| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2021 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Position vector at time t (constant velocity) |
| Difficulty | Easy -1.3 This is a straightforward kinematics question requiring basic vector arithmetic to find velocity (displacement/time), then verifying the stone passes through O by checking if position vector equals zero at some time, and finally calculating speed as magnitude of velocity. All steps are routine applications of standard formulas with no problem-solving insight needed. |
| Spec | 1.10e Position vectors: and displacement1.10f Distance between points: using position vectors |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| Attempts to compare two position vectors; e.g. \((-24\mathbf{i}-10\mathbf{j}) = -2\times(12\mathbf{i}+5\mathbf{j})\) | M1 | Allow attempt using two of \(\overrightarrow{AO}\), \(\overrightarrow{OB}\) or \(\overrightarrow{AB}\) |
| States \(\overrightarrow{AO}\) is parallel to \(\overrightarrow{OB}\), therefore stone passes through point \(O\) | A1 | Must state parallel AND conclude passes through \(O\); alternatively show point \((0,0)\) lies on the line |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \(AB = \sqrt{(12+24)^2+(10+5)^2}\) | M1 | Correct method for distance \(AB\); condone slips but expect \(\sqrt{a^2+b^2}\) where \(a\) or \(b\) is correct |
| Speed \(= \dfrac{\sqrt{(12+24)^2+(10+5)^2}}{4}\) | dM1 | Dependent on previous mark; look for \(\dfrac{\text{distance }AB}{4}\) |
| Speed \(= 9.75\text{ ms}^{-1}\) | A1 | Requires units |
# Question 4:
## Part (a):
| Working/Answer | Marks | Guidance |
|---|---|---|
| Attempts to compare two position vectors; e.g. $(-24\mathbf{i}-10\mathbf{j}) = -2\times(12\mathbf{i}+5\mathbf{j})$ | M1 | Allow attempt using two of $\overrightarrow{AO}$, $\overrightarrow{OB}$ or $\overrightarrow{AB}$ |
| States $\overrightarrow{AO}$ is parallel to $\overrightarrow{OB}$, therefore stone passes through point $O$ | A1 | Must state parallel AND conclude passes through $O$; alternatively show point $(0,0)$ lies on the line |
## Part (b):
| Working/Answer | Marks | Guidance |
|---|---|---|
| $AB = \sqrt{(12+24)^2+(10+5)^2}$ | M1 | Correct method for distance $AB$; condone slips but expect $\sqrt{a^2+b^2}$ where $a$ or $b$ is correct |
| Speed $= \dfrac{\sqrt{(12+24)^2+(10+5)^2}}{4}$ | dM1 | Dependent on previous mark; look for $\dfrac{\text{distance }AB}{4}$ |
| Speed $= 9.75\text{ ms}^{-1}$ | A1 | Requires units |
---\begin{enumerate}
\item \hspace{0pt} [In this question the unit vectors $\mathbf { i }$ and $\mathbf { j }$ are due east and due north respectively.]
\end{enumerate}
A stone slides horizontally across ice.\\
Initially the stone is at the point $A ( - 24 \mathbf { i } - 10 \mathbf { j } ) \mathrm { m }$ relative to a fixed point $O$.\\
After 4 seconds the stone is at the point $B ( 12 \mathbf { i } + 5 \mathbf { j } )$ m relative to the fixed point $O$.\\
The motion of the stone is modelled as that of a particle moving in a straight line at constant speed.
Using the model,\\
(a) prove that the stone passes through $O$,\\
(b) calculate the speed of the stone.
\hfill \mbox{\textit{Edexcel AS Paper 1 2021 Q4 [5]}}