First slide

Theory of expressions

Question

Set of all real values of a such that $f (x) = \frac{(2 a - 1) x^{2} + 2 (a + 1) x + (2 a - 1)}{x^{2} - 2 x + 40}$ is always negative is

Moderate

A

$(- \infty, 0)$
B

$(0, \infty)$
C

$(- \infty, 1 / 2)$
D

None

Solution

$f (x) = \frac{(2 a - 1) x^{2} + 2 (a + 1) x + (2 a - 1)}{x^{2} - 2 x + 40}$
$\begin{array}{l} Since x^{2} - 2 x + 40 > 0 for all real x, \\ (2 a - 1) x^{2} + 2 (a + 1) x + (2 a - 1) < 0 \forall x \in R \end{array}$
$\begin{array}{l} ∴ 2 a - 1 < 0 or a < \frac{1}{2} \\ and D < 0 (1) \\ \Rightarrow 4 (a + 1)^{2} - 4 (2 a - 1)^{2} < 0 \\ \Rightarrow a (a - 2) > 0 \\ \Rightarrow a < 0 or a > 2 (2) \end{array}$
From (1) and (2), a <0

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