The number of integral values of a, $a \in (6, 100)$ for which the equation $[t a n x]^{2} + t a n x - a = 0$ has real roots (where [.] denotes greatest integer function) is:

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Solution

The correct option is A 7
$a ϵ (6, 100), [t a n x]^{2} + t a n x - a = 0 [t a n x]^{2} + t a n x = a, f o r a t o b e i n t e g r a l, t a n x s h o u l d b e i n t e g r a l . t a n x = 1 a n d 2, a = 2 a n d 6, r e j e c t e d . t a n x = 3, 4, 5, 6, 7, 8, 9; a = 12, 20, 30, 42, 56, 72, 90. S o, 7 v a l u e s$

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