First slide

Introduction to linear inequalities

Question

$Let a > 2 be a constant. If there are just 18 positive integers satisfying the inequality (x - a) (x - 2 a) (x - a^{2}) < 0 then the value of a is$

Moderate

A

4
B

5
C

6
D

7

Solution

$\begin{aligned} as a > 2 hence \\ a^{2} > 2 a > a > 2 \end{aligned}$
$now (x - a) (x - 2 a) (x - a^{2}) < 0 \Rightarrow the solution set is as shown$

$\begin{array}{l} between (0, a) there are (a - 1) {positive integers and between (2a,a}^{2} {), there are a}^{2} -1-2a positive integers \\ ∴ a^{2} - 1 - 2 a + a - 1 = 18 \\ \Rightarrow a^{2} - a - 20 = 0 \\ \Rightarrow (a - 5) (a + 4) = 0 \\ ∴ a = 5 \end{array}$

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