Sets, Relations & Functions
Domain of Logarithmic Functions
nta_pyq_2025_apr
Grade 11
Question:
If the domain of the function $\log_5(18x - x^2 - 77)$ is $(\alpha, \beta)$ and the domain of the function $\log_{(x-1)}\!\left(\frac{2x^2+3x-2}{x^2-3x-4}\right)$ is $(\gamma, \delta)$, then $\alpha^2 + \beta^2 + \gamma^2$ is equal to:
Step-by-Step Solution
Key Concept: For first function: solve $18x-x^2-77>0$. For second: base $x-1>0$, $x-1\neq1$, and argument $>0$.
$18x-x^2-77>0 \Rightarrow x^2-18x+77<0 \Rightarrow x\in(7,11)$, so $\alpha=7,\beta=11$. For second: $x>1$, $x\neq2$, $\frac{(2x-1)(x+2)}{(x-4)(x+1)}>0$ with $x>1$ gives $x\in(4,\infty)$, $\gamma=4$. $\alpha^2+\beta^2+\gamma^2 = 49+121+16 = 186$.
Correct Answer: 186