Question

Let $x^{2} - 10x + 85 = \lambda$

∴ Domain for first term

$$\begin{array}{r} \lambda > 0\#(1) \end{array}$$

$$\begin{aligned} & 7 - \log_{2}\lambda > 0 \Rightarrow \lambda < 2^{7}\#(2) \end{aligned}$$

$$\begin{aligned} & \log_{5}\left( 7 - \log_{2}\lambda \right) > 0 \Rightarrow \lambda < 2^{6}\#(3) \end{aligned}$$

∴ from (1), (2) & (3)

$${0 < \lambda < 2^{6} }{0 < x^{2} - 10x + 85 < 64}$$

$$\begin{array}{r} \Rightarrow x \in (3,7)\#(A) \end{array}$$

& domain for second term $- 1 \leq \frac{3x - 7}{x - 17} \leq 1$

$$\begin{array}{r} \Rightarrow x \in \lbrack - 5,6\rbrack\#(B) \end{array}$$

From (A) & (B), domain of function will be ( 3,6 ]

$${\Rightarrow \alpha = 3,\beta = 6 }{\Rightarrow \alpha + \beta = 9}$$