 \sin (A + B) \cos (A + B) - 25 \cos^2 (A + B)$ is

Let $x, y, z$ be non-zero real numbers such that $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=7$ and $\frac{y}{x}+\frac{z}{y}+\frac{x}{z}=9$. Then,the value of $\frac{x^3}{y^3}+\frac{y^3}{z^3}+\frac{z^3}{x^3}-3$ is equal to

Let $p, q, r \in R^+$. If $27pqr \geq (p + q + r)^3$ and $3p + 4q + 5r = 12$,then find the value of $p^3 + q^4 + r^5$.

If $x$ is real,then the maximum and minimum values of the expression $\frac{x^2 - 3x + 4}{x^2 + 3x + 4}$ are

$A$ student,while solving a quadratic equation in $x$,copied its constant term incorrectly and obtained the roots as $5$ and $9$. Another student copied the constant term and the coefficient of $x^2$ of the same equation correctly as $12$ and $4$ respectively. If $s$,$p$,and $\Delta$ denote the sum of the roots,the product of the roots,and the discriminant of the correct equation respectively,then find the value of $\frac{\Delta}{3p+s}$.