First slide

Theory of expressions

Question

The values of $' a'$ for which the expression $x^{2} - (3 a - 1) x + 2 a^{2} + 2 a - 11$ is always positive are given

Moderate

A

$5 < a < 9$
B

$6 < a < 8$
C

$a < 5$
D

$a > 9$

Solution

The given expression is $x^{2} - (3 a - 1) x + 2 a^{2} + 2 a - 11$ …. ( 1 )
From Eq.( 1 ): the coefficient of $x^{2} > 0.$
Hence, the given expression will assume positive values only if $Δ < 0$
From Eq.( 1 ): $a = 1, b = - (3 a - 1), c = 2 a^{2} + 2 a - 11$
$b^{2} - 4 a c < 0$
$\Rightarrow {- {(3 a - 1)}^{2}} - 4 \times 1 \times (2 a^{2} + 2 a - 11) < 0$
$\Rightarrow 9 a^{2} - 6 a + 1 - 8 a^{2} - 8 a + 44 < 0$
$\Rightarrow a^{2} - 14 a + 45 < 0$
$\Rightarrow (a - 5) (a - 9) < 0$

$\Rightarrow 5 < a < 9.$

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