| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2019 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Cross Product & Distances |
| Type | Volume of tetrahedron using scalar triple product |
| Difficulty | Challenging +1.2 This question requires finding intersection points of lines with a plane, computing a quadrilateral area using vectors, and applying the scalar triple product formula for parallelepiped volume. While it involves multiple steps and Further Maths content (scalar triple product), each component is a standard application of taught techniques without requiring novel insight or complex problem-solving strategies. |
| Spec | 4.04b Plane equations: cartesian and vector forms4.04g Vector product: a x b perpendicular vector8.04e Scalar triple product: volumes of tetrahedra and parallelepipeds |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Area \(APQC\) = Area \(ABC\) – Area \(PBQ\); or Area \(APC\) + Area \(CPQ\); or Area \(APQ\) + Area \(AQC\); or Area \(APQC = \frac{1}{2} | \mathbf{AQ} \times \mathbf{PC} | \) |
| Line \(AB\): \(r = \begin{pmatrix}3\\4\\5\end{pmatrix} + \lambda\begin{pmatrix}7\\-5\\0\end{pmatrix}\) or Line \(BC\): \(r = \begin{pmatrix}10\\-1\\5\end{pmatrix} + \mu\begin{pmatrix}6\\-8\\14\end{pmatrix}\) | M1 | Correct attempt to find equation of line \(AB\) or line \(BC\) |
| \(4(3+7\lambda) - 8(4-5\lambda) + 5 = 2 \Rightarrow \lambda = \ldots \Rightarrow P\) is \(\ldots\) or \(4(10+6\mu)-8(-1-8\mu)+5+14\mu = 2 \Rightarrow \mu = \ldots \Rightarrow Q\) is \(\ldots\) | M1 | Uses at least one line with equation of given plane to determine at least one parameter and coordinates of \(P\) or \(Q\) |
| \(P(4.75,\ 2.75,\ 5)\) and \(Q(7,\ 3,\ -2)\) | A1 | Both coordinates correct |
| Area \(ABC = \frac{1}{2}\begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k}\\7 & -5 & 0\\6 & -8 & 14\end{vmatrix} = \frac{1}{2}\sqrt{70^2+98^2+26^2}\) | M1 | Uses all required information to calculate appropriate areas correctly leading to area of quadrilateral; needs to be complete method |
| Area \(APQC = \frac{1}{2}\begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k}\\-4 & 1 & 7\\0.75 & -4.25 & 14\end{vmatrix} = \frac{1}{2}\sqrt{43.75^2+61.25^2+16.25^2}\) | ||
| Area \(ABC\) – Area \(PBQ = 38.5\)* | A1* | Reaches 38.5 with no errors |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\overrightarrow{AB} = \begin{pmatrix}7\\-5\\0\end{pmatrix},\ \overrightarrow{AC} = \begin{pmatrix}1\\3\\-14\end{pmatrix},\ \overrightarrow{AD} = \begin{pmatrix}k-3\\0\\-6\end{pmatrix}\) | M1 | Adopts correct strategy by finding suitable vectors and forming scalar triple product |
| \(\overrightarrow{AB} \times \overrightarrow{AC} \cdot \overrightarrow{AD} = \begin{vmatrix}7 & -5 & 0\\1 & 3 & -14\\k-3 & 0 & -6\end{vmatrix} = \ldots\) | ||
| \(\overrightarrow{AB} \times \overrightarrow{AC} \cdot \overrightarrow{AD} = 7\times{-18} + 5(-6+14k-42)\) | A1 | Correct expression for triple product in terms of \(k\); should be \(\pm(70k - 366)\); ignore presence/absence of "\(\frac{1}{6}\)" |
| \(7\times{-18} + 5(-6+14k-42) = \pm 226 \Rightarrow k = \ldots\) | dM1 | Realises \(\pm 226\) is possible for value of triple product and attempts to solve for 2 values of \(k\); dependent on previous M mark |
| \(k = 2\) or \(\frac{296}{35}\) | A1 | Correct values (must be exact) |
