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Home » The number of ways to choose 2 distinct gaps is: - Dyverse

The number of ways to choose 2 distinct gaps is: - Dyverse

📅 March 9, 2026 👤 scraface

Mar 09, 2026

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📌 Hence, the truth-weight parameter is $ oxed{6} $.Question: Find $ x $ such that the vectors $ egin{pmatrix} x \ 2 \end{pmatrix} $ and $ egin{pmatrix} 3 \ -x \end{pmatrix} $ are orthogonal.

📌 Solution: Two vectors are orthogonal if their dot product is zero. Compute the dot product: $ x \cdot 3 + 2 \cdot (-x) = 3x - 2x = x $. Set $ x = 0 $. Thus, $ oxed{0} $ is the solution.

📌 Question: Determine the number of real solutions to the equation $ \sin^2(2x) + 3\sin(2x) + 2 = 0 $.

📌 Solution: Let $ y = \sin(2x) $. The equation becomes $ y^2 + 3y + 2 = 0 $, which factors as $ (y + 1)(y + 2) = 0 $. Thus, $ y = -1 $ or $ y = -2 $. Since $ \sin(2x) $ ranges between $-1$ and $1$, $ y = -2 $ is invalid. For $ y = -1 $, $ \sin(2x) = -1 $ has infinitely many solutions (e.g., $ 2x = rac{3\pi}{2} + 2\pi k $, $ k \in \mathbb{Z} $). However, if restricted to a specific interval (not stated here), the count would depend on the domain. Assuming $ x \in \mathbb{R} $, there are infinitely many solutions. But if the question implies a general count, the answer is oxed{ ext{infinitely many}}.

📌 Question: Find the minimum value of $ (\cos x + \sec x)^2 + (\sin x + \csc x)^2 $.

📌 Solution: Expand the expression: $ \cos^2x + 2 + \sec^2x + \sin^2x + 2 + \csc^2x $. Simplify using $ \cos^2x + \sin^2x = 1 $ and $ \sec^2x = 1 + an^2x $, $ \csc^2x = 1 + \cot^2x $: $ 1 + 2 + 1 + an^2x + 1 + \cot^2x + 2 = 7 + an^2x + \cot^2x $. Let $ t = an^2x $, so $ \cot^2x = rac{1}{t} $. The expression becomes $ 7 + t + rac{1}{t} $. By AM-GM, $ t + rac{1}{t} \geq 2 $, so the minimum is $ 7 + 2 = 9 $. Thus, the minimum value is $ oxed{9} $.

📌 Question: Among all roots of $ z^6 + z^4 + z^2 + 1 = 0 $, the maximum imaginary part can be expressed as $ \sin heta $, where $ 0 < heta < rac{\pi}{2} $. Find $ heta $.

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