| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Line intersection with line |
| Difficulty | Standard +0.3 This is a straightforward multi-part vectors question requiring standard techniques: finding midpoint, dividing a line segment in a given ratio, forming line equations, finding intersection by equating parameters, and using dot product for angles. All steps are routine applications of A-level vector methods with no novel insight required, making it slightly easier than average. |
| Spec | 1.10c Magnitude and direction: of vectors1.10f Distance between points: using position vectors1.10h Vectors in kinematics: uniform acceleration in vector form |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State that the position vector of \(M\) is \(3\mathbf{i} + \mathbf{j}\) | B1 | |
| Use a correct method to find the position vector of \(N\) | M1 | |
| Obtain answer \(\dfrac{10}{3}\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}\) | A1 | |
| Use a correct method to form an equation for \(MN\) | M1 | |
| Obtain correct answer in any form, e.g. \(\mathbf{r} = 3\mathbf{i} + \mathbf{j} + \lambda\!\left(\dfrac{1}{3}\mathbf{i} + \mathbf{j} + 2\mathbf{k}\right)\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply \(\mathbf{r} = \mu(2\mathbf{i} + 2\mathbf{j} + 3\mathbf{k})\) as equation for \(OB\) | B1 | |
| Equate sufficient components of \(MN\) and \(OB\) and solve for \(\lambda\) or for \(\mu\) | M1 | |
| Obtain \(\lambda = 3\) or \(\mu = 2\) and position vector \(4\mathbf{i} + 4\mathbf{j} + 6\mathbf{k}\) for \(P\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Carry out correct process for evaluating the scalar product of direction vectors for \(OP\) and \(MP\), or equivalent | M1 | |
| Using the correct process for the moduli, divide the scalar product by the product of the moduli and evaluate the inverse cosine of the result | M1 | |
| Obtain answer \(21.6°\) | A1 |
## Question 10:
### Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State that the position vector of $M$ is $3\mathbf{i} + \mathbf{j}$ | B1 | |
| Use a correct method to find the position vector of $N$ | M1 | |
| Obtain answer $\dfrac{10}{3}\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}$ | A1 | |
| Use a correct method to form an equation for $MN$ | M1 | |
| Obtain correct answer in any form, e.g. $\mathbf{r} = 3\mathbf{i} + \mathbf{j} + \lambda\!\left(\dfrac{1}{3}\mathbf{i} + \mathbf{j} + 2\mathbf{k}\right)$ | A1 | |
**Total: 5 marks**
### Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply $\mathbf{r} = \mu(2\mathbf{i} + 2\mathbf{j} + 3\mathbf{k})$ as equation for $OB$ | B1 | |
| Equate sufficient components of $MN$ and $OB$ and solve for $\lambda$ or for $\mu$ | M1 | |
| Obtain $\lambda = 3$ or $\mu = 2$ and position vector $4\mathbf{i} + 4\mathbf{j} + 6\mathbf{k}$ for $P$ | A1 | |
**Total: 3 marks**
### Part (c):
| Answer | Mark | Guidance |
|--------|------|----------|
| Carry out correct process for evaluating the scalar product of direction vectors for $OP$ and $MP$, or equivalent | M1 | |
| Using the correct process for the moduli, divide the scalar product by the product of the moduli and evaluate the inverse cosine of the result | M1 | |
| Obtain answer $21.6°$ | A1 | |
**Total: 3 marks**10 With respect to the origin $O$, the points $A$ and $B$ have position vectors given by $\overrightarrow { O A } = 6 \mathbf { i } + 2 \mathbf { j }$ and $\overrightarrow { O B } = 2 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }$. The midpoint of $O A$ is $M$. The point $N$ lying on $A B$, between $A$ and $B$, is such that $A N = 2 N B$.
\begin{enumerate}[label=(\alph*)]
\item Find a vector equation for the line through $M$ and $N$.\\
The line through $M$ and $N$ intersects the line through $O$ and $B$ at the point $P$.
\item Find the position vector of $P$.
\item Calculate angle $O P M$, giving your answer in degrees.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2020 Q10 [11]}}