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If $a = \sin \frac{π}{18} . \sin \frac{5 π}{18} . \sin \frac{7 π}{18}$ and x is the solution of the equation $y = 2 [x] + 2$ and $y = 3 [x - 2]$ , where $[x]$ denotes the integral part of x, then a is equal to

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$[x]$

$2 [x]$

$\frac{1}{[x]}$

${[x]}^{2}$

answer is D.

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Detailed Solution

$a = \sin 10 ° \sin 50 ° . \sin 70 °$

$= \frac{1}{4} \sin 3 (10 °) = \frac{1}{4} \sin 30 ° = \frac{1}{4} . \frac{1}{2} = \frac{1}{8}$

$a = \frac{1}{8} \Rightarrow \frac{1}{a} = 8 \dots \dots (1)$

$2 [x] + 2 = 3 [x - 2]$

$2 [x] + 2 = 3 [x] + 3 (- 2)$

$2 [x] + 2 = 3 [x] - 6$

$[x] = 8$

$a = \frac{1}{[x]} (∵ from (1))$

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