Sets, Relations & Functions
Domain of Composite Logarithmic and Inverse Trig Function
nta_pyq_2026_jan
Grade 11
Question:
Let the domain of the function $f(x)=\log_3\log_5\!\left(7-\log_2(x^2-10x+85)\right)+\sin^{-1}\!\left(\left|\dfrac{3x-7}{17-x}\right|\right)$ be $(\alpha,\beta]$. Then $\alpha+\beta$ is equal to:
Step-by-Step Solution
Key Concept: For $\log_3\log_5(\cdot)$: need $\log_5(\cdot)>0\Rightarrow7-\log_2(x^2-10x+85)>1\Rightarrow\log_2(x^2-10x+85)<6\Rightarrow x^2-10x+85<64\Rightarrow(x-3)(x-7)<0\Rightarrow3<x<7$.
Domain $(3,6]$. $\alpha+\beta=9$.
Correct Answer: 1