If the domain of the function \(f(x) = \sin^{-1}\left( \frac{x - 1}{2x + 3} \right)\) is \(\mathbb{R} - (\alpha, \beta)\), then \(12 \alpha \beta\) is equal to:
- 36
- 24
- 40
- 32
The Correct Option is D
Approach Solution - 1
Given function:
\( f(x) = \sin^{-1}\left(\frac{x - 1}{2x + 3}\right) \)
For \( f(x) \) to be defined, the following conditions must hold:
\( 2x + 3 \neq 0 \) and \( -1 \leq \frac{x - 1}{2x + 3} \leq 1 \)
⇒ \( \frac{x - 1 - (2x + 3)}{2x + 3} \leq 0 \leq \frac{x - 1 + (2x + 3)}{2x + 3} \)
⇒ \( \frac{-x - 4}{2x + 3} \leq 0 \leq \frac{3x + 2}{2x + 3} \)
Now, solving \( \frac{-x - 4}{2x + 3} \leq 0 \):
Critical points: \( x = -4, -\frac{3}{2} \)
Sign analysis gives: \( x \in (-\infty, -4] \cup \left(-\frac{3}{2}, \infty \right) \)
Similarly, for \( \frac{3x + 2}{2x + 3} \geq 0 \):
Critical points: \( x = -\frac{3}{2}, -\frac{2}{3} \)
Sign analysis gives: \( x \in (-\infty, -\frac{3}{2}) \cup \left[-\frac{2}{3}, \infty \right) \)
Taking the intersection of the two intervals, the domain is:
\( x \in (-\infty, -4] \cup \left(-\frac{2}{3}, \infty \right) \)
Thus, the range is:
\( R = \left(-4, -\frac{2}{3}\right] \)
Now, let \( \alpha = -4 \) and \( \beta = -\frac{2}{3} \)
Then, \( 12 \times (-4) \times \left(-\frac{2}{3}\right) = 32 \)
Hence, the final answer is:
\( 32 \)
Approach Solution -2
Step 1: Conditions for the domain of \( f(x) \) The argument of \( \sin^{-1}(x) \), \( \frac{x-1}{2x+3} \), must satisfy two conditions:
- \( 2x + 3 \neq 0 \) (denominator cannot be zero), so \( x \neq -\frac{3}{2} \).
- \( \left| \frac{x-1}{2x+3} \right| \leq 1 \).
Step 2: Solve \( \left| \frac{x-1}{2x+3} \right| \leq 1 \) Split the inequality into two cases:
1. For \( \frac{x-1}{2x+3} \geq -1 \):
\( x-1 \geq -(2x+3) \implies x-1 \geq -2x-3 \).
Simplify:
\( 3x \geq -2 \implies x \geq -\frac{2}{3} \).
2. For \( \frac{x-1}{2x+3} \leq 1 \):
\( x-1 \leq 2x+3 \implies -x \leq 4 \).
Simplify:
\( x \geq -4 \).
Thus, combining the results:
\( x \in [-4, -\frac{2}{3}] \) and exclude \( x = -\frac{3}{2} \).
Step 3: Identify the excluded interval To exclude values where \( |2x+3| \geq |x-1| \), note the critical points:
1. Solve \( |x-1| = |2x+3| \), which gives:
\( x = -4, \; x = -\frac{2}{3} \).
Using these results and the behavior of the function, the domain of \( f(x) \) is:
\( x \in (-\infty, -4] \cup \left(-\frac{2}{3}, \infty\right) \).
Step 4: Determine \( \alpha \) and \( \beta \) From the excluded interval \( \left(-\frac{3}{2}, -\frac{2}{3}\right) \):
\( \alpha = -4, \; \beta = -\frac{2}{3} \).
Step 5: Compute \( 12\alpha\beta \):
\( 12\alpha\beta = 12 \times (-4) \times \left(-\frac{2}{3}\right) \).
Simplify:
\( 12\alpha\beta = 12 \times 8/3 = 32 \).
Final Answer is Option (4): 32.
Top JEE Main Mathematics Questions
- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
- If , then the general value of is:
- The general solution of
- If the constant term in the binomial expansion of $\left(\frac{x^{\frac{3}{2}}}{2}-\frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3 l}$ is $2^\alpha \beta$, where $\beta<0$ is an odd number, then $|\alpha l-\beta|$ is equal to_____
Top JEE Main Inverse Trigonometric Functions Questions
- If \(a=sin^{-1}(sin\ 5)\) and \(b=cos^{-1}(cos\ 5)\), then \(a^2+b^2=\)
- Considering only the principal values of inverse trigonometric functions, the number of positive real values of \( x \) satisfying \[ \tan^{-1}(x) + \tan^{-1}(2x) = \frac{\pi}{4} \] is:
- Let the inverse trigonometric functions take principal values. The number of real solutions of the equation \[ 2 \sin^{-1} x + 3 \cos^{-1} x = \frac{2\pi}{5}, \] is ______ .
- 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:
- Given the inverse trigonometric function assumes principal values only. Let \( x, y \) be any two real numbers in \( [-1, 1] \) such that \[ \cos^{-1}x - \sin^{-1}y = \alpha, \, -\frac{\pi}{2} \leq \alpha \leq \pi. \] Then, the minimum value of \( x^2 + y^2 + 2xy \sin \alpha \) is:
Top JEE Main Questions
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
A black body is at a temperature of 2880 K. The energy of radiation emitted by this body with wavelength between 499 nm and 500 nm is U1, between 999 nm and 1000 nm is U2 and between 1499 nm and 1500 nm is U3. The Wien's constant, b = 2.88×106 nm-K. Then,