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Domain of cos⁻¹ with Greatest Integer: 12(α+β) = 6 | JEE 2026

JEE Maths question with a full step-by-step solution.

Question
Let [⋅][\cdot] denote the greatest integer function. If the domain of f(x)=cos⁡−1(4x+2[x]3)f(x)=\cos^{-1}\left(\dfrac{4x+2[x]}{3}\right) is [α,β][\alpha,\beta], then 12(α+β)12(\alpha+\beta) is equal to
A66correct
B88
C99
D44
Solution
Step 1: For cos⁡−1\cos^{-1}, require −1≤4x+2[x]3≤1-1\le\dfrac{4x+2[x]}{3}\le1, i.e.
−3−4x≤2[x]≤3−4x.-3-4x\le 2[x]\le 3-4x.
 Step 2: Analysing the step function 2[x]2[x] against the two lines gives the domain
x∈[−14, 34].x\in\left[-\frac14,\ \frac34\right].
 Step 3: So α=−14, β=34\alpha=-\dfrac14,\ \beta=\dfrac34, and
12(α+β)=12(−14+34)=12⋅12=6.12(\alpha+\beta)=12\left(-\frac14+\frac34\right)=12\cdot\frac12=6.
 Correct answer: (1)
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