Question:
Let \( S = \{ x : \cos^{-1} x = \pi + \sin^{-1} x + \sin^{-1} (2x + 1) \ \). Then
\[
\sum_{x \in S} (2x - 1)^2 \text{ is equal to:}
\]
Let \( S = \{ x : \cos^{-1} x = \pi + \sin^{-1} x + \sin^{-1} (2x + 1) \ \). Then
\[
\sum_{x \in S} (2x - 1)^2 \text{ is equal to:}
\]
Show Hint
For solving trigonometric inverse equations, use the known identities and properties of inverse trigonometric functions to simplify the expression.
Updated On: Mar 24, 2025
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Correct Answer: 16
Solution and Explanation
Step 1: Solve the equation \( \cos^{-1} x = \pi + \sin^{-1} x + \sin^{-1} (2x + 1) \).
Step 2: Find the possible values of \( x \) that satisfy the equation.
Step 3: Calculate the sum \( \sum_{x \in S} (2x - 1)^2 \) for these values. The result is 16.
Step 2: Find the possible values of \( x \) that satisfy the equation.
Step 3: Calculate the sum \( \sum_{x \in S} (2x - 1)^2 \) for these values. The result is 16.
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