Higher November 2021 Paper 1 Q14
14 \(ABCDEF\) and \(GHIJKL\) are regular hexagons each with centre \(O\).
Diagram NOT accurately drawn
\(GHIJKL\) is an enlargement of \(ABCDEF\), with centre \(O\) and scale factor 2
\(\overrightarrow{OA} = \mathbf{a}\qquad \overrightarrow{OB} = \mathbf{b}\)
(a) Write the following vectors, in terms of a and b.
Simplify your answers.
Simplify your answers.
(i) \(\overrightarrow{AB}\) (1)
(ii) \(\overrightarrow{KI}\) (2)
(iii) \(\overrightarrow{LD}\) (2)
The triangle \(OAB\) has an area of 5 cm2
(b) Calculate the area of the shaded region. (3)
| Scheme | Marks |
|---|---|
| \(\mathbf{b} - \mathbf{a}\) | B1 |
| (1) |
Notes
B1: oe
| Scheme | Marks |
|---|---|
| E.g. \((KI = KJ + JI =)\) \(2(\mathbf{b} - \mathbf{a}) + 2\mathbf{b}\) oe | M1ft |
| Correct answer scores full marks (unless from obvious incorrect working) Answer: \(4\mathbf{b} - 2\mathbf{a}\) | A1 |
| (2) |
Notes
M1ft: (i) for any valid correct path (oe) in capitals or lower case letters
A1: oe simplified
| Scheme | Marks |
|---|---|
| E.g. \((LD = LF + FE + ED =)\) \((\mathbf{b} - \mathbf{a}) + (\mathbf{b} - \mathbf{a}) - \mathbf{a}\) oe | M1ft |
| Correct answer scores full marks (unless from obvious incorrect working) Answer: \(2\mathbf{b} - 3\mathbf{a}\) | A1 |
| (2) |
Notes
M1ft: (i) for any valid correct path (oe) in capitals or lower case letters
A1: oe simplified
| Scheme | Marks |
|---|---|
| \((GHIJKL =)\ 6 \times 5 \times 2^2\ (= 120)\) or \((GABH =)\ 5 \times 2^2 - 5\ (= 15)\) or \(3 \times 5\ (= 15)\) or (Number of triangles in shaded region =) \((6 \times 4) - 6\ (= 18)\) | M1 |
| \(\text{``}{120}\text{''} - (6 \times 5)\) or \(6 \times \text{``}{15}\text{''}\) or \(\text{``}{18}\text{''} \times 5\) | M1 |
| Correct answer scores full marks (unless from obvious incorrect working) Answer: 90 | A1 |
| (3) | |
| (8 marks) |