<p><strong>70.</strong> If \(y = \sin(mx)\) and \(y_n = \dfrac{d^n y}{dx^n}\), then the determinant \[\begin{vmatrix} y & y_1 & y_2 \\ y_3 & y_4 & y_5 \\ y_6 & y_7 & y_8 \end{vmatrix} =\] ______.</p>
Step-by-Step Solution
Key Concept: Find the pattern of derivatives of y = sin(mx): each derivative introduces a factor of m and cycles through sin/cos with phase shifts. Recognize that all entries are linear combinations of sin(mx) and cos(mx), making rows linearly dependent.
<p><strong>Step 1: Find the derivatives of y = sin(mx)</strong></p><p>y₀ = sin(mx)<br>y₁ = m·cos(mx)<br>y₂ = -m²·sin(mx)<br>y₃ = -m³·cos(mx)<br>y₄ = m⁴·sin(mx)<br>y₅ = m⁵·cos(mx)<br>y₆ = -m⁶·sin(mx)<br>y₇ = -m⁷·cos(mx)<br>y₈ = m⁸·sin(mx)</p><p><strong>Step 2: Observe the recurrence relation</strong></p><p>Notice that yₙ₊₂ = -m²·yₙ for all n ≥ 0<br>This means: y₂ = -m²·y₀, y₄ = -m²·y₂, y₆ = -m²·y₄, y₈ = -m²·y₆</p><p><strong>Step 3: Identify linear dependence</strong></p><p>Row 3: [y₆, y₇, y₈] = -m⁶[sin(mx), m·cos(mx), -m²·sin(mx)]<br>Row 3 = -m⁶ × Row 1 (after factoring)<br><br>Similarly, the three rows are linearly dependent because they all satisfy the same differential recurrence relation with period 4 in the sin/cos pattern.</p><p><strong>Step 4: Conclude</strong></p><p>When rows are linearly dependent, the determinant equals zero.</p><p>∴ Answer: <strong>0</strong></p>
Correct Answer: 0