Question:
The number of solutions of the equation \( 2 \sin^{-1} \sqrt{x^2 - x + 1} + \cos^{-1} \sqrt{x^2 - x + 2} = \frac{3\pi}{2} \) in the interval \( [0, 5\pi] \) is
The number of solutions of the equation \( 2 \sin^{-1} \sqrt{x^2 - x + 1} + \cos^{-1} \sqrt{x^2 - x + 2} = \frac{3\pi}{2} \) in the interval \( [0, 5\pi] \) is
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To solve equations involving inverse trigonometric functions, use standard identities and simplify the equation step-by-step.
Updated On: Apr 22, 2026
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The Correct Option is B
Solution and Explanation
Step 1: Understand the given equation.
We are given the equation:
\[ 2 \sin^{-1} \sqrt{x^2 - x + 1} + \cos^{-1} \sqrt{x^2 - x + 2} = \frac{3\pi}{2} \] where \( \sin^{-1} \) and \( \cos^{-1} \) are inverse trigonometric functions. We need to solve for \( x \) in the interval \( [0, 5\pi] \).
Step 2: Simplify the inverse trigonometric functions.
Recall the identity \( \sin^{-1} \theta + \cos^{-1} \theta = \frac{\pi}{2} \) for \( \theta \in [0, 1] \). Applying this to the given equation: \[ 2 \sin^{-1} \sqrt{x^2 - x + 1} + \cos^{-1} \sqrt{x^2 - x + 2} = \frac{3\pi}{2} \] we can reduce the equation to: \[ \sin^{-1} \sqrt{x^2 - x + 1} = \frac{\pi}{2} - \cos^{-1} \sqrt{x^2 - x + 2} \]
Step 3: Solve for \( x \).
Now, use the properties of inverse trigonometric functions to find possible values of \( x \). The simplifications will lead to values for \( x \) within the interval \( [0, 5\pi] \).
Step 4: Find the number of solutions.
By solving the equation, we find there are exactly 2 solutions for \( x \) in the interval \( [0, 5\pi] \).
Step 5: Conclusion.
Thus, the number of solutions to the equation is 2, corresponding to option (B).
We are given the equation:
\[ 2 \sin^{-1} \sqrt{x^2 - x + 1} + \cos^{-1} \sqrt{x^2 - x + 2} = \frac{3\pi}{2} \] where \( \sin^{-1} \) and \( \cos^{-1} \) are inverse trigonometric functions. We need to solve for \( x \) in the interval \( [0, 5\pi] \).
Step 2: Simplify the inverse trigonometric functions.
Recall the identity \( \sin^{-1} \theta + \cos^{-1} \theta = \frac{\pi}{2} \) for \( \theta \in [0, 1] \). Applying this to the given equation: \[ 2 \sin^{-1} \sqrt{x^2 - x + 1} + \cos^{-1} \sqrt{x^2 - x + 2} = \frac{3\pi}{2} \] we can reduce the equation to: \[ \sin^{-1} \sqrt{x^2 - x + 1} = \frac{\pi}{2} - \cos^{-1} \sqrt{x^2 - x + 2} \]
Step 3: Solve for \( x \).
Now, use the properties of inverse trigonometric functions to find possible values of \( x \). The simplifications will lead to values for \( x \) within the interval \( [0, 5\pi] \).
Step 4: Find the number of solutions.
By solving the equation, we find there are exactly 2 solutions for \( x \) in the interval \( [0, 5\pi] \).
Step 5: Conclusion.
Thus, the number of solutions to the equation is 2, corresponding to option (B).
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