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:
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:
Show 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.
Updated On: Mar 24, 2025
- \( 8 \)
- \( 6 \)
- \( 2 \)
- \( 4 \)
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The Correct Option is D
Solution and Explanation
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: \( n(A \cup B) = 4 \).
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