Sets, Relations & Functions
Domain of Function — Combined Logarithm and Inverse Trig
nta_pyq_2024_jan
Grade 11
Question:
If the domain of the function $f(x)=\log_e\left(\dfrac{2x+3}{4x^2+x-3}\right)+\cos^{-1}\left(\dfrac{2x-1}{x+2}\right)$ is $(\alpha,\beta]$, then the value of $5\beta-4\alpha$ is equal to
Step-by-Step Solution
Key Concept: For the log: $\frac{2x+3}{(4x-3)(x+1)}>0$. For $\cos^{-1}$: $-1\leq\frac{2x-1}{x+2}\leq1$. Find the intersection of these domains.
Domain $=(3/4,3]$. $\alpha=3/4,\beta=3$. $5\beta-4\alpha=15-3=12$.
Correct Answer: 2