If the domain of the function $f(x)=\sin ^{-1}\left(\frac{x-1}{2 x+3}\right)$ is $R-(\alpha, \beta)$ then $12 \alpha \beta$ is equal to :

JEE Main · 2024 · 09 Apr 2024 · Shift 1 · Inverse Trigonometric Functions

If the domain of the function

f (x) = sin^{- 1} (\frac{x - 1}{2 x + 3})

R - (α, β)

then

12 α β

is equal to :

Answer: (D)

Need

\frac{x - 1}{2 x + 3} \leq 1

with

2 x + 3 \neq = 0

. Squaring:

(2 x + 3)^{2} - (x - 1)^{2} \geq 0 \Rightarrow (x + 4) (3 x + 2) \geq 0

, so

x \leq - 4

x \geq - \frac{2}{3}

. The domain is

R - (- 4, - \frac{2}{3})

, giving

α = - 4, β = - \frac{2}{3}

and

12 α β = 12 (- 4) (- \frac{2}{3}) = 32

— option (4).

JEE Main · 2024 · 09 Apr 2024 · Shift 1 · Single correct option

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