Functions Question 10
Question 10 - 2024 (30 Jan Shift 2)
If the domain of the function $f(x)=\log _e\left(\frac{2 x+3}{4 x^{2}+x-3}\right)+\cos ^{-1}\left(\frac{2 x-1}{x+2}\right)$ is $(\alpha, \beta]$, then the value of $5 \beta-4 \alpha$ is equal to
(1) 10
(2) 12
(3) 11
(4) 9
Show Answer
Answer (2)
Solution
$\frac{2 x+3}{4 x^{2}+x-3}>0$ and $-1 \leq \frac{2 x-1}{x+2} \leq 1$
$\frac{2 \times+3}{(4 x-3)(x+1)}>0 \quad \frac{3 x+1}{x+2} \geq 0 \& \frac{x-3}{x+2} \leq 0$
$(-\infty / 2+-1$
$(-2,3] \ldots\left[\frac{-1}{3}, \infty\right)$
$\left[\frac{-1}{3}, 3\right]$…(3)
$(1) \cap(2) \cap(3)$
$\left(\frac{3}{4}, 3\right]$
$\alpha=\frac{3}{4} \beta=3$
$5 \beta-4 \alpha=15-3=12$
