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Domain of a function with greatest integer terms

This type of question, asked 5 times: 2021 Paper 9, 2021 Paper 11, 11 Apr 2023, 4 Apr 2026 S1, 6 Apr 2026 S1. All 5 questions below, oldest first.

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  1. Question 1

    2021 Paper 9

    74
    The real valued function f(x)=csc⁡−1xx−[x]f(x) =\frac{\csc^{- 1}x}{\sqrt{x - \lbrack x\rbrack}}, where [x] denotes the greatest integer less than or equal to xx, is defined for all xx belonging to:

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  2. Question 2

    2021 Paper 11

    50
    Let [x]\lbrack x\rbrack denote the greatest integer ≤x\leq x, where x∈Rx\in R. If the domain of the real valued function f(x)=∣[x]∣−2∣[x]∣−3f\left( x \right)=\sqrt{\frac{\left| \left\lbrack x \right\rbrack \right|- 2}{\left| \left\lbrack x \right\rbrack \right|- 3}} is (−∞,a)∪[b,c)∪[4,∞),a<b<c( - \infty,a) \cup \lbrack b,c) \cup \lbrack 4,\infty),a<b<c, then the value of a+b+ca+b+c is:

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  3. Question 3

    11 Apr 2023

    159
    The domain of the function f(x)=1[x]2−3[x]−10f(x) =\frac{1}{\sqrt{\lbrack x\rbrack^{2}- 3\lbrack x\rbrack - 10}} is (where [x] denotes the greatest integer less than or equal to xx )

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  4. Question 4

    4 Apr 2026 S1

    51
    Let [ • ] denote the greatest integer function. If the domain of the function f(x)=cos⁡−1(4x+2[x]3)f(x) =\cos^{- 1}\left( \frac{4x+ 2\lbrack x\rbrack}{3} \right) is [α,β]\lbrack\alpha,\beta\rbrack, then 12(α+β)12(\alpha+\beta) is equal to :

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  5. Question 5

    6 Apr 2026 S1

    51
    Let [.] denote the greatest integer function. If the domain of the function f(x)=sin⁡−1(x+[x]3)f(x) =\sin^{- 1}\left( \frac{x+ \lbrack x\rbrack}{3} \right) is [α,β)\lbrack\alpha,\beta), then α2+β2\alpha^{2}+\beta^{2} is equal to:

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