Question

Let \( A = \left\{ x \in (0, \pi) \mid - \log\left(\frac{2}{\pi}\right)\sin x + \log\left(\frac{2}{\pi}\right)\cos x = 2 \right\} \) and

\[ B = \left\{ x \geq 0 : \sqrt{x}(\sqrt{x - 4}) - 3\sqrt{x - 2} + 6 = 0 \right\}. \]

Then \( n(A \cup B) \) is equal to:

• A. \( 8 \)
• B. \( 6 \)
• C. \( 2 \)
• D. \( 4 \)

Hint:

When solving for the union of two sets, simplify the equations for each set, find the solutions, and count the total number of unique elements in the union.

Answer 1

The Correct Option is D

We first solve for the set \( A \) by simplifying the given equation and finding the range of \( x \) that satisfies it. Next, we solve for the set \( B \) using the given equation. After determining the elements in both sets, we calculate \( n(A \cup B) \), the number of elements in the union of sets \( A \) and \( B \).

Final Answer: \( n(A \cup B) = 4 \).