Question

For $x\in R$ the expression $\frac{x^2+2x+1}{x^2+2x+7}$ lies in the interval

• A. $[0,-1]$
• B. $(-\infty ,0]\cup [1,\infty )$
• C. $[0,1)$
• D. None of these

Answer 1

The correct option is C $[0,1)$

Let $\frac{x^2+2x+1}{x^2+2x+7}=\lambda$
$\Rightarrow (1-\lambda )x^2+(2-2\lambda )x+(1-7\lambda )=0$
For real $x$, $D\ge 0$
$(2-2\lambda {)}^2-4(1-\lambda \left)\right(1-7\lambda )\ge 0$
$\Rightarrow -24{\lambda }^2+24\lambda \ge 0$
$\Rightarrow \lambda (\lambda -1)\le 0$
$\Rightarrow \lambda \in [0,1]$
Hence, the given expression lies in $[0,1]$.