If the domain of the function $\sin^{-1}\left(\frac{3x-22}{2x-19}\right)+\log_{e}\left(\frac{3x^{2}-8x+5}{x^{2}-3x-10}\right)$ is $(\alpha, \beta]$, then $3\alpha+10\beta$ is equal to

JEE Main · 2024 · 04 Apr 2024 · Shift 1 · Inverse Trigonometric Functions

If the domain of the function

sin^{- 1} (\frac{3 x - 22}{2 x - 19}) + lo g_{e} (\frac{3 x ^{2} - 8 x + 5}{x ^{2} - 3 x - 10})

(α, β]

, then

3 α + 10 β

is equal to

Answer: (C)

sin^{- 1}

part:**

- 1 \leq \frac{3 x - 22}{2 x - 19} \leq 1

gives

x \geq \frac{41}{5}

(with

x \neq = \frac{19}{2}

), i.e.

x \in [\frac{41}{5}, \infty)

.

log part:

\frac{3 x ^{2} - 8 x + 5}{x ^{2} - 3 x - 10} > 0

, i.e.

\frac{( 3 x - 5 ) ( x - 1 )}{( x - 5 ) ( x + 2 )} > 0

, which holds for

x > 5

(among the relevant region).

Intersection:

(5, \frac{41}{5}]

, so

α = 5, β = \frac{41}{5}

3 α + 10 β = 15 + 82 = 97.

Option C.

JEE Main · 2024 · 04 Apr 2024 · Shift 1 · Single correct option

If the domain of the function $f (x) = cos^{- 1} (\frac{2 x - 5}{11 - 3 x}) + sin^{- 1} (2 x^{2} - 3 x + 1)$ is the interval $[α, β]$ , then $α + 2 β$ is equal to :
JEE Main 2026View solution →
The number of solutions of $tan^{- 1} 4 x + tan^{- 1} 6 x = \frac{π}{6}$ , where $- \frac{1}{2 6} < x < \frac{1}{2 6}$ is equal to
JEE Main 2026View solution →
If the domain of the function $f (x) = sin^{- 1} (\frac{1}{x ^{2} - 2 x - 2})$ , is $(- \infty, α] \cup [β, γ] \cup [δ, \infty)$ , then $α + β + γ + δ$ is equal to
JEE Main 2026View solution →
If $k = tan (\frac{π}{4} + \frac{1}{2} cos^{- 1} (\frac{2}{3})) + tan (\frac{1}{2} sin^{- 1} (\frac{2}{3}))$ then the number of solutions of the equation …
JEE Main 2026View solution →
Considering the principal values of inverse trigonometric functions, the value of the expression $tan (2 sin^{- 1} (\frac{2}{13}) - 2 cos^{- 1} (\frac{3}{10}))$ is equal to :
JEE Main 2026View solution →
Let [ • ] denote the greatest integer function. If the domain of the function $f (x) = cos^{- 1} (\frac{4 x + 2 [ x ]}{3})$ is $[α, β]$ , then $12 (α + β)$ is equal to :
JEE Main 2026View solution →