Question

For real values of $x,$ the value of the expression $\frac{11 x^{2} - 12 x - 6}{x^{2} + 4 x + 2}$

• A. lies between $-17$ and $-3$
• B. does not lie between $-17$ and $-3$
• C. lies between $3$ and $17$
• D. does not lie between $3$ and $17$

Answer 1

Answer: (B)

Let $y=\frac{11 x^{2} - 12 x - 6}{x^{2} + 4 x + 2}$
$yx^{2}+4yx+2y=11x^{2}-12x-6$
$\left(y - 11\right)x^{2}+\left(4 y + 12\right)x+\left(2 y + 6\right)=0$
$x\in R\Rightarrow D\geq 0$
$\Rightarrow \left(4 \left(y + 3\right)\right)^{2}-4\left(y - 11\right)2\left(y + 3\right)\geq 0$
$\Rightarrow 8\left(y + 3\right)\left\{2 \left(y + 3\right) - \left(y - 11\right)\right\}\geq 0$
$\Rightarrow 8\left(y + 3\right)\left(y + 17\right)\geq 0$
$\Rightarrow y\leq -17$ or $y\geq -3$
$\Rightarrow y$ does not lie between $-17$ and $-3$