| Exam Board | Edexcel |
|---|---|
| Module | Paper 1 (Paper 1) |
| Year | 2020 |
| Session | October |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Parallel and perpendicular lines |
| Difficulty | Moderate -0.5 This is a straightforward vectors question requiring basic vector subtraction for part (a) and showing parallel vectors for part (b). The trapezium proof only needs demonstrating one pair of parallel sides (OC parallel to AB) by showing direction vectors are scalar multiples. Below average difficulty as it's routine application of standard techniques with no problem-solving insight required. |
| Spec | 1.10d Vector operations: addition and scalar multiplication1.10g Problem solving with vectors: in geometry |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\overrightarrow{AB} = (3\mathbf{i}-3\mathbf{j}-4\mathbf{k})-(2\mathbf{i}+5\mathbf{j}-6\mathbf{k})\) | M1 | Attempts to subtract either way around. Implied by two of \(\pm 1\mathbf{i} \pm 8\mathbf{j} \pm 2\mathbf{k}\) |
| \(= \mathbf{i}-8\mathbf{j}+2\mathbf{k}\) | A1 | Accept column vector form but not \((1,-8,2)\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| States \(\overrightarrow{OC} = 2 \times \overrightarrow{AB}\) | M1 | Compares \(\mathbf{i}-8\mathbf{j}+2\mathbf{k}\) with \(2\mathbf{i}-16\mathbf{j}+4\mathbf{k}\) by stating any one of: \(\overrightarrow{OC}=2\times\overrightarrow{AB}\), or column vector equality, or \(\overrightarrow{OC}=\lambda\times\overrightarrow{AB}\). May be awarded if AB subtracted wrong way or one numerical slip. |
| Explains that as \(OC\) is parallel to \(AB\), so \(OABC\) is a trapezium | A1 | Full explanation required. Requires fully correct calculations so part (a) must be \(\overrightarrow{AB}=(\mathbf{i}-8\mathbf{j}+2\mathbf{k})\). Needs reason and minimal conclusion. |
## Question 3:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\overrightarrow{AB} = (3\mathbf{i}-3\mathbf{j}-4\mathbf{k})-(2\mathbf{i}+5\mathbf{j}-6\mathbf{k})$ | M1 | Attempts to subtract either way around. Implied by two of $\pm 1\mathbf{i} \pm 8\mathbf{j} \pm 2\mathbf{k}$ |
| $= \mathbf{i}-8\mathbf{j}+2\mathbf{k}$ | A1 | Accept column vector form but not $(1,-8,2)$ |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| States $\overrightarrow{OC} = 2 \times \overrightarrow{AB}$ | M1 | Compares $\mathbf{i}-8\mathbf{j}+2\mathbf{k}$ with $2\mathbf{i}-16\mathbf{j}+4\mathbf{k}$ by stating any one of: $\overrightarrow{OC}=2\times\overrightarrow{AB}$, or column vector equality, or $\overrightarrow{OC}=\lambda\times\overrightarrow{AB}$. May be awarded if AB subtracted wrong way or one numerical slip. |
| Explains that as $OC$ is parallel to $AB$, so $OABC$ is a trapezium | A1 | Full explanation required. Requires fully correct calculations so part (a) must be $\overrightarrow{AB}=(\mathbf{i}-8\mathbf{j}+2\mathbf{k})$. Needs reason and minimal conclusion. |
---\begin{enumerate}
\item Relative to a fixed origin $O$
\end{enumerate}
\begin{itemize}
\item point $A$ has position vector $2 \mathbf { i } + 5 \mathbf { j } - 6 \mathbf { k }$
\item point $B$ has position vector $3 \mathbf { i } - 3 \mathbf { j } - 4 \mathbf { k }$
\item point $C$ has position vector $2 \mathbf { i } - 16 \mathbf { j } + 4 \mathbf { k }$\\
(a) Find $\overrightarrow { A B }$\\
(b) Show that quadrilateral $O A B C$ is a trapezium, giving reasons for your answer.
\end{itemize}
\hfill \mbox{\textit{Edexcel Paper 1 2020 Q3 [4]}}