# Question 7:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Area $APQC$ = Area $ABC$ – Area $PBQ$; or Area $APC$ + Area $CPQ$; or Area $APQ$ + Area $AQC$; or Area $APQC = \frac{1}{2}|\mathbf{AQ} \times \mathbf{PC}|$ | M1 | Identifies correct strategy; attempt does not need to be complete |
| Line $AB$: $r = \begin{pmatrix}3\\4\\5\end{pmatrix} + \lambda\begin{pmatrix}7\\-5\\0\end{pmatrix}$ or Line $BC$: $r = \begin{pmatrix}10\\-1\\5\end{pmatrix} + \mu\begin{pmatrix}6\\-8\\14\end{pmatrix}$ | M1 | Correct attempt to find equation of line $AB$ or line $BC$ |
| $4(3+7\lambda) - 8(4-5\lambda) + 5 = 2 \Rightarrow \lambda = \ldots \Rightarrow P$ is $\ldots$ or $4(10+6\mu)-8(-1-8\mu)+5+14\mu = 2 \Rightarrow \mu = \ldots \Rightarrow Q$ is $\ldots$ | M1 | Uses at least one line with equation of given plane to determine at least one parameter and coordinates of $P$ or $Q$ |
| $P(4.75,\ 2.75,\ 5)$ and $Q(7,\ 3,\ -2)$ | A1 | Both coordinates correct |
| Area $ABC = \frac{1}{2}\begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k}\\7 & -5 & 0\\6 & -8 & 14\end{vmatrix} = \frac{1}{2}\sqrt{70^2+98^2+26^2}$ | M1 | Uses all required information to calculate appropriate areas correctly leading to area of quadrilateral; needs to be complete method |
| Area $APQC = \frac{1}{2}\begin{vmatrix}\mathbf{i} & \mathbf{j} & \mathbf{k}\\-4 & 1 & 7\\0.75 & -4.25 & 14\end{vmatrix} = \frac{1}{2}\sqrt{43.75^2+61.25^2+16.25^2}$ | | |
| Area $ABC$ – Area $PBQ = 38.5$* | A1* | Reaches 38.5 with no errors |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\overrightarrow{AB} = \begin{pmatrix}7\\-5\\0\end{pmatrix},\ \overrightarrow{AC} = \begin{pmatrix}1\\3\\-14\end{pmatrix},\ \overrightarrow{AD} = \begin{pmatrix}k-3\\0\\-6\end{pmatrix}$ | M1 | Adopts correct strategy by finding suitable vectors and forming scalar triple product |
| $\overrightarrow{AB} \times \overrightarrow{AC} \cdot \overrightarrow{AD} = \begin{vmatrix}7 & -5 & 0\\1 & 3 & -14\\k-3 & 0 & -6\end{vmatrix} = \ldots$ | | |
| $\overrightarrow{AB} \times \overrightarrow{AC} \cdot \overrightarrow{AD} = 7\times{-18} + 5(-6+14k-42)$ | A1 | Correct expression for triple product in terms of $k$; should be $\pm(70k - 366)$; ignore presence/absence of "$\frac{1}{6}$" |
| $7\times{-18} + 5(-6+14k-42) = \pm 226 \Rightarrow k = \ldots$ | dM1 | Realises $\pm 226$ is possible for value of triple product and attempts to solve for 2 values of $k$; dependent on previous M mark |
| $k = 2$ or $\frac{296}{35}$ | A1 | Correct values (must be exact) |\begin{enumerate}
\item With respect to a fixed origin $O$, the points $A$, $B$ and $C$ have coordinates $( 3,4,5 ) , ( 10 , - 1,5 )$ and ( $4,7 , - 9$ ) respectively.
\end{enumerate}
The plane $\Pi$ has equation $4 x - 8 y + z = 2$\\
The line segment $A B$ meets the plane $\Pi$ at the point $P$ and the line segment $B C$ meets the plane $\Pi$ at the point $Q$.\\
(a) Show that, to 3 significant figures, the area of quadrilateral $A P Q C$ is 38.5
The point $D$ has coordinates $( k , 4 , - 1 )$, where $k$ is a constant.\\
Given that the vectors $\overrightarrow { A B } , \overrightarrow { A C }$ and $\overrightarrow { A D }$ form three edges of a parallelepiped of volume 226\\
(b) find the possible values of the constant $k$.
\hfill \mbox{\textit{Edexcel FP1 2019 Q7 [10]}}