Higher June 2024 Paper 1R Q22
22 The diagram shows a cuboid \(ABCDEFGH\) with horizontal base \(ADEH\)
Diagram NOT accurately drawn
\(AB = 8\) cm \(\qquad AD = 12\) cm \(\qquad DE = 20\) cm
\(M\) is the midpoint of the base \(ADEH\) and \(P\) is the midpoint of the edge \(CF\)
Work out the size of angle \(BMP\)
Give your answer correct to one decimal place.
(6)
| Scheme | Marks |
|---|---|
| eg \(MP = \sqrt{8^2 + 6^2}\;(= 10)\) | M1 |
| eg \(BM = \sqrt{\left(\sqrt{10^2 + 6^2}\right)^2 + 8^2}\;\left(= \sqrt{200} = 10\sqrt{2} = 14.1...\right)\) | M1 |
| eg \(BP = \sqrt{12^2 + 10^2}\;\left(= \sqrt{244} = 2\sqrt{61} = 15.6...\right)\) | M1 |
eg \(\text{``}{244}\text{''} = \text{``}{10}\text{''}^2 + \text{``}{200}\text{''} - 2 \times \text{``}{10}\text{''} \times \text{``}{\sqrt{200}}\text{''}\cos BMP\) or \(\cos BMP = \dfrac{\text{``}{10}\text{''}^2 + \text{``}{200}\text{''} - \text{``}{244}\text{''}}{2 \times \text{``}{10}\text{''} \times \text{``}{\sqrt{200}}\text{''}}\) oe | M1 |
| angle \(BMP = \cos^{-1}\left(\dfrac{\text{``}{10}\text{''}^2 + \text{``}{200}\text{''} - \text{``}{244}\text{''}}{2 \times \text{``}{10}\text{''} \times \text{``}{\sqrt{200}}\text{''}}\right)\) oe | M1 |
| Correct answer scores full marks (unless from obvious incorrect working) Answer: 78.6 | A1 |
| (6) | |
| (6 marks) |
Notes
M1: for a method to find \(MP\)
may be seen in subsequent working eg in the correct place in the cosine rule
10 seen without a correct method or without being identified as \(MP\) scores M0
M1: for a method to find \(BM\)
M1: for a method to find \(BP\)
M1: for correct substitution into the cosine rule
M1: for a complete correct method to find angle \(BMP\)
A1: accept 78.5 – 78.7