| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Perpendicular distance point to line |
| Difficulty | Standard +0.3 This is a straightforward 3D vectors question requiring standard techniques: finding position vectors from a diagram, writing a line equation in vector form, and calculating perpendicular distance using the formula |a×b|/|b|. The setup is clear with a simple pyramid geometry, and part (b) is a 'show that' which confirms students are on track. Slightly above average due to 3D visualization and the vector product calculation, but all techniques are routine for Further Maths Pure students. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10b Vectors in 3D: i,j,k notation1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State \(\overrightarrow{OM} = 4\mathbf{i} + 2\mathbf{j}\) | B1 | |
| Use a correct method to find \(\overrightarrow{ON}\) | M1 | |
| Obtain answer \(3\mathbf{j} + \mathbf{k}\) | A1 | |
| Use a correct method to find a line equation for \(MN\) | M1 | |
| Obtain answer \(\mathbf{r} = 3\mathbf{j} + \mathbf{k} + \lambda(4\mathbf{i} - \mathbf{j} - \mathbf{k})\), or equivalent | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Taking a general point \(P\) on \(MN\), form an equation in \(\lambda\) by either equating a relevant scalar product to zero or equating the derivative of \(\overrightarrow{OP}\) to zero or using Pythagoras in triangle \(OPM\) or \(OPN\) | M1 | |
| Obtain \(\lambda = \frac{2}{9}\) | A1 | OE |
| Use correct method to find \(OP\) | M1 | |
| Obtain the given answer correctly | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use a scalar product to find the projection of \(OM\) (or \(ON\)) on \(MN\) | M1 | |
| Obtain answer \(\frac{14}{\sqrt{18}}\) \(\left(\text{or } \frac{4}{\sqrt{18}}\right)\) | A1 | |
| Use Pythagoras to obtain the perpendicular | M1 | |
| Obtain the given answer correctly | A1 |
## Question 8(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State $\overrightarrow{OM} = 4\mathbf{i} + 2\mathbf{j}$ | B1 | |
| Use a correct method to find $\overrightarrow{ON}$ | M1 | |
| Obtain answer $3\mathbf{j} + \mathbf{k}$ | A1 | |
| Use a correct method to find a line equation for $MN$ | M1 | |
| Obtain answer $\mathbf{r} = 3\mathbf{j} + \mathbf{k} + \lambda(4\mathbf{i} - \mathbf{j} - \mathbf{k})$, or equivalent | A1 | |
## Question 8(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Taking a general point $P$ on $MN$, form an equation in $\lambda$ by either equating a relevant scalar product to zero or equating the derivative of $\overrightarrow{OP}$ to zero or using Pythagoras in triangle $OPM$ or $OPN$ | M1 | |
| Obtain $\lambda = \frac{2}{9}$ | A1 | OE |
| Use correct method to find $OP$ | M1 | |
| Obtain the given answer correctly | A1 | |
**Alternative method:**
| Answer | Mark | Guidance |
|--------|------|----------|
| Use a scalar product to find the projection of $OM$ (or $ON$) on $MN$ | M1 | |
| Obtain answer $\frac{14}{\sqrt{18}}$ $\left(\text{or } \frac{4}{\sqrt{18}}\right)$ | A1 | |
| Use Pythagoras to obtain the perpendicular | M1 | |
| Obtain the given answer correctly | A1 | |8\\
\includegraphics[max width=\textwidth, alt={}, center]{bbe57fc0-a8a5-4fe5-a637-4f9db00bdc13-10_588_789_260_678}
In the diagram, $O A B C D$ is a pyramid with vertex $D$. The horizontal base $O A B C$ is a square of side 4 units. The edge $O D$ is vertical and $O D = 4$ units. The unit vectors $\mathbf { i } , \mathbf { j }$ and $\mathbf { k }$ are parallel to $O A , O C$ and $O D$ respectively.
The midpoint of $A B$ is $M$ and the point $N$ on $C D$ is such that $D N = 3 N C$.
\begin{enumerate}[label=(\alph*)]
\item Find a vector equation for the line through $M$ and $N$.
\item Show that the length of the perpendicular from $O$ to $M N$ is $\frac { 1 } { 3 } \sqrt { 82 }$.\\
$9 \quad$ Let $\mathrm { f } ( x ) = \frac { 1 } { ( 9 - x ) \sqrt { x } }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2021 Q8 [9]}}