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JEE Advanced
Exam
JEE Advanced
Subject
Mathematics
Chapter
Trigonometry

JEE Advanced · Mathematics · Trigonometry

The number of integral values of a ; a ∈(6,100) for which the...

The number of integral values of a;a∈(6,100)a ; a \in(6,100) for which the equation [tan⁡x]2+tan⁡x−a=0[\tan x]^2+\tan x-a=0 has real roots; where [.] denotes greatest integer function is __________ .

Correct answer

7

Step-by-step solution

[tan⁡x]2+tan⁡x−a=0[\tan \mathrm{x}]^2+\tan \mathrm{x}-\mathrm{a}=0 ∵[tan⁡x]2\because[\tan \mathrm{x}]^2 and a are integral in nature ⇒tan⁡x\Rightarrow \tan \mathrm{x} is integer. ⇒\Rightarrow equation reduces to tan⁡2x+tan⁡x−a=0\tan ^2 \mathrm{x}+\tan \mathrm{x}-\mathrm{a}=0 ⇒tan⁡x(tan⁡x+1)=a\Rightarrow \tan \mathrm{x}(\tan \mathrm{x}+1)=\mathrm{a} ∴\therefore a is product of consecutive integer Here a=12,20,30,42,56,72,90\mathrm{a}=12,20,30,42,56,72,90 ⇒\Rightarrow a has 7 values.
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