<p>If <i>z = (3 + 7i)(λ + iμ)</i>, when <i>λ, μ ∈ ℤ \ {0}</i> and <i>i = √−1</i>, is purely imaginary then minimum value of |<i>z</i>|<sup>2</sup> is</p>
Step-by-Step Solution
Key Concept: For z = (3 + 7i)(λ + iμ) to be purely imaginary, its real part must equal zero. This constraint relates λ and μ, and we minimize |z|² subject to this constraint with λ, μ ∈ ℤ \ {0}.
<p><strong>Step 1: Expand the product.</strong></p><p>z = (3 + 7i)(λ + iμ) = 3λ + 3iμ + 7iλ + 7i²μ</p><p>= 3λ + 3iμ + 7iλ - 7μ</p><p>= (3λ - 7μ) + i(3μ + 7λ)</p><p><strong>Step 2: Apply the purely imaginary condition.</strong></p><p>For z to be purely imaginary, its real part must be zero:</p><p>3λ - 7μ = 0</p><p>Therefore: 3λ = 7μ, which gives λ/μ = 7/3</p><p><strong>Step 3: Find integer solutions.</strong></p><p>Since λ, μ ∈ ℤ \ {0} and λ = 7k, μ = 3k for some non-zero integer k.</p><p>The minimum non-zero values occur at k = ±1:</p><p>• When k = 1: λ = 7, μ = 3</p><p>• When k = -1: λ = -7, μ = -3</p><p><strong>Step 4: Calculate |z|² for the minimum case.</strong></p><p>The imaginary part is: 3μ + 7λ = 3(3) + 7(7) = 9 + 49 = 58</p><p>Since the real part is 0, we have:</p><p>|z|² = 0² + 58² = 3364</p><p>Verification with k = -1: imaginary part = 3(-3) + 7(-7) = -9 - 49 = -58, so |z|² = (-58)² = 3364</p><p><strong>∴ Answer: D</strong></p>
Correct Answer: D