Let the domain of the function $\mathrm{f}(\mathrm{x})=\log _{3} \log _{5}\left(7-\log _{2}\left(\mathrm{x}^{2}-10 \mathrm{x}+85\right)\right)+\sin ^{-1}\left(\left|\frac{3 \mathrm{x}-7}{17-\mathrm{x}}\right|\right)$ be $(\alpha, \beta]$. Then $\alpha+\beta$ is equal to :

JEE Main · 2026 · 22 Jan 2026 · Shift 2 · Sets Relations and Functions

Let the domain of the function

f (x) = lo g_{3} lo g_{5} (7 - lo g_{2} (x^{2} - 10 x + 85)) + sin^{- 1} (\frac{3 x - 7}{17 - x})

(α, β]

. Then

α + β

is equal to :

Answer: (C)

For the logarithmic part, with

λ = x^{2} - 10 x + 85 = (x - 5)^{2} + 60 > 0

, the nesting needs

lo g_{5} (7 - lo g_{2} λ) > 0

, i.e.

7 - lo g_{2} λ > 1

, so

λ < 64

. This gives

x^{2} - 10 x + 21 < 0

, i.e.

x \in (3, 7)

. For

sin^{- 1} \frac{3 x - 7}{17 - x}

one needs

\frac{3 x - 7}{17 - x} \leq 1

; on

(3, 7)

this reduces to

3 x - 7 \leq 17 - x

, i.e.

x \leq 6

. The intersection is

(3, 6]

, so

α = 3, β = 6

and

α + β = 9

JEE Main · 2026 · 22 Jan 2026 · Shift 2 · Single correct option