JEE Mathematics 2026 Question with Solution

Given: $f(x)=\log_3\!\left[\log_5(7-\log_2(x^2-10x+85))\right] +\sin^{-1}\!\left(\frac{3x-7}{17-x}\right)$

Find: The value of $\alpha+\beta$ if the domain is $(\alpha,\beta)$.

For the domain, intersect the valid intervals obtained from the logarithmic part and the inverse sine part.

Step 1: Domain from logarithmic terms

$$\log_5(7-\log_2(x^2-10x+85))>0$$ $$\Rightarrow 7-\log_2(x^2-10x+85)>1$$ $$\log_2(x^2-10x+85)<6$$ $$\Rightarrow x^2-10x+85<64$$ $$\Rightarrow x^2-10x+21<0$$ $$\Rightarrow 3$$

Given: The function is the sum of a logarithmic expression and an inverse sine expression.

Find: The interval $(\alpha,\beta)$ and then $\alpha+\beta$.

The domain of the whole function is the intersection of the domains of both parts.

For $\log_3\!\left[\log_5(7-\log_2(x^2-10x+85))\right]$, the inner quantity of $\log_3$ must be positive. The extracted working gives:

$$\log_5(7-\log_2(x^2-10x+85))>0$$ $$\Rightarrow 7-\log_2(x^2-10x+85)>1$$ $$\Rightarrow \log_2(x^2-10x+85)<6$$ $$\Rightarrow x^2-10x+85<64$$ $$\Rightarrow x^2-10x+21<0$$ $$\Rightarrow 3$$