| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Point on line satisfying condition |
| Difficulty | Standard +0.3 This is a straightforward multi-part vectors question requiring standard techniques: finding a line equation from two points, using scalar multiples of vectors, and solving a distance equation. Part (c) involves solving a quadratic but follows a routine method. All parts are textbook-standard with no novel insight required, making it slightly easier than average. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10b Vectors in 3D: i,j,k notation1.10d Vector operations: addition and scalar multiplication1.10e Position vectors: and displacement |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Obtain direction vector \(-\mathbf{i} + \mathbf{j} + 2\mathbf{k}\), or equivalent | B1 | Accept answers as column vectors throughout |
| Use a correct method to form a vector equation | M1 | |
| State answer \(\mathbf{r} = \mathbf{i} + 2\mathbf{j} - \mathbf{k} + \lambda(-\mathbf{i} + \mathbf{j} + 2\mathbf{k})\), or equivalent correct form | A1 | e.g. \(\mathbf{r} = \begin{pmatrix}0\\3\\1\end{pmatrix} + \mu\begin{pmatrix}1\\-1\\-2\end{pmatrix}\). Allow \(\begin{pmatrix}x\\y\\z\end{pmatrix}\) for \(\mathbf{r}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use a correct method to find the position vector of \(C\) | M1 | e.g. \(\mathbf{OC} = \mathbf{OA} + \mathbf{AC} = \begin{pmatrix}1-3\\2+3\\-1+6\end{pmatrix}\) |
| Obtain answer \(-2\mathbf{i} + 5\mathbf{j} + 5\mathbf{k}\), or equivalent | A1 | Accept as coordinates |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| State \(\overrightarrow{OP}\) in component form | B1 FT | |
| Form an equation in \(\lambda\) by equating the modulus of \(OP\) to \(\sqrt{14}\), or equivalent | M1 | |
| Simplify and obtain \(3\lambda^2 - \lambda - 4 = 0\), or equivalent | A1 | \(3\lambda^2 + \lambda - 4 = 0\) if using \(\mathbf{i}-\mathbf{j}-2\mathbf{k}\) in (a). \(3\mu^2 + 5\mu - 2 = 0\) if using \(-\mathbf{i}+\mathbf{j}+2\mathbf{k}\) in (a) and \(OB\) |
| Solve a 3-term quadratic and find a position vector | M1 | \(\left(\lambda = -1, \frac{4}{3}\text{ or }\lambda=1,-\frac{4}{3}\text{ or }\mu=\frac{1}{3},-2\text{ or }\mu=-\frac{1}{3},2\right)\) |
| Obtain answers \(2\mathbf{i} + \mathbf{j} - 3\mathbf{k}\) and \(-\frac{1}{3}\mathbf{i} + \frac{10}{3}\mathbf{j} + \frac{5}{3}\mathbf{k}\), or equivalent | A1 | Accept as coordinates |
## Question 10(a):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Obtain direction vector $-\mathbf{i} + \mathbf{j} + 2\mathbf{k}$, or equivalent | B1 | Accept answers as column vectors throughout |
| Use a correct method to form a vector equation | M1 | |
| State answer $\mathbf{r} = \mathbf{i} + 2\mathbf{j} - \mathbf{k} + \lambda(-\mathbf{i} + \mathbf{j} + 2\mathbf{k})$, or equivalent correct form | A1 | e.g. $\mathbf{r} = \begin{pmatrix}0\\3\\1\end{pmatrix} + \mu\begin{pmatrix}1\\-1\\-2\end{pmatrix}$. Allow $\begin{pmatrix}x\\y\\z\end{pmatrix}$ for $\mathbf{r}$ |
---
## Question 10(b):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use a correct method to find the position vector of $C$ | M1 | e.g. $\mathbf{OC} = \mathbf{OA} + \mathbf{AC} = \begin{pmatrix}1-3\\2+3\\-1+6\end{pmatrix}$ |
| Obtain answer $-2\mathbf{i} + 5\mathbf{j} + 5\mathbf{k}$, or equivalent | A1 | Accept as coordinates |
---
## Question 10(c):
| Answer | Marks | Guidance |
|--------|-------|----------|
| State $\overrightarrow{OP}$ in component form | B1 FT | |
| Form an equation in $\lambda$ by equating the modulus of $OP$ to $\sqrt{14}$, or equivalent | M1 | |
| Simplify and obtain $3\lambda^2 - \lambda - 4 = 0$, or equivalent | A1 | $3\lambda^2 + \lambda - 4 = 0$ if using $\mathbf{i}-\mathbf{j}-2\mathbf{k}$ in (a). $3\mu^2 + 5\mu - 2 = 0$ if using $-\mathbf{i}+\mathbf{j}+2\mathbf{k}$ in (a) and $OB$ |
| Solve a 3-term quadratic and find a position vector | M1 | $\left(\lambda = -1, \frac{4}{3}\text{ or }\lambda=1,-\frac{4}{3}\text{ or }\mu=\frac{1}{3},-2\text{ or }\mu=-\frac{1}{3},2\right)$ |
| Obtain answers $2\mathbf{i} + \mathbf{j} - 3\mathbf{k}$ and $-\frac{1}{3}\mathbf{i} + \frac{10}{3}\mathbf{j} + \frac{5}{3}\mathbf{k}$, or equivalent | A1 | Accept as coordinates |
---10 With respect to the origin $O$, the position vectors of the points $A$ and $B$ are given by $\overrightarrow { O A } = \left( \begin{array} { r } 1 \\ 2 \\ - 1 \end{array} \right)$ and $\overrightarrow { O B } = \left( \begin{array} { l } 0 \\ 3 \\ 1 \end{array} \right)$.
\begin{enumerate}[label=(\alph*)]
\item Find a vector equation for the line $l$ through $A$ and $B$.
\item The point $C$ lies on $l$ and is such that $\overrightarrow { A C } = 3 \overrightarrow { A B }$.
Find the position vector of $C$.
\item Find the possible position vectors of the point $P$ on $l$ such that $O P = \sqrt { 14 }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2021 Q10 [10]}}