| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2020 |
| Session | March |
| 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 standard 3D vectors question involving a cuboid setup. Parts (a) and (b) require routine position vector calculations and forming a line equation. Part (c) asks for the foot of perpendicular from a point to a line, which is a textbook technique (using the dot product condition that DP ⊥ direction vector). While it has multiple parts and requires careful coordinate work, it follows standard procedures without requiring novel insight. Slightly easier than average due to the structured cuboid context making visualization straightforward. |
| Spec | 1.07s Parametric and implicit differentiation |
| Answer | Marks |
|---|---|
| Obtain \(\overrightarrow{OM} = 2\mathbf{i}+\mathbf{j}\) | B1 |
| Use a correct method to find \(\overrightarrow{MN}\) | M1 |
| Obtain \(\overrightarrow{MN} = -\mathbf{i}+2\mathbf{j}+2\mathbf{k}\) | A1 |
| Answer | Marks |
|---|---|
| Use a correct method to form an equation for \(MN\) | M1 |
| Obtain \(\mathbf{r} = 2\mathbf{i}+\mathbf{j}+\lambda(-\mathbf{i}+2\mathbf{j}+2\mathbf{k})\), or equivalent | A1 |
| Answer | Marks |
|---|---|
| Find \(\overrightarrow{DP}\) for a point \(P\) on \(MN\) with parameter \(\lambda\), e.g. \((2-\lambda, 1+2\lambda, -2+2\lambda)\) | B1 |
| Equate scalar product of \(\overrightarrow{DP}\) and a direction vector for \(MN\) to zero and solve for \(\lambda\) | M1 |
| Obtain \(\lambda = \frac{4}{9}\) | A1 |
| State that the position vector of \(P\) is \(\frac{14}{9}\mathbf{i}+\frac{17}{9}\mathbf{j}+\frac{8}{9}\mathbf{k}\) | A1 |
## Question 8(a):
| Obtain $\overrightarrow{OM} = 2\mathbf{i}+\mathbf{j}$ | B1 | |
| Use a correct method to find $\overrightarrow{MN}$ | M1 | |
| Obtain $\overrightarrow{MN} = -\mathbf{i}+2\mathbf{j}+2\mathbf{k}$ | A1 | |
## Question 8(b):
| Use a correct method to form an equation for $MN$ | M1 | |
| Obtain $\mathbf{r} = 2\mathbf{i}+\mathbf{j}+\lambda(-\mathbf{i}+2\mathbf{j}+2\mathbf{k})$, or equivalent | A1 | |
## Question 8(c):
| Find $\overrightarrow{DP}$ for a point $P$ on $MN$ with parameter $\lambda$, e.g. $(2-\lambda, 1+2\lambda, -2+2\lambda)$ | B1 | |
| Equate scalar product of $\overrightarrow{DP}$ and a direction vector for $MN$ to zero and solve for $\lambda$ | M1 | |
| Obtain $\lambda = \frac{4}{9}$ | A1 | |
| State that the position vector of $P$ is $\frac{14}{9}\mathbf{i}+\frac{17}{9}\mathbf{j}+\frac{8}{9}\mathbf{k}$ | A1 | |8\\
\includegraphics[max width=\textwidth, alt={}, center]{8f81a526-783c-4321-b540-c9deccfee17b-12_639_713_262_715}
In the diagram, $O A B C D E F G$ is a cuboid in which $O A = 2$ units, $O C = 3$ units and $O D = 2$ units. Unit vectors $\mathbf { i } , \mathbf { j }$ and $\mathbf { k }$ are parallel to $O A , O C$ and $O D$ respectively. The point $M$ on $A B$ is such that $M B = 2 A M$. The midpoint of $F G$ is $N$.
\begin{enumerate}[label=(\alph*)]
\item Express the vectors $\overrightarrow { O M }$ and $\overrightarrow { M N }$ in terms of $\mathbf { i } , \mathbf { j }$ and $\mathbf { k }$.
\item Find a vector equation for the line through $M$ and $N$.
\item Find the position vector of $P$, the foot of the perpendicular from $D$ to the line through $M$ and $N$. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2020 Q8 [9]}}