First slide

Introduction to sets

Question

$If (x - 3 a) (x - a - 3) < 0 \forall x \in [1, 3] then exhaustive set of values of a is$

Moderate

A

$(0, \frac{1}{3})$
B

$(- 2, 3)$
C

$(\frac{1}{3}, 3)$
D

$(- 2, 0)$

Solution

$\begin{array}{l} (x - 3 a) (x - a - 3) < 0 \forall x \in [1, 3] \\ \Rightarrow (1 - 3 a) (1 - a - 3) < 0 \end{array}$
$\Rightarrow (a - \frac{1}{3}) (a + 2) < 0$
$\Rightarrow - 2 < a < \frac{1}{3} - - - - (1)$
$also, (3 - 3 a) (3 - a - 3) < 0$
$\Rightarrow a (a - 1) < 0$
$\Rightarrow 0 < a < 1 - - - - (2)$
$Finally, 0 < a < \frac{1}{3} (From (i) and (ii) common values)$

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