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JEE Main 2026 · 4 Apr · Shift 1, Mathematics question 1

Let [⋅][\cdot] denote the greatest integer function. If the domain of the function 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:

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Answer and worked solution

Official answer: A

Condition. −3≤4x+2[x]≤3-3\le4x+2[x]\le3. Check each integer part:

  • [x]=0[x]=0, x∈[0,1)x\in[0,1): 4x≤34x\le3, so x∈[0,34]x\in[0,\frac34].
  • [x]=−1[x]=-1, x∈[−1,0)x\in[-1,0): 4x−2≥−34x-2\ge-3, so x∈[−14,0)x\in[-\frac14,0).
  • [x]=1[x]=1 needs 4x≤14x\le1, and [x]=−2[x]=-2 needs 4x≥14x\ge1: both impossible.

Domain [−14,34]\left[-\frac14,\frac34\right], so 12(α+β)=12⋅12=612(\alpha+\beta)=12\cdot\frac12=6.

(A)

Topics:Domain and rangeInverse trigonometric functions
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