| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2022 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Cross Product & Distances |
| Type | Find parameter value for geometric condition |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question requiring cross products, angle conditions, and geometric applications. While each part uses standard techniques (dot product for angles, cross product properties, area formula), the combination of three distinct parts requiring different vector methods, plus the algebraic manipulation needed (especially solving the quadratic in part c), places this above average difficulty for A-level but not exceptionally hard for FP1 students. |
| Spec | 1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication4.04c Scalar product: calculate and use for angles4.04g Vector product: a x b perpendicular vector |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(( | OB | =)\sqrt{4^2+(2p)^2+1^2}\left(=\sqrt{17+4p^2}\right)\) |
| \(\cos 45 = \frac{4}{\sqrt{17+4p^2}} \Rightarrow p = \ldots\) | M1 | Complete method to find value of \(p\); sets \(\cos 45 = 4/ |
| \(p = \pm\frac{\sqrt{15}}{2}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\overrightarrow{OA} \times \overrightarrow{OB} = \begin{pmatrix}2+2p\\-6\\4p-8\end{pmatrix}\) | B1 | Correct vector product (allow if seen anywhere in question) |
| E.g. Sets \(\begin{pmatrix}2+2p\\-6\\4p-8\end{pmatrix} = \begin{pmatrix}4\lambda\\-p\lambda\\2\lambda\end{pmatrix}\) and solves to find value of \(p\) | M1 | Complete method to find value of \(p\) |
| \(p = 3\) only | A1 | Complete argument leading to \(p=3\) only; where method gives more than one value, other values must be rejected |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{1}{2} | \overrightarrow{OA}\times\overrightarrow{OB} | = 3\sqrt{2} \Rightarrow (2+2p)^2+(-6)^2+(4p-8)^2 = (6\sqrt{2})^2\) |
| Solves 3TQ: \(20p^2 - 56p + 32 = 0 \Rightarrow p = \ldots\) | dM1 | Dependent on M1. Solves 3TQ to find value for \(p\) |
| \(p = 2,\ \frac{4}{5}\) | A1 | Note: allow if \(\overrightarrow{OA}\times\overrightarrow{OB}\) was correct apart from j component sign |
## Question 3:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(|OB|=)\sqrt{4^2+(2p)^2+1^2}\left(=\sqrt{17+4p^2}\right)$ | B1 | Correct expression for magnitude of $\overrightarrow{OB}$ |
| $\cos 45 = \frac{4}{\sqrt{17+4p^2}} \Rightarrow p = \ldots$ | M1 | Complete method to find value of $p$; sets $\cos 45 = 4/|\overrightarrow{OB}|$ and solves |
| $p = \pm\frac{\sqrt{15}}{2}$ | A1 | |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\overrightarrow{OA} \times \overrightarrow{OB} = \begin{pmatrix}2+2p\\-6\\4p-8\end{pmatrix}$ | B1 | Correct vector product (allow if seen anywhere in question) |
| E.g. Sets $\begin{pmatrix}2+2p\\-6\\4p-8\end{pmatrix} = \begin{pmatrix}4\lambda\\-p\lambda\\2\lambda\end{pmatrix}$ and solves to find value of $p$ | M1 | Complete method to find value of $p$ |
| $p = 3$ only | A1 | Complete argument leading to $p=3$ only; where method gives more than one value, other values must be rejected |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{1}{2}|\overrightarrow{OA}\times\overrightarrow{OB}| = 3\sqrt{2} \Rightarrow (2+2p)^2+(-6)^2+(4p-8)^2 = (6\sqrt{2})^2$ | M1 | Complete method to set up polynomial in $p$; sets half magnitude of vector product $= 3\sqrt{2}$ and reaches quadratic expression in $p$ |
| Solves 3TQ: $20p^2 - 56p + 32 = 0 \Rightarrow p = \ldots$ | dM1 | Dependent on M1. Solves 3TQ to find value for $p$ |
| $p = 2,\ \frac{4}{5}$ | A1 | Note: allow if $\overrightarrow{OA}\times\overrightarrow{OB}$ was correct apart from **j** component sign |
---\begin{enumerate}
\item With respect to a fixed origin $O$, the points $A$ and $B$ have coordinates $( 2,2 , - 1 )$ and ( $4,2 p , 1$ ) respectively, where $p$ is a constant.
\end{enumerate}
For each of the following, determine the possible values of $p$ for which,\\
(a) $O B$ makes an angle of $45 ^ { \circ }$ with the positive $x$-axis\\
(b) $\overrightarrow { O A } \times \overrightarrow { O B }$ is parallel to $\left( \begin{array} { r } 4 \\ - p \\ 2 \end{array} \right)$\\
(c) the area of triangle $O A B$ is $3 \sqrt { 2 }$
\hfill \mbox{\textit{Edexcel FP1 2022 Q3 [9]}}