| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2006 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Point on line satisfying condition |
| Difficulty | Challenging +1.2 This is a multi-part 3D coordinate geometry question from FP3 requiring vector methods and spatial reasoning. Part (i) involves finding a plane equation using two points and a parallel line (non-routine but systematic). Part (ii) is trivial parametric form recall. Part (iii) requires finding a common perpendicular to two skew lines, which is conceptually demanding but follows standard FP3 techniques. The 10-mark total and need for careful vector manipulation place this above average difficulty, but it's still a standard FP3 question type without requiring exceptional insight. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04b Plane equations: cartesian and vector forms |
| Answer | Marks | Guidance |
|---|---|---|
| (i) EITHER \(a = [2, 3, 5]\), \(b = [2, 2, 0]\); \(\mathbf{n} = a \times b = \pm k[-10, 10, -2]\); Use (2, 1, 5) OR (0, -1, 5); \(\Rightarrow 5x - 5y + z = 10\) OR \(a = [2, 3, 5]\), \(b = [2, 2, 0]\); e.g. \(r = [2, 1, 5] + \lambda[2, 2, 0] + \mu[2, 3, 5]\); \([x, y, z] = [2+2\lambda+2\mu, 1+2\lambda+3\mu, 5+5\mu]\) | B1; M1; A1√; M1; A1 | For stating 2 vectors in the plane; For finding perpendicular to plane; For correct n. f.t. from incorrect b; For substituting a point into equation \(ax+by+cz=d\) where \([a,b,c] =\) their n; For correct cartesian equation AEF; For stating 2 vectors in the plane; For stating parametric equation of plane; For writing 3 equations in \(x, y, z\) f.t. from incorrect b; For eliminating \(\lambda\) and \(\mu\); For correct cartesian equation AEF |
| (ii) \([2r, 3r-4, 5r-9]\) | B1 1 | For stating a point A on \(l_1\) with parameter \(t\) AEF |
| (iii) \(\pm[2r+5, 3t-7, 5t-13]\); \(\pm[2r+5, 3t-7, 5t-13] \cdot [2, 3, 5] = 0\); \(\Rightarrow t = 2\); \(\frac{x+5}{9} = \frac{y-3}{-1} = \frac{z-4}{-3}\) OR \(\frac{x-4}{9} = \frac{y-2}{-1} = \frac{z-1}{-3}\) | M1; M1; A1; A1 4 [10] | For finding direction of \(l_2\) from A and \((-5, 3, 4)\); For using scalar product for perpendicularity with any vector involving \(t\); For correct value of \(t\); For a correct equation AEFcartesian |
**(i)** EITHER $a = [2, 3, 5]$, $b = [2, 2, 0]$; $\mathbf{n} = a \times b = \pm k[-10, 10, -2]$; Use (2, 1, 5) **OR** (0, -1, 5); $\Rightarrow 5x - 5y + z = 10$ **OR** $a = [2, 3, 5]$, $b = [2, 2, 0]$; e.g. $r = [2, 1, 5] + \lambda[2, 2, 0] + \mu[2, 3, 5]$; $[x, y, z] = [2+2\lambda+2\mu, 1+2\lambda+3\mu, 5+5\mu]$ | B1; M1; A1√; M1; A1 | For stating 2 vectors in the plane; For finding perpendicular to plane; For correct **n**. f.t. from incorrect **b**; For substituting a point into equation $ax+by+cz=d$ where $[a,b,c] =$ their **n**; For correct cartesian equation **AEF**; For stating 2 vectors in the plane; For stating parametric equation of plane; For writing 3 equations in $x, y, z$ f.t. from incorrect **b**; For eliminating $\lambda$ and $\mu$; For correct cartesian equation **AEF**
**(ii)** $[2r, 3r-4, 5r-9]$ | B1 1 | For stating a point A on $l_1$ with parameter $t$ **AEF**
**(iii)** $\pm[2r+5, 3t-7, 5t-13]$; $\pm[2r+5, 3t-7, 5t-13] \cdot [2, 3, 5] = 0$; $\Rightarrow t = 2$; $\frac{x+5}{9} = \frac{y-3}{-1} = \frac{z-4}{-3}$ **OR** $\frac{x-4}{9} = \frac{y-2}{-1} = \frac{z-1}{-3}$ | M1; M1; A1; A1 4 [10] | For finding direction of $l_2$ from A and $(-5, 3, 4)$; For using scalar product for perpendicularity with any vector involving $t$; For correct value of $t$; For a correct equation **AEF**cartesian
**SR** For $2p + 3q + 5r = 0$ and no further progress award B1
---A line $l_1$ has equation $\frac{x}{2} = \frac{y + 4}{3} = \frac{z + 9}{5}$.
\begin{enumerate}[label=(\roman*)]
\item Find the cartesian equation of the plane which is parallel to $l_1$ and which contains the points $(2, 1, 5)$ and $(0, -1, 5)$. [5]
\item Write down the position vector of a point on $l_1$ with parameter $t$. [1]
\item Hence, or otherwise, find an equation of the line $l_2$ which intersects $l_1$ at right angles and which passes through the point $(-5, 3, 4)$. Give your answer in the form $\frac{x - a}{p} = \frac{y - b}{q} = \frac{z - c}{r}$. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR FP3 2006 Q5 [10]}}