Higher June 2025 Paper 2 Q23
23 \(ABCD\) is a quadrilateral.
\(AC\) and \(BD\) intersect at \(P\).
\[\begin{aligned} \overrightarrow{AB} &= 9\mathbf{b} - 2\mathbf{a} \\ \overrightarrow{BC} &= 5\mathbf{a} \\ \overrightarrow{DC} &= 10\mathbf{b} \end{aligned}\]
Not drawn accurately
\[\begin{aligned} BP : PD &= 3 : 2 \\ AP : PC &= 1 : k \end{aligned}\]Work out the value of \(k\).
You must show your working. [5 marks]
| Answer | Mark | Comments |
|---|---|---|
| Two of \((\overrightarrow{AP} =)\ \mathbf{a} + 3\mathbf{b}\) \((\overrightarrow{PC} =)\ 2\mathbf{a} + 6\mathbf{b}\) \((\overrightarrow{AC} =)\ 3\mathbf{a} + 9\mathbf{b}\) and answer 2 | B5 | B4 two of \((\overrightarrow{AP} =)\ \mathbf{a} + 3\mathbf{b}\) \((\overrightarrow{PC} =)\ 2\mathbf{a} + 6\mathbf{b}\) \((\overrightarrow{AC} =)\ 3\mathbf{a} + 9\mathbf{b}\) B3 \((\overrightarrow{AP} =)\ \mathbf{a} + 3\mathbf{b}\) or \((\overrightarrow{PC} =)\ 2\mathbf{a} + 6\mathbf{b}\) B2 \((\overrightarrow{BP} =)\ 3\mathbf{a} - 6\mathbf{b}\) or \((\overrightarrow{PD} =)\ 2\mathbf{a} - 4\mathbf{b}\) B1 \((\overrightarrow{AC} =)\ 3\mathbf{a} + 9\mathbf{b}\) or \((\overrightarrow{BD} =)\ 5\mathbf{a} - 10\mathbf{b}\) or \((\overrightarrow{AD} =)\ 3\mathbf{a} - \mathbf{b}\) |
Additional guidance
| Accept eg \((\overrightarrow{PA} =)\ -\mathbf{a} - 3\mathbf{b}\) for \((\overrightarrow{AP} =)\ \mathbf{a} + 3\mathbf{b}\) | |
| Accept eg \(9\mathbf{b} + 3\mathbf{a}\) or \(3(\mathbf{a} + 3\mathbf{b})\) for \(3\mathbf{a} + 9\mathbf{b}\) but do not accept \(9\mathbf{b} - 2\mathbf{a} + 5\mathbf{a}\) | |
| Accept eg \(AC\) or \(A \to C\) or \(ABC\) or \(A \to B \to C\) for \(\overrightarrow{AC}\) | |
| Condone upper case letters for lower case letters and vice-versa | |
| Ignore incorrect vectors and award marks for any correct vectors | |
| Vectors may be seen on the diagram | |
| Correct vectors may be seen in ratios and/or divisions eg \(\mathbf{a} + 3\mathbf{b} : 2\mathbf{a} + 6\mathbf{b}\) Answer \(\dfrac{1}{2}\) | B4 |