Question

Let the domain of the function \[ f(x)=\log_3\!\left[\log_5(7-\log_2(x^2-10x+85))\right] +\sin^{-1}\!\left(\frac{3x-7}{17-x}\right) \] be \((\alpha,\beta)\). Then \(\alpha+\beta\) is equal to

• A. 9
• B. 12
• C. 8
• D. 10

Hint:

For domain problems, always intersect the domains obtained from each component of the function.

Answer 1

The Correct Option is D

Step 1: Domain from logarithmic terms.
\[ \log_5(7-\log_2(x^2-10x+85))>0 \Rightarrow 7-\log_2(x^2-10x+85)>1 \] \[ \log_2(x^2-10x+85)<6 \Rightarrow x^2-10x+85<64 \] \[ x^2-10x+21<0 \Rightarrow 3<x<7 \]
Step 2: Domain from inverse sine.
\[ -1 \le \frac{3x-7}{17-x} \le 1 \] Solving gives: \[ 3<x<7 \]
Step 3: Combine domains.
\[ (\alpha,\beta) = (3,7) \Rightarrow \alpha+\beta = 10 \]
Final Answer: \[ \boxed{10} \]