| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2022 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Ratio division of line segment |
| Difficulty | Moderate -0.8 This is a straightforward vector question testing basic operations: vector subtraction (finding QR from two position vectors), magnitude calculation, and ratio division of a line segment using the standard formula. All techniques are routine for AS-level with no problem-solving insight required, making it easier than average but not trivial since it requires multiple steps and careful arithmetic. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication1.10g Problem solving with vectors: in geometry |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\overrightarrow{QR} = \overrightarrow{PR} - \overrightarrow{PQ} = 13\mathbf{i} - 15\mathbf{j} - (3\mathbf{i}+5\mathbf{j})\) | M1 | Attempts subtraction either way; cannot be awarded for adding the two vectors; may be implied by one correct component |
| \(= 10\mathbf{i} - 20\mathbf{j}\) | A1 | Allow \(\begin{pmatrix}10\\-20\end{pmatrix}\) but not \(\begin{pmatrix}10\mathbf{i}\\-20\mathbf{j}\end{pmatrix}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\lvert\overrightarrow{QR}\rvert = \sqrt{10^2 + (-20)^2}\) | M1 | Correct use of Pythagoras; attempts to square and add before square rooting; follow through on their \(\overrightarrow{QR}\) |
| \(= 10\sqrt{5}\) | A1ft | Follow through of the form \(\pm10\mathbf{i} \pm 20\mathbf{j}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\overrightarrow{PS} = \overrightarrow{PQ} + \tfrac{3}{5}\overrightarrow{QR} = 3\mathbf{i}+5\mathbf{j}+\tfrac{3}{5}(10\mathbf{i}-20\mathbf{j})=\ldots\) or \(\overrightarrow{PS} = \overrightarrow{PR}+\tfrac{2}{5}\overrightarrow{RQ} = 13\mathbf{i}-15\mathbf{j}+\tfrac{2}{5}(-10\mathbf{i}+20\mathbf{j})=\ldots\) | M1 | Full attempt at finding \(\overrightarrow{PS}\); must attempt \(\overrightarrow{PQ}\pm\tfrac{3}{5}\overrightarrow{QR}\) or \(\overrightarrow{PR}\pm\tfrac{2}{5}\overrightarrow{RQ}\); cannot score for just stating the expression; follow through on their \(\overrightarrow{QR}\) |
| \(= 9\mathbf{i} - 7\mathbf{j}\) | A1 | Allow \(\begin{pmatrix}9\\-7\end{pmatrix}\); only withhold if mark already withheld in (a) for \(\begin{pmatrix}10\mathbf{i}\\-20\mathbf{j}\end{pmatrix}\) |
## Question 3:
### Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\overrightarrow{QR} = \overrightarrow{PR} - \overrightarrow{PQ} = 13\mathbf{i} - 15\mathbf{j} - (3\mathbf{i}+5\mathbf{j})$ | M1 | Attempts subtraction either way; cannot be awarded for adding the two vectors; may be implied by one correct component |
| $= 10\mathbf{i} - 20\mathbf{j}$ | A1 | Allow $\begin{pmatrix}10\\-20\end{pmatrix}$ but not $\begin{pmatrix}10\mathbf{i}\\-20\mathbf{j}\end{pmatrix}$ |
### Part (b)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\lvert\overrightarrow{QR}\rvert = \sqrt{10^2 + (-20)^2}$ | M1 | Correct use of Pythagoras; attempts to square and add before square rooting; follow through on their $\overrightarrow{QR}$ |
| $= 10\sqrt{5}$ | A1ft | Follow through of the form $\pm10\mathbf{i} \pm 20\mathbf{j}$ |
### Part (c)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\overrightarrow{PS} = \overrightarrow{PQ} + \tfrac{3}{5}\overrightarrow{QR} = 3\mathbf{i}+5\mathbf{j}+\tfrac{3}{5}(10\mathbf{i}-20\mathbf{j})=\ldots$ or $\overrightarrow{PS} = \overrightarrow{PR}+\tfrac{2}{5}\overrightarrow{RQ} = 13\mathbf{i}-15\mathbf{j}+\tfrac{2}{5}(-10\mathbf{i}+20\mathbf{j})=\ldots$ | M1 | Full attempt at finding $\overrightarrow{PS}$; must attempt $\overrightarrow{PQ}\pm\tfrac{3}{5}\overrightarrow{QR}$ or $\overrightarrow{PR}\pm\tfrac{2}{5}\overrightarrow{RQ}$; cannot score for just stating the expression; follow through on their $\overrightarrow{QR}$ |
| $= 9\mathbf{i} - 7\mathbf{j}$ | A1 | Allow $\begin{pmatrix}9\\-7\end{pmatrix}$; only withhold if mark already withheld in (a) for $\begin{pmatrix}10\mathbf{i}\\-20\mathbf{j}\end{pmatrix}$ |\begin{enumerate}
\item The triangle $P Q R$ is such that $\overrightarrow { P Q } = 3 \mathbf { i } + 5 \mathbf { j }$ and $\overrightarrow { P R } = 13 \mathbf { i } - 15 \mathbf { j }$\\
(a) Find $\overrightarrow { Q R }$\\
(b) Hence find $| \overrightarrow { Q R } |$ giving your answer as a simplified surd.
\end{enumerate}
The point $S$ lies on the line segment $Q R$ so that $Q S : S R = 3 : 2$\\
(c) Find $\overrightarrow { P S }$
\hfill \mbox{\textit{Edexcel AS Paper 1 2022 Q3 [6]}}