JEE Main 2026 — Jan 22, Shift 2 — Mathematics solution

JEE Main2026Jan 22, Shift 2Mathematics

Let the domain of the function $f (x) = lo g_{3} lo g_{5} (7 - lo g_{2} (x^{2} - 10 x + 85)) + sin^{- 1} (\frac{3 x - 7}{17 - x})$ be $(α, β]$ . Then $α + β$ is equal to :

A
10
B
12
C
9
D
8

Solution

Set $μ = x^{2} - 10 x + 85$ . The chained logs require $μ > 0$ , $7 - lo g_{2} μ > 0$ (i.e., $μ < 2^{7}$ ), and $lo g_{5} (7 - lo g_{2} μ) > 0$ (i.e., $μ < 2^{6}$ ).

So $0 < μ < 64$ , i.e., $x^{2} - 10 x + 85 < 64 \Rightarrow x^{2} - 10 x + 21 < 0 \Rightarrow x \in (3, 7)$ .

For $sin^{- 1}$ : $\frac{3 x - 7}{17 - x} \leq 1 \Rightarrow - (17 - x) \leq 3 x - 7 \leq 17 - x$ , giving $x \in [- 5, 6]$ .

Intersection: $x \in (3, 6]$ , so $α = 3$ , $β = 6$ , $α + β = 9$ .

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