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

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a

4

b

5

c

6

d

7

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detailed solution

Correct option is B

as a>2 hence a2>2a>a>2 now (x−a)(x−2a)x−a2<0⇒ the solution set is as shown between (0,a) there are (a−1) positive integers and between (2a,a2), there are a2 -1-2a positive integers∴ a2−1-2a+a−1=18⇒a2−a−20=0⇒(a−5)(a+4)=0∴ a=5

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The solution of $x - 1 = (x - [x]) (x - {x})$ (where $[x]$ and ${x}$ are the integral and fractional part of $x$ ) is :

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