Let $z$ be a complex number such that $\left|2z + \frac{1}{z}\right| = 1$ and $\arg(z) = 0$. Then minimum value of $\sin^2\theta$ is ____.
Step-by-Step Solution
Key Concept: The sum of areas of one copy each of non-congruent rectangles exceeds the square's area, forcing duplication.
To find the minimum number of non-congruent rectangles with minimum area fitting in a $6 \times 6$ square, we enumerate rectangles by area: $1×1$ (area 1), $1×2$ (area 2), $1×3$ (area 3), $1×4$ and $2×2$ (area 4), $1×5$ (area 5), $1×6$ and $2×3$ (area 6), $1×7$ (area 7, doesn't fit), and $2×4$ (area 8). The sum $1+2+3+4+5+6+8 = 39 > 36$, so at least 2 congruent rectangles are required.
Correct Answer: 0.875