First slide

Theory of expressions

Question

The values of ‘a’ for which $(a^{2} - 1) x^{2} + 2 (a - 1) x + 2$ is positive for any $x,$ are

Moderate

A

$a \geq 1$
B

$a \leq 1$
C

$a > - 3$
D

$a \leq - 3$ or $a \geq 1$

Solution

We know that the expression $a x^{2} + b x + c > 0$ for
all $x,$ if $a > 0$ and $b^{2} < 4 a c$
$∴ (a^{2} - 1) x^{2} + 2 (a - 1) x + 2 \geq 0$ for all $x$ .
$\Rightarrow a^{2} - 1 \geq 0$ and $4 (a - 1)^{2} - 8 (a^{2} - 1) \leq 0$
$\Rightarrow a^{2} - 1 \geq 0$ and $- 4 (a - 1) (a + 3) \leq 0$
$\Rightarrow a^{2} - 1 \geq 0$ and $(a - 1) (a + 3) \geq 0$
$\Rightarrow a \leq - 1$ or, $a \geq 1$ and $a \leq - 3$ or, $a \geq 1$
$\Rightarrow a \leq - 3$ or, $a \geq 1$

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