| Exam Board | OCR |
|---|---|
| Module | H240/01 (Pure Mathematics) |
| Year | 2020 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Ratio division of line segment |
| Difficulty | Moderate -0.3 This is a standard vectors question testing ratio division and collinearity. Part (a) requires routine position vector manipulation, part (b) is essentially given (just explaining why vectors on a line are parallel), and part (c) involves equating two expressions for PR and solving for a ratio. While multi-part, each step follows textbook methods with no novel insight required, making it slightly easier than average. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10b Vectors in 3D: i,j,k notation1.10d Vector operations: addition and scalar multiplication1.10e Position vectors: and displacement1.10f Distance between points: using position vectors1.10g Problem solving with vectors: in geometry |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\overrightarrow{BQ}=\frac{1}{2}(\mathbf{a}-\mathbf{b})\) | B1 | Correct \(\overrightarrow{BQ}\) or \(\overrightarrow{QB}\); or any correct vector involving \(Q\), but must be clear which vector it is |
| \(\overrightarrow{PQ}=\frac{1}{4}\mathbf{b}+\frac{1}{2}(\mathbf{a}-\mathbf{b})=\frac{1}{2}\mathbf{a}-\frac{1}{4}\mathbf{b}\) | B1 | Correct \(\overrightarrow{PQ}\); must be simplified to two terms; SC allow B1 if correct unsimplified \(PQ\) is seen but individual vectors not explicit |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\overrightarrow{PR}\) has the same direction as \(\overrightarrow{PQ}\), so vector must be a multiple of \(\overrightarrow{PQ}\) | B1 | Explain parallel (or collinear) vectors have direction vectors that are multiples of each other; allow 'gradient' for 'direction', or 'they are on the same straight line', but must state or use 'multiple'; clear detail of scaling factor |
| So \(\overrightarrow{PR}=\lambda\left(\frac{1}{2}\mathbf{a}-\frac{1}{4}\mathbf{b}\right)=\frac{1}{4}\lambda(2\mathbf{a}-\mathbf{b})\) \(=k(2\mathbf{a}-\mathbf{b})\) A.G. | B1 | Show given answer convincingly |
| [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\overrightarrow{AR}=-\mathbf{a}+\frac{3}{4}\mathbf{b}+k(2\mathbf{a}-\mathbf{b})\) | B1 | Correct expression for \(\overrightarrow{AR}\) (or \(\overrightarrow{OR}\)), in terms of \(k\); could use \(A\) to \(Q\) to \(R\) (condone if \(k\) still used) |
| \(\overrightarrow{AR}\) multiple of \(\mathbf{a}\) only, \(\frac{3}{4}\mathbf{b}-k\mathbf{b}=0\) | M1 | Use coefficient of \(\mathbf{b}=0\); must be used in \(\overrightarrow{AR}\) or \(\overrightarrow{OR}\) |
| Obtain \(k=\frac{3}{4}\) | A1 | Obtain correct value for \(k\); may get different value for their \(k\) |
| ratio \(OA:AR=2:1\) | A1 | Correct ratio (allow \(1:\frac{1}{2}\)) oe; answer only is 0, as question says 'determine' |
| [4] |
# Question 5:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\overrightarrow{BQ}=\frac{1}{2}(\mathbf{a}-\mathbf{b})$ | B1 | Correct $\overrightarrow{BQ}$ or $\overrightarrow{QB}$; or any correct vector involving $Q$, but must be clear which vector it is |
| $\overrightarrow{PQ}=\frac{1}{4}\mathbf{b}+\frac{1}{2}(\mathbf{a}-\mathbf{b})=\frac{1}{2}\mathbf{a}-\frac{1}{4}\mathbf{b}$ | B1 | Correct $\overrightarrow{PQ}$; must be simplified to two terms; SC allow B1 if correct unsimplified $PQ$ is seen but individual vectors not explicit |
| **[2]** | | |
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\overrightarrow{PR}$ has the same direction as $\overrightarrow{PQ}$, so vector must be a multiple of $\overrightarrow{PQ}$ | B1 | Explain parallel (or collinear) vectors have direction vectors that are multiples of each other; allow 'gradient' for 'direction', or 'they are on the same straight line', but must state or use 'multiple'; clear detail of scaling factor |
| So $\overrightarrow{PR}=\lambda\left(\frac{1}{2}\mathbf{a}-\frac{1}{4}\mathbf{b}\right)=\frac{1}{4}\lambda(2\mathbf{a}-\mathbf{b})$ $=k(2\mathbf{a}-\mathbf{b})$ **A.G.** | B1 | Show given answer convincingly |
| **[2]** | | |
## Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\overrightarrow{AR}=-\mathbf{a}+\frac{3}{4}\mathbf{b}+k(2\mathbf{a}-\mathbf{b})$ | B1 | Correct expression for $\overrightarrow{AR}$ (or $\overrightarrow{OR}$), in terms of $k$; could use $A$ to $Q$ to $R$ (condone if $k$ still used) |
| $\overrightarrow{AR}$ multiple of $\mathbf{a}$ only, $\frac{3}{4}\mathbf{b}-k\mathbf{b}=0$ | M1 | Use coefficient of $\mathbf{b}=0$; must be used in $\overrightarrow{AR}$ or $\overrightarrow{OR}$ |
| Obtain $k=\frac{3}{4}$ | A1 | Obtain correct value for $k$; may get different value for their $k$ |
| ratio $OA:AR=2:1$ | A1 | Correct ratio (allow $1:\frac{1}{2}$) oe; answer only is 0, as question says 'determine' |
| **[4]** | | |5\\
\includegraphics[max width=\textwidth, alt={}, center]{febe231d-200a-4957-b41b-de5b9be98b0a-5_424_583_255_244}
The diagram shows points $A$ and $B$, which have position vectors $\mathbf { a }$ and $\mathbf { b }$ with respect to an origin $O$. $P$ is the point on $O B$ such that $O P : P B = 3 : 1$ and $Q$ is the midpoint of $A B$.
\begin{enumerate}[label=(\alph*)]
\item Find $\overrightarrow { P Q }$ in terms of $\mathbf { a }$ and $\mathbf { b }$.
The line $O A$ is extended to a point $R$, so that $P Q R$ is a straight line.
\item Explain why $\overrightarrow { P R } = k ( 2 \mathbf { a } - \mathbf { b } )$, where $k$ is a constant.
\item Hence determine the ratio $O A : A R$.
\end{enumerate}
\hfill \mbox{\textit{OCR H240/01 2020 Q5 [8]}}