Higher June 2024 Paper 2 Q19
19 The diagram shows quadrilateral \(OACB\).
\(M\) is the midpoint of \(OA\).
\(N\) is the point on \(BC\) such that \(BN : NC = 4 : 5\)
\(\overrightarrow{OA} = \mathbf{a} \qquad \overrightarrow{OB} = \mathbf{b} \qquad \overrightarrow{AC} = k\mathbf{b}\) where \(k\) is a positive integer.
(a) Express \(\overrightarrow{MN}\) in terms of \(k\), \(\mathbf{a}\) and \(\mathbf{b}\).
Give your answer in its simplest form. (4)
Give your answer in its simplest form. (4)
(b) Is \(MN\) parallel to \(OB\)?
Give a reason for your answer. (1)
Give a reason for your answer. (1)
| Answer | Mark | Mark scheme |
|---|---|---|
| \(\dfrac{1}{18}(8k\mathbf{b} + 10\mathbf{b} - \mathbf{a})\) | P1 | for a correct expression for \(\overrightarrow{CB}\) or \(\overrightarrow{BC}\) eg, \(\overrightarrow{CB} = -k\mathbf{b} - \mathbf{a} + \mathbf{b}\) or \(\overrightarrow{BC} = -\mathbf{b} + \mathbf{a} + k\mathbf{b}\) |
| P1 | for a correct expression for \(\overrightarrow{CN}\) or \(\overrightarrow{BN}\) or \(\overrightarrow{NC}\) or \(\overrightarrow{NB}\) eg, \(\overrightarrow{CN} = \dfrac{5}{9}(-k\mathbf{b} - \mathbf{a} + \mathbf{b})\) or \(\overrightarrow{BN} = \dfrac{4}{9}(-\mathbf{b} + \mathbf{a} + k\mathbf{b})\) or \(\overrightarrow{NC} = \dfrac{5}{9}(-\mathbf{b} + \mathbf{a} + k\mathbf{b})\) or \(\overrightarrow{NB} = \dfrac{4}{9}(-k\mathbf{b} - \mathbf{a} + \mathbf{b})\) | |
| P1 | for a correct unsimplified expression for \(\overrightarrow{MN}\) eg \(\dfrac{1}{2}\mathbf{a} + k\mathbf{b} + \dfrac{5}{9}(-k\mathbf{b} - \mathbf{a} + \mathbf{b})\) oe or \(-\dfrac{1}{2}\mathbf{a} + \mathbf{b} + \dfrac{4}{9}(-\mathbf{b} + \mathbf{a} + k\mathbf{b})\) oe | |
| A1 | for \(\dfrac{1}{18}(8k\mathbf{b} + 10\mathbf{b} - \mathbf{a})\) oe eg \(\dfrac{5}{9}\mathbf{b} + \dfrac{4}{9}k\mathbf{b} - \dfrac{1}{18}\mathbf{a}\) |
Additional guidance
All vectors must be clearly identified
This mark implies the previous one
Must have a maximum of 3 vector terms, \(\mathbf{a}\), \(\mathbf{b}\), and \(k\mathbf{b}\)
| Answer | Mark | Mark scheme |
|---|---|---|
| No, with explanation | C1 | No with supporting reason ft (a) Acceptable reasons: No, since \(\dfrac{1}{18}(8k\mathbf{b} + 10\mathbf{b} - \mathbf{a})\) is not a multiple of \(\mathbf{b}\) No, as \(N\) is not the midpoint of \(BC\) No, they are not multiples of each other No, does not factorise to \(\mathbf{b}\) Not a multiple of \(OB\) \(OB\) doesn’t have an \(\mathbf{a}\) Not acceptable reasons: Yes… No, they don’t share the same multiples \(OB\) doesn’t go into \(MN\) |