2026 MAY TZA P1 Q9

Eleanor uses the handle on a heavy suitcase to pull it so that it moves horizontally along the floor.
The handle makes an angle of \(\theta\) radians with the horizontal, as shown in the following diagram.

The magnitude, \(F\) Newtons, of the minimum force that Eleanor needs to apply to make the suitcase move horizontally when the handle makes an angle of \(\theta\) radians is given by

\(F(\theta) = \frac{A}{\cos{\theta} + \mu \sin{\theta}}\), where \(0 \leq \theta \leq \frac{\pi}{2}\) and \(\mu \in \mathbb{R}\).

When the handle is horizontal, \(F = 300\).

1. Show that \(A = 300\).

   [2]

In the case when \(\theta = \frac{\pi}{6}\), \(F = 100\sqrt{3}\).

2. Show that \(\mu = \sqrt{3}\).

   [4]

Eleanor wishes to minimize the magnitude of the force she needs to apply to make the suitcase move horizontally. In order to achieve this, she needs to hold the handle at the angle \(\alpha\), where \(0 < \alpha < \frac{\pi}{2}\).

3. [9]

   1. Show that \(\frac{\mathrm{d}F}{\mathrm{d}\theta} = \frac{300 \left(\sin{\theta} - \sqrt{3}\cos{\theta} \right)}{ \left(\cos{\theta} + \sqrt{3}\sin{\theta} \right)^2}\).

   2. Hence, show that \(\alpha = \frac{\pi}{3}\) and find the corresponding value of \(F(\alpha)\).