First slide

Introduction to sets

Question

If set $A = \{x ∣ \frac{x^{2} (5 - x) (1 - 2 x)}{(5 x + 1) (x + 2)} < 0\}$ and set $B = \{x ∣ \frac{3 x + 1}{6 x^{3} + x^{2} - x} > 0\}$ , then $A \cap B$ does not contain

Moderate

A

(1, 4)
B

(5, 11)
C

$(- \frac{3}{2}, \frac{- 1}{2})$
D

none of these

Solution

We have $\frac{x^{2} (x - 5) (2 x - 1)}{(5 x + 1) (x + 2)} < 0$
From the sign scheme, solution is
$(- 2, - \frac{1}{5}) \cup (\frac{1}{2}, 5)$ (i)
Now $\frac{3 x + 1}{x (6 x^{2} + x - 1)} > 0$
$\Rightarrow \frac{3 x + 1}{x (3 x - 1) (2 x + 1)} > 0$
From the sign scheme solution is
$(- \infty, - \frac{1}{2}) \cup (- \frac{1}{3}, 0) \cup (\frac{1}{3}, \infty)$ (ii)
Clearly, from (i) and (ii), option(s) (1) and (3) are correct.

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