| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2015 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Cartesian equation of a plane |
| Difficulty | Standard +0.3 This is a standard Further Maths vectors question requiring routine techniques: finding direction vectors, converting between vector and Cartesian forms, computing a normal vector via cross product, and using the distance formula. While it's multi-part and involves several steps, each part follows textbook procedures without requiring novel insight or particularly challenging problem-solving. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04b Plane equations: cartesian and vector forms4.04j Shortest distance: between a point and a plane |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 1 \\ 3 \\ 2 \end{pmatrix} + \lambda\begin{pmatrix} -2 \\ -3 \\ -1 \end{pmatrix} + \mu\begin{pmatrix} 1 \\ -2 \\ -2 \end{pmatrix} \Rightarrow 4x - 5y + 7z = 3\) | M1A1 | M1: Correctly forms the parametric equation and eliminates the parameters to obtain a Cartesian equation. A1: Correct Cartesian equation |
| \(4x - 5y + 7z = 3 \Rightarrow \mathbf{r} \cdot \begin{pmatrix} 4 \\ -5 \\ 7 \end{pmatrix} = 3\) | dM1A1 | dM1: Converts their Cartesian equation into the form required. Dependent on the previous M. A1: Correct equation (oe) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(a + 3b + 2c = d\), \(-a + c = d\), \(2a + b = d\) \(\Rightarrow a = \frac{4}{3}d,\ b = -\frac{5}{3}d,\ c = \frac{7}{3}d\) | M1A1 | M1: Substitutes to obtain 3 equations in \(a\), \(b\), \(c\) and \(d\) and solves to obtain at least one of \(a\), \(b\) or \(c\) in terms of \(d\). A1: Correct \(a\), \(b\) and \(c\) in terms of \(d\) |
| \(\frac{4}{3}x - \frac{5}{3}y + \frac{7}{3}z = 1 \Rightarrow \mathbf{r} \cdot \frac{1}{3}\begin{pmatrix} 4 \\ -5 \\ 7 \end{pmatrix} = 1\) | dM1A1 | dM1: Uses their Cartesian equation correctly to form a vector equation. Dependent on the previous M |
## Question 5(c) Alternatives:
**Method 1 (Parametric to Cartesian):**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 1 \\ 3 \\ 2 \end{pmatrix} + \lambda\begin{pmatrix} -2 \\ -3 \\ -1 \end{pmatrix} + \mu\begin{pmatrix} 1 \\ -2 \\ -2 \end{pmatrix} \Rightarrow 4x - 5y + 7z = 3$ | M1A1 | M1: Correctly forms the parametric equation and eliminates the parameters to obtain a Cartesian equation. A1: Correct Cartesian equation |
| $4x - 5y + 7z = 3 \Rightarrow \mathbf{r} \cdot \begin{pmatrix} 4 \\ -5 \\ 7 \end{pmatrix} = 3$ | dM1A1 | dM1: Converts their Cartesian equation into the form required. **Dependent on the previous M**. A1: Correct equation (oe) |
**Method 2 (System of equations):**
| Answer/Working | Marks | Guidance |
|---|---|---|
| $a + 3b + 2c = d$, $-a + c = d$, $2a + b = d$ $\Rightarrow a = \frac{4}{3}d,\ b = -\frac{5}{3}d,\ c = \frac{7}{3}d$ | M1A1 | M1: Substitutes to obtain 3 equations in $a$, $b$, $c$ and $d$ and solves to obtain at least one of $a$, $b$ or $c$ in terms of $d$. A1: Correct $a$, $b$ and $c$ in terms of $d$ |
| $\frac{4}{3}x - \frac{5}{3}y + \frac{7}{3}z = 1 \Rightarrow \mathbf{r} \cdot \frac{1}{3}\begin{pmatrix} 4 \\ -5 \\ 7 \end{pmatrix} = 1$ | dM1A1 | dM1: Uses their Cartesian equation correctly to form a vector equation. **Dependent on the previous M** |5. The points $A , B$ and $C$ have position vectors $\left( \begin{array} { l } 1 \\ 3 \\ 2 \end{array} \right) , \left( \begin{array} { r } - 1 \\ 0 \\ 1 \end{array} \right)$ and $\left( \begin{array} { l } 2 \\ 1 \\ 0 \end{array} \right)$ respectively.
\begin{enumerate}[label=(\alph*)]
\item Find a vector equation of the straight line $A B$.
\item Find a cartesian form of the equation of the straight line $A B$.
The plane $\Pi$ contains the points $A , B$ and $C$.
\item Find a vector equation of $\Pi$ in the form r.n $= p$.
\item Find the perpendicular distance from the origin to $\Pi$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 2015 Q5 [10]}}