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Let \(a\) be the set of solutions of the equation \[ \cos(\cos^{-1}x)+\sin^{-1}\!\left[\sin\left(\frac{1+x^2}{2}\right)\right]=2\sec^{-1}(\sec x), \] where \([\,\cdot\,]\) denotes the greatest integer function. Then the possible values of \(\left|\,1^{10}a\,\right|\) are

1. 1
2. 5
3. 10
4. Both (a) and (c)

Correct answer: Both (a) and (c)

Solution

The equation simplifies to finding values of x that satisfy the given trigonometric identities, leading to integer solutions. The greatest integer function limits the possible outcomes, resulting in a set of solutions that can yield either 1 or 10 as the absolute value of the product of 1 and the number of solutions.

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