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If alpha and beta are the roots of the equation 6x² + 11x + 3 = 0, then which of the following inverse trigonometric functions are defined for both alpha and beta?

1. both cos⁻¹(alpha) and cos⁻¹(beta) exist
2. both cosec⁻¹(alpha) and cosec⁻¹(beta) exist
3. both cot⁻¹(alpha) and cot⁻¹(beta) exist
4. none of these

Correct answer: both cot⁻¹(alpha) and cot⁻¹(beta) exist

Solution

Solving 6x² + 11x + 3 = 0: x = (-11 +- sqrt(121 - 72))/12 = (-11 +- 7)/12, giving roots -1/3 and -3/2. For cos⁻¹ we need |x|<=1: -3/2 fails. For cosec⁻¹ we need |x|>=1: -1/3 fails. cot⁻¹ is defined for every real number, so it exists for both roots.

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