Question:

<p><strong>Solution:</strong></p><p>The curve <span class="math">y = 2e^{2x}</span> intersects the Y-axis at <span class="math">(0, 2)</span>.</p><p>Slope: <span class="math">\frac{dy}{dx} = 4e^{2x}</span></p><p>At <span class="math">(0, 2)</span>: <span class="math">\frac{dy}{dx}\bigg|_{(0,2)} = 4</span></p><p>Angle of intersection with Y-axis: <span class="math">\theta = \tan^{-1}(4) = \cot^{-1}\left(\frac{1}{4}\right)</span></p><p>Given: <span class="math">\cot^{-1}\left|\frac{8n-4}{3}\right| = \cot^{-1}\left(\frac{1}{4}\right)</span></p><p>Therefore: <span class="math">\left|\frac{8n-4}{3}\right| = \frac{1}{4}</span></p><p><span class="math">8n - 4 = \pm\frac{3}{4}</span></p><p><span class="math">8n = 4 \pm \frac{3}{4} \Rightarrow n = 2 \text{ or } n = -1</span></p><p>∴ Answer: (r, q)</p>