First slide

Theory of equations

Question

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

Moderate

A

$\frac{1}{3} \leq k \leq 3$
B

$k \geq 5$
C

$k \leq 0$
D

None of these

Solution

From $k = \frac{x^{2} - x + 1}{x^{2} + x + 1}$
We have $x^{2} (k - 1) + x (k + 1) + k - 1 = 0$
As given, x is real $\Rightarrow {(k + 1)}^{2} - 4 {(k - 1)}^{2} \geq 0$
$\Rightarrow 3 k^{2} - 10 k + 3 \leq 0$
Which is possible only when the value of k lies between the roots of the equation $3 k^{2} - 10 k + 3 = 0$
That is, when $\frac{1}{3} \leq k \leq 3$ { Since roots are $\frac{1}{3} and 3}$

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