Question

Let \( S = \{ x : \cos^{-1} x = \pi + \sin^{-1} x + \sin^{-1} (2x + 1) \ \). Then \[ \sum_{x \in S} (2x - 1)^2 \text{ is equal to:} \]

Hint:

For solving trigonometric inverse equations, use the known identities and properties of inverse trigonometric functions to simplify the expression.

Answer 1

Correct Answer: 16

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.