If x is real and K= x^2-x+1 x^2+x+1 , then - Vidyadip

MCQ

If $x$ is real and $\text{K}=\frac{\text{x}^2-\text{x}+1}{\text{x}^2+\text{x}+1},$ then

✓
$\text{K}\in\Big[\frac{1}{3,3}\Big]$
B
$K > 3$
C
$\text{K}<\frac{1}{3}$
D
None of these.

✓

Answer

Correct option: A.

$\text{K}\in\Big[\frac{1}{3,3}\Big]$

$\text{K}=\frac{\text{x}^2-\text{x}+1}{\text{x}^2+\text{x}+1}$
$\Rightarrow kx^2+ kx + K = x^2 - x + 1$
$\Rightarrow (k - 1)x^2 + (k + 1) x + k - 1 = 0$
For real values of $x,$ the discriminant of $(k - 1)x^2 + (k + 1) x + k - 1 = 0$ should be greater than or equal to zero.
$\therefore$ If $\text{k}\neq1$
$(k + 1)^2 - 4(k - 1) (k - 1) > 0$
$\Rightarrow(\text{k}+1)^2-\big\{2(\text{k}-1)\big\}^2>0$
$\Rightarrow (3k - 1) (k - 3) < 0$
$\Rightarrow\frac{1}{3}<\text{K}<3$
And if $k = 1,$ then,
$x = 0,$ which is real $...(ii)$
So, from $(i)$ and $(ii)$, we get,
$\text{k}\in\Big[\frac{1}{3},3\Big]$

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