First slide

Theory of expressions

Question

If x is real, then the value of $\frac{x^{2} + 34 x - 71}{x^{2} + 2 x - 7}$ does not lie between

Moderate

A

-9 and -5
B

-5 and 9
C

0 and 9
D

5 and 9

Solution

Let $y = \frac{x^{2} + 34 x - 71}{x^{2} + 2 x - 7}$
$\Rightarrow x^{2} (y - 1) + 2 (y - 17) x + (71 - 7 y) = 0$
For real values of x, its discriminant $D ⩾ 0$
$\Rightarrow 4 {(y - 17)}^{2} - 4 (y - 1) (71 - 7 y) ⩾ 0$
$\Rightarrow (y^{2} - 3 + y + 289) - (71 y - 7 y^{2} - 71 + 7 y) ⩾ 0$
$\Rightarrow y^{2} - 14 y + 45 ⩾ 0 \Rightarrow (y - 5) (y - 9) ⩾ 0$
It is possible when both $y - 5$ and $y - 9$ are negative or both positive. Let $y - 5 \leq 0 \Rightarrow y \leq 5$ and $y - 9 ⩽ 0 \Rightarrow y \leq 9 .$
Hence $y \leq 5 . . . (i)$
If $y - 5 \geq 0 \Rightarrow y ⩾ 5$ and $y - 9 \geq 0 \Rightarrow y \geq 9 .$
Hence $y \geq 9 .$
Therefore y does not lie between 5 and 9.

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