| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Line intersection with line |
| Difficulty | Moderate -0.3 This is a straightforward multi-part vectors question requiring standard techniques: finding midpoints and points dividing lines in given ratios, forming vector equations of lines, and solving simultaneous equations for line intersection. All steps are routine A-level procedures with no novel insight required, making it slightly easier than average. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10e Position vectors: and displacement1.10g Problem solving with vectors: in geometry |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State \(\overrightarrow{OM} = \begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix}\) | B1 | |
| Use a correct method to find \(\overrightarrow{ON}\) | M1 | |
| Obtain answer \(\begin{pmatrix} 3 \\ -2 \\ -1 \end{pmatrix}\) | A1 | |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Carry out a correct method to form a vector equation for \(MN\) | M1 | |
| Obtain a correct equation in any form, e.g. \(\mathbf{r} = \begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ -3 \\ -1 \end{pmatrix}\) | A1 | OE |
| Total | 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State a correct vector equation for \(AB\) in any form, e.g. \(\mathbf{r} = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} + \mu \begin{pmatrix} -1 \\ 5 \\ 1 \end{pmatrix}\) | B1 | |
| Equate components of \(AB\) and \(MN\) and solve for \(\lambda\) or for \(\mu\) | M1 | |
| Obtain \(\lambda = -3\) or \(\mu = 2\) | A1 | |
| Obtain position vector \(\begin{pmatrix} -1 \\ 10 \\ 3 \end{pmatrix}\), or equivalent, for \(Q\) | A1 | |
| Total | 4 |
## Question 9(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State $\overrightarrow{OM} = \begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix}$ | B1 | |
| Use a correct method to find $\overrightarrow{ON}$ | M1 | |
| Obtain answer $\begin{pmatrix} 3 \\ -2 \\ -1 \end{pmatrix}$ | A1 | |
| **Total** | **3** | |
---
## Question 9(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Carry out a correct method to form a vector equation for $MN$ | M1 | |
| Obtain a correct equation in any form, e.g. $\mathbf{r} = \begin{pmatrix} 2 \\ 1 \\ 0 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\ -3 \\ -1 \end{pmatrix}$ | A1 | OE |
| **Total** | **2** | |
---
## Question 9(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State a correct vector equation for $AB$ in any form, e.g. $\mathbf{r} = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} + \mu \begin{pmatrix} -1 \\ 5 \\ 1 \end{pmatrix}$ | B1 | |
| Equate components of $AB$ and $MN$ and solve for $\lambda$ or for $\mu$ | M1 | |
| Obtain $\lambda = -3$ or $\mu = 2$ | A1 | |
| Obtain position vector $\begin{pmatrix} -1 \\ 10 \\ 3 \end{pmatrix}$, or equivalent, for $Q$ | A1 | |
| **Total** | **4** | |
---9 With respect to the origin $O$, the position vectors of the points $A , B$ and $C$ are given by
$$\overrightarrow { O A } = \left( \begin{array} { l }
0 \\
5 \\
2
\end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { l }
1 \\
0 \\
1
\end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r }
4 \\
- 3 \\
- 2
\end{array} \right)$$
The midpoint of $A C$ is $M$ and the point $N$ lies on $B C$, between $B$ and $C$, and is such that $B N = 2 N C$.
\begin{enumerate}[label=(\alph*)]
\item Find the position vectors of $M$ and $N$.
\item Find a vector equation for the line through $M$ and $N$.
\item Find the position vector of the point $Q$ where the line through $M$ and $N$ intersects the line through $A$ and $B$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2022 Q9 [9]}}