2026 May TZC P2 Q7

All lengths in this question are in metres.

A sphere has diameter \([\mathrm{RS}]\), where \(\mathrm{R} \ (5, -3, 2)\) and \(\mathrm{S} \ (-3, 3, -4)\).

1. Determine

   1. the coordinates of the centre of the sphere;

   2. the radius of the sphere.

      [4]

2. Find the volume of the sphere.

   [2]

A cuboid has a rectangular base of length \(x\), width \(x^2\) and height \(y\). The cuboiud has a volume of \(1000\) cubic metres and surface area \(A\).

3. 1. Write down an equation for the volume of the cuboid in terns of \(x\) and \(y\),

   2. Show that \(A = 2 \left( x^3 + \frac{1000}{x} + \frac{1000}{x^2} \right) \).

      [4]

The surface area has a minimum value when \(x = k\).

4. 1. Find the minimum value of \(A\).

   2. Find the value of \( \frac{\mathrm{d}^2A}{\mathrm{d}x^2}\) when \(x = k\).

   3. Use your answer to part (d)(ii) to justify that \(A\) is a minimum when \(x = k\).

      [5]