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Domain combining a log and an inverse sine or cosine of rational expressions

This type of question, asked 6 times: 8 Apr 2023, 15 Apr 2023, 30 Jan 2024, 4 Apr 2024, 3 Apr 2025, 22 Jan 2026 S1. All 7 questions below, oldest first.

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  1. Question 1

    8 Apr 2023

    174
    If domain of the function log⁡e(6x2+5x+12x−1)+cos⁡−1(2x2−3x+43x−5)\log_{e}\left( \frac{6x^{2}+ 5x + 1}{2x - 1} \right)+\cos^{- 1}\left( \frac{2x^{2}- 3x + 4}{3x - 5} \right) is (α,β)∪(γ,δ](\alpha,\beta) \cup (\gamma,\delta\rbrack, then 18(α2+β2+γ2+δ2)18\left( \alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2} \right) is equal to

  2. Question 2

    15 Apr 2023

    80
    If the domain of the function
    f(x)=log⁡e(4x2+11x+6)+sin⁡−1f(x) =\log_{e}\left( 4x^{2}+ 11x + 6 \right)+\sin^{- 1}
    (4x+3)+cos⁡−1(10x+63)(4x + 3) +\cos^{- 1}\left( \frac{10x + 6}{3} \right) is (α,β](\alpha,\beta\rbrack, Then 36∣α+β∣36|\alpha + \beta| is equal to :

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  3. Question 3

    30 Jan 2024

    168
    If the domain of the function f(x)=log⁡ef(x) =\log_{e} (2x+34x2+x−3)+cos⁡−1(2x−1x+2)\left( \frac{2x + 3}{4x^{2}+ x - 3} \right)+\cos^{- 1}\left( \frac{2x - 1}{x + 2} \right) is  (α,β]\ (\alpha,\beta\rbrack, then the value of 5β−4α5\beta - 4\alpha is equal to

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  4. Question 4

    30 Jan 2024

    72
    If the domain of the function f(x)=cos⁡−1(2−∣x∣4)+(log⁡e(3−x))−1f(x)=\cos^{-1}\left(\frac{2-|x|}{4}\right)+\left(\log_{e}(3-x)\right)^{-1} is [−α,β)−{γ}[-\alpha,\beta)-\{\gamma\}, then α+β+γ\alpha+\beta+\gamma is equal to:

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  5. Question 5

    4 Apr 2024

    78
    If the domain of the function sin⁡−1(3x−222x−19)+log⁡e(3x2−8x+5x2−3x−10)\sin^{- 1}\left( \frac{3x - 22}{2x - 19} \right)+\log_{e}\left( \frac{3x^{2}- 8x + 5}{x^{2}- 3x - 10} \right) is  (α,β]\ (\alpha,\beta\rbrack, then 3α+10β3\alpha + 10\beta is equal to :

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  6. Question 6

    3 Apr 2025

    58
    If the domain of the function f(x)=log⁡e(2x−35+4x)+sin⁡−1(4+3x2−x)f(x) =\log_{e}\left( \frac{2x- 3}{5 + 4x} \right)+\sin^{- 1}\left( \frac{4 + 3x}{2 -x} \right) is [α,β)\lbrack\alpha,\beta), then α2+4β\alpha^{2}+ 4\beta is equal to.

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  7. Question 7

    22 Jan 2026 S1

    64
    If the domain of the function f(x)=sin⁡−1(5−x3+2x)+1log⁡e(10−x)f(x) =\sin^{- 1}\left( \frac{5 - x}{3 + 2x} \right)+\frac{1}{\log_{e}(10 - x)} is (−∞,α]∪[β,γ)−{δ}\mathbf{( - \infty,\alpha\rbrack \cup \lbrack\beta,\gamma) - \{\delta\}}, then 6(α+β+γ+δ)6(\alpha + \beta + \gamma + \delta) is equal to

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