| Exam Board | AQA |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2021 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Distance between two moving objects |
| Difficulty | Standard +0.8 This is a multi-step vectors problem requiring understanding of parallel lines, distance between skew/parallel lines, and geometric reasoning. Part (c) asks students to find the perpendicular distance between parallel lines given distances at two time points, requiring visualization and application of Pythagoras' theorem or perpendicular distance formula. The 5-mark allocation and 'fully justify' requirement indicate this goes beyond routine calculation, requiring problem-solving insight about the geometry of parallel lines in motion. |
| Spec | 1.10e Position vectors: and displacement1.10f Distance between points: using position vectors1.10h Vectors in kinematics: uniform acceleration in vector form |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| When two vectors are parallel, one is a scalar multiple of the other | E1 (AO 2.4) | Must refer to scalar multiple or show algebraically in form \(\mathbf{a} = k\mathbf{b}\). Do not accept 'factor' |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Amba's velocity \(= 0.8(2.8\mathbf{i} + 9.6\mathbf{j})\); Amba's speed \(= \sqrt{2.24^2 + 7.68^2} = 8\) m s\(^{-1}\) | B1 (AO 1.1b) | Verifies that \(k = 0.8\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\mathbf{s} = 4 \times \begin{bmatrix}2.24\\7.68\end{bmatrix} = \begin{bmatrix}8.96\\30.72\end{bmatrix}\) | B1 (AO 1.1b) | Finds displacement when \(t = 4\) using \(\mathbf{s} = \mathbf{u}t\); PI by correct position vector |
| \(\mathbf{r} = \begin{bmatrix}2\\-7\end{bmatrix} + \begin{bmatrix}8.96\\30.72\end{bmatrix}\) | M1 (AO 3.4) | Uses displacement with \(\begin{bmatrix}2\\-7\end{bmatrix}\) to find Amba's position vector |
| \(\mathbf{r} = \begin{bmatrix}10.96\\23.72\end{bmatrix}\) metres | A1 (AO 1.1b) | OE; condone missing or incorrect units |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Jo's speed \(= \sqrt{2.8^2 + 9.6^2} = 10\) m s\(^{-1}\) | M1 (AO 1.1a) | Finds Jo's speed; PI by obtaining 40 m for Jo's distance |
| Jo's distance \(= 4 \times 10 = 40\) m | A1 (AO 1.1b) | Finds distance travelled by Jo |
| Amba's distance \(= 8 \times 4 = 32\) m | B1 (AO 1.1b) | Finds distance travelled by Amba over 4 seconds |
| Appropriate method to determine required distance | M1 (AO 3.1b) | e.g. using triangle with sides 5, 32, 40 |
| \(\sqrt{5^2 - \left[\dfrac{40-32}{2}\right]^2} = 3\) m | A1 (AO 3.2a) | Distance \(= 3\) metres; must state units |
## Question 19(a)(i):
| Answer/Working | Marks | Guidance |
|---|---|---|
| When two vectors are parallel, one is a scalar multiple of the other | E1 (AO 2.4) | Must refer to scalar multiple or show algebraically in form $\mathbf{a} = k\mathbf{b}$. Do not accept 'factor' |
---
## Question 19(a)(ii):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Amba's velocity $= 0.8(2.8\mathbf{i} + 9.6\mathbf{j})$; Amba's speed $= \sqrt{2.24^2 + 7.68^2} = 8$ m s$^{-1}$ | B1 (AO 1.1b) | Verifies that $k = 0.8$ |
---
## Question 19(b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\mathbf{s} = 4 \times \begin{bmatrix}2.24\\7.68\end{bmatrix} = \begin{bmatrix}8.96\\30.72\end{bmatrix}$ | B1 (AO 1.1b) | Finds displacement when $t = 4$ using $\mathbf{s} = \mathbf{u}t$; PI by correct position vector |
| $\mathbf{r} = \begin{bmatrix}2\\-7\end{bmatrix} + \begin{bmatrix}8.96\\30.72\end{bmatrix}$ | M1 (AO 3.4) | Uses displacement with $\begin{bmatrix}2\\-7\end{bmatrix}$ to find Amba's position vector |
| $\mathbf{r} = \begin{bmatrix}10.96\\23.72\end{bmatrix}$ metres | A1 (AO 1.1b) | OE; condone missing or incorrect units |
---
## Question 19(c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Jo's speed $= \sqrt{2.8^2 + 9.6^2} = 10$ m s$^{-1}$ | M1 (AO 1.1a) | Finds Jo's speed; PI by obtaining 40 m for Jo's distance |
| Jo's distance $= 4 \times 10 = 40$ m | A1 (AO 1.1b) | Finds distance travelled by Jo |
| Amba's distance $= 8 \times 4 = 32$ m | B1 (AO 1.1b) | Finds distance travelled by Amba over 4 seconds |
| Appropriate method to determine required distance | M1 (AO 3.1b) | e.g. using triangle with sides 5, 32, 40 |
| $\sqrt{5^2 - \left[\dfrac{40-32}{2}\right]^2} = 3$ m | A1 (AO 3.2a) | Distance $= 3$ metres; must state units |19
\begin{enumerate}[label=(\alph*)]
\item (ii) Verify that $k = 0.8$ \\[0pt]
[1 mark] $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ \\
19
\item Find the position vector of Amba when $t = 4$ \\[0pt]
[3 marks] $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ $\_\_\_\_$ \\
19
\item At both $t = 0$ and $t = 4$ there is a distance of 5 metres between Jo and Amba's positions.
Determine the shortest distance between their two parallel lines of motion.\\
Fully justify your answer.\\
\includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-32_2492_1721_217_150}
\end{enumerate}
\hfill \mbox{\textit{AQA Paper 2 2021 Q19 [9]}}