Higher June 2024 Paper 2R Q24
24 The diagram shows a solid, S, made from a cone and a hemisphere.
Diagram NOT accurately drawn
The centre of the circular face of the cone coincides with the centre of the flat surface of the hemisphere.
The radius of the circular face of the cone, \(x\) cm, is equal to the radius of the hemisphere.
The total height of S is 4 × the radius of the hemisphere.
A separate sphere has radius \(kx\) cm.
The volume of this sphere is 12.5 × the volume of S
A solid, T, is similar to solid S
The volume of T is 512 × the volume of S
The total surface area of T is \(d\) × the total surface area of S
| Scheme | Marks |
|---|---|
eg \(\dfrac{1}{3} \times \pi \times x^2 \times 3x\,(= \pi x^3)\) oe or \(\dfrac{1}{2} \times \dfrac{4}{3} \times \pi \times x^3\left(= \dfrac{4}{6}\pi x^3 = \dfrac{2}{3}\pi x^3\right)\) oe or \(\dfrac{4}{3} \times \pi \times (kx)^3\) oe | M1 |
eg \(\dfrac{4}{3} \times \pi \times (kx)^3 = 12.5 \times \left(\dfrac{1}{2} \times \dfrac{4}{3}\pi x^3 + \dfrac{1}{3}\pi x^2(3x)\right)\) oe or \(\dfrac{4}{3} \times \pi \times (kx)^3 = 12.5\left(\dfrac{2}{3}\pi x^3 + \pi x^3\right)\) oe or \(\dfrac{4}{3} \times \pi \times (kx)^3 = 12.5 \times \dfrac{5}{3}\pi x^3\) oe or \(\dfrac{4}{3} \times \pi \times (kx)^3 = \dfrac{125}{6}\pi x^3\) oe | M1 |
eg \((k^3 =)\, \dfrac{\frac{125}{6}\pi}{\frac{4}{3}\pi}\) oe or \((k^3 =)\, \dfrac{125}{8}\) oe or \((k =)\sqrt[3]{\dfrac{125}{8}}\) oe \(k^3x^3 = \dfrac{12.5 \times \frac{5}{3}\pi x^3}{\frac{4}{3}\pi}\) oe or \(kx = \dfrac{\sqrt[3]{12.5 \times \frac{5}{3}\pi x^3}}{\sqrt[3]{\frac{4}{3}\pi}}\) or \(kx = \sqrt[3]{\dfrac{125x^3}{8}}\) oe | M1 |
| Working not required, so correct answer scores full marks (unless from obvious incorrect working) Answer: 2.5 | A1 |
| (4) |
Notes
M1: for an expression for the volume of the cone or the hemisphere or the sphere
NB Ignore missing brackets around \(kx\) for this mark
Allow \(r\) for \(x\) for all M marks
M1: for a correct equation for the volumes
NB If \((kx)^3\) not expanded at this stage then must see brackets
M1: for a correct calculation for \(k\) or \(k^3\)
or
for a correct equation for \(kx\) or \(k^3x^3\)
A1: oe
| Scheme | Marks |
|---|---|
| 64 | B1 |
| (1) | |
| (5 marks) |