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All x satisfying the inequality (cot⁻¹ x)² − 7(cot⁻¹ x) + 10 > 0, lie in the interval:

1. (cot 2, ∞)
2. (−∞, cot 5) ∪ (cot 2, ∞)
3. (cot 5, cot 4)
4. (−∞, cot 5) ∪ (cot 4, cot 2)

Correct answer: (cot 2, ∞)

Solution

The inequality represents a quadratic function in terms of cot⁻¹ x, and solving it reveals that the roots are at cot 2 and cot 5. The quadratic opens upwards, indicating that the values satisfying the inequality are those greater than the larger root, which is cot 2.

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