| Exam Board | Edexcel |
|---|---|
| Module | AS Paper 1 (AS Paper 1) |
| Year | 2019 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Angle between two vectors |
| Difficulty | Standard +0.3 Part (i) requires geometric insight about parallel vectors (when triangle inequality becomes equality), which is conceptually interesting but accessible. Part (ii) is a standard application of the cosine rule in vector form with given magnitudes and angle, requiring systematic algebraic manipulation but no novel techniques. Overall slightly easier than average due to being a routine multi-step problem with clear methods. |
| Spec | 1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication1.10g Problem solving with vectors: in geometry |
| Answer | Marks | Guidance |
|---|---|---|
| Explains that a and b lie in the same direction oe | B1 | Accept any valid response e.g. "The lines are collinear." Condone "They are parallel." Mark positively. ISW after a correct answer. Do not accept "the length of line \(a+b\) is the same as the length of line \(a\) + the length of line \(b\)". Do not accept \( |
| Answer | Marks | Guidance |
|---|---|---|
| A triangle showing \(3\), \(6\) and \(30°\) in the correct positions, with \( | \mathbf{m} | =3\), \( |
| Answer | Marks | Guidance |
|---|---|---|
| Attempts \(\dfrac{\sin 30°}{6} = \dfrac{\sin\theta}{3}\) | M1 | Correct sine rule statement with sides and angles in correct positions. If a triangle is drawn, angles and sides must be in correct positions. Allow recovery from negative lengths. If no diagram drawn, correct sine rule gives M1 M1. Do not accept \(\dfrac{\sin 30°}{6} = \dfrac{\sin\theta}{-3}\) (M0). |
| Answer | Marks | Guidance |
|---|---|---|
| \(\theta = \text{awrt } 14.5°\) | A1 | \(\theta = \text{awrt } 14.5°\) |
| Answer | Marks | Guidance |
|---|---|---|
| Angle between vector m and vector \(\mathbf{m}-\mathbf{n}\) is awrt \(135.5°\) | A1 | CSO awrt \(135.5°\) |
## Question 16:
### Part (i):
Explains that **a** and **b** lie in the same direction oe | B1 | Accept any valid response e.g. "The lines are collinear." Condone "They are parallel." Mark positively. ISW after a correct answer. Do not accept "the length of line $a+b$ is the same as the length of line $a$ + the length of line $b$". Do not accept $|\mathbf{a}|$ and $|\mathbf{b}|$ are parallel.
**Total: (1)**
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### Part (ii):
A triangle showing $3$, $6$ and $30°$ in the correct positions, with $|\mathbf{m}|=3$, $|\mathbf{m}-\mathbf{n}|=6$, angle of $30°$ between $\mathbf{m}$ and $-\mathbf{n}$ | M1 | Look for $6$ opposite $30°$ with another side of $3$. Condone the triangle not being obtuse angled and not being to scale. Do not condone negative lengths in the triangle (automatically M0).
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Attempts $\dfrac{\sin 30°}{6} = \dfrac{\sin\theta}{3}$ | M1 | Correct sine rule statement with sides and angles in correct positions. If a triangle is drawn, angles and sides must be in correct positions. Allow recovery from negative lengths. If no diagram drawn, correct sine rule gives M1 M1. Do not accept $\dfrac{\sin 30°}{6} = \dfrac{\sin\theta}{-3}$ (M0).
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$\theta = \text{awrt } 14.5°$ | A1 | $\theta = \text{awrt } 14.5°$
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Angle between vector **m** and vector $\mathbf{m}-\mathbf{n}$ is awrt $135.5°$ | A1 | CSO awrt $135.5°$
**Total: (4)**
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**(5 marks)**\begin{enumerate}
\item (i) Two non-zero vectors, $\mathbf { a }$ and $\mathbf { b }$, are such that
\end{enumerate}
$$| \mathbf { a } + \mathbf { b } | = | \mathbf { a } | + | \mathbf { b } |$$
Explain, geometrically, the significance of this statement.\\
(ii) Two different vectors, $\mathbf { m }$ and $\mathbf { n }$, are such that $| \mathbf { m } | = 3$ and $| \mathbf { m } - \mathbf { n } | = 6$
The angle between vector $\mathbf { m }$ and vector $\mathbf { n }$ is $30 ^ { \circ }$\\
Find the angle between vector $\mathbf { m }$ and vector $\mathbf { m } - \mathbf { n }$, giving your answer, in degrees, to one decimal place.
\hfill \mbox{\textit{Edexcel AS Paper 1 2019 Q16 [5]}}