| Exam Board | OCR |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2009 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Angle between two planes |
| Difficulty | Standard +0.8 This is a multi-part 3D vectors question requiring finding a plane equation from three points, calculating angle between planes using normal vectors, and finding intersection of line with plane. While systematic, it demands solid spatial reasoning, coordinate work with a cuboid, and multiple vector techniques across three connected parts—above average for A-level but standard for FP3. |
| Spec | 4.04b Plane equations: cartesian and vector forms4.04d Angles: between planes and between line and plane4.04f Line-plane intersection: find point |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Use 2 of \([-4,2,0],[0,0,3],[-4,2,3],[4,-2,3]\) or multiples | M1 | For finding vector product of 2 appropriate vectors in plane \(ACGE\) |
| \(\mathbf{n} = k[1,2,0]\) | A1 | For correct n |
| Use \(A[4,0,0]\), \(C[0,2,0]\), \(G[0,2,3]\) or \(E[4,0,3]\) | M1 | For substituting a point in the plane |
| \(\mathbf{r}.[1,2,0] = 4\) | A1 4 | For correct equation. AEF in this form |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\mathbf{r} = [4,0,0] + \lambda[-4,2,0] + \mu[0,0,3]\) | M1 | For writing plane in 2-parameter form |
| \(\Rightarrow x = 4 - 4\lambda\), \(y = 2\lambda\), \(z = 3\mu\) | A1 | For 3 correct equations |
| \(x + 2y = 4\) | M1 | For eliminating \(\lambda\) (and \(\mu\)) |
| \(\Rightarrow \mathbf{r}.[1,2,0] = 4\) | A1 | For correct equation. AEF in this form |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\theta = \cos^{-1}\frac{ | [3,0,-4]\cdot[1,2,0] | }{\sqrt{3^2+0^2+4^2}\sqrt{1^2+2^2+0^2}}\) |
| \(\theta = \cos^{-1}\frac{3}{5\sqrt{5}} = 74.4°\) | A1 4 | For correct angle |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(AM\): \((\mathbf{r} =)[4,0,0] + t[-2,2,3]\) (or \([2,2,3] + t[-2,2,3]\)) | M1, A1 | For obtaining parametric expression for \(AM\); For correct expression seen or implied |
| \(3(4-2t) - 4(3t) = 0\) (or \(3(2-2t) - 4(3+3t) = 0\)) | M1 | For finding intersection of \(AM\) with \(ACGE\) |
| \(t = \frac{2}{3}\) (or \(t = -\frac{1}{3}\)) OR \(\mathbf{w} = \left[\frac{8}{3}, \frac{4}{3}, 2\right]\) | A1 | For correct \(t\) OR position vector |
| \(AW : WM = 2:1\) | A1 5 | For correct ratio |
# Question 6:
## Part (i) Method 1:
| Answer/Working | Mark | Guidance |
|---|---|---|
| Use 2 of $[-4,2,0],[0,0,3],[-4,2,3],[4,-2,3]$ or multiples | M1 | For finding vector product of 2 appropriate vectors in plane $ACGE$ |
| $\mathbf{n} = k[1,2,0]$ | A1 | For correct **n** |
| Use $A[4,0,0]$, $C[0,2,0]$, $G[0,2,3]$ or $E[4,0,3]$ | M1 | For substituting a point in the plane |
| $\mathbf{r}.[1,2,0] = 4$ | A1 **4** | For correct equation. **AEF** in this form |
## Part (i) Method 2:
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\mathbf{r} = [4,0,0] + \lambda[-4,2,0] + \mu[0,0,3]$ | M1 | For writing plane in 2-parameter form |
| $\Rightarrow x = 4 - 4\lambda$, $y = 2\lambda$, $z = 3\mu$ | A1 | For 3 correct equations |
| $x + 2y = 4$ | M1 | For eliminating $\lambda$ (and $\mu$) |
| $\Rightarrow \mathbf{r}.[1,2,0] = 4$ | A1 | For correct equation. **AEF** in this form |
## Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\theta = \cos^{-1}\frac{|[3,0,-4]\cdot[1,2,0]|}{\sqrt{3^2+0^2+4^2}\sqrt{1^2+2^2+0^2}}$ | B1$\sqrt{}$, M1, M1 | For using correct vectors (allow multiples), f.t. from **n**; For using scalar product; For multiplying both moduli in denominator |
| $\theta = \cos^{-1}\frac{3}{5\sqrt{5}} = 74.4°$ | A1 **4** | For correct angle |
## Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $AM$: $(\mathbf{r} =)[4,0,0] + t[-2,2,3]$ (or $[2,2,3] + t[-2,2,3]$) | M1, A1 | For obtaining parametric expression for $AM$; For correct expression seen or implied |
| $3(4-2t) - 4(3t) = 0$ (or $3(2-2t) - 4(3+3t) = 0$) | M1 | For finding intersection of $AM$ with $ACGE$ |
| $t = \frac{2}{3}$ (or $t = -\frac{1}{3}$) OR $\mathbf{w} = \left[\frac{8}{3}, \frac{4}{3}, 2\right]$ | A1 | For correct $t$ OR position vector |
| $AW : WM = 2:1$ | A1 **5** | For correct ratio |
**Total: 13 marks**
---6\\
\begin{tikzpicture}[thick, >=Stealth]
% Projection parameters: depth direction in 2D
\pgfmathsetmacro{\ax}{0.7} % x-component per unit depth
\pgfmathsetmacro{\ay}{0.5} % y-component per unit depth
% --- Vertices ---
% Bottom face
\coordinate (O) at (0, 0);
\coordinate (A) at (4, 0);
\coordinate (B) at ({4 + 2*\ax}, {2*\ay});
\coordinate (C) at ({2*\ax}, {2*\ay});
% Top face
\coordinate (D) at (0, 3);
\coordinate (E) at (4, 3);
\coordinate (F) at ({4 + 2*\ax}, {3 + 2*\ay});
\coordinate (G) at ({2*\ax}, {3 + 2*\ay});
% Midpoint of edge GF
\coordinate (M) at ({0.5*(4 + 4*\ax)}, {3 + 2*\ay});
% --- Hidden edges (dashed) ---
\draw[dashed] (O) -- (C) -- (B);
\draw[dashed] (C) -- (G);
% --- Visible edges (solid) ---
% Bottom
\draw (O) -- (A) -- (B);
% Top
\draw (D) -- (G) -- (F) -- (E) -- (D);
% Verticals
\draw (O) -- (D);
\draw (A) -- (E);
\draw (B) -- (F);
% --- Vertex labels ---
\node[below left] at (O) {$O$};
\node[below] at (A) {$A$};
\node[right] at (B) {$B$};
\node[above left, xshift=2pt] at (C) {$C$};
\node[left] at (D) {$D$};
\node[below right] at (E) {$E$};
\node[above right] at (F) {$F$};
\node[above] at (G) {$G$};
\node[above] at (M) {$M$};
% --- Dots at key points ---
\fill (M) circle (1.5pt);
% --- Dimension arrows ---
\draw[->] (O) -- node[pos=1,below] {$4\mathbf{i}$} ($(O)!0.7!(A)$);
\draw[->] (O) -- node[left, pos=1] {$3\mathbf{k}$} ($(O)!0.6!(D)$);
\draw[->] (O) -- node[above] {$2\mathbf{j}$} (C);
\end{tikzpicture}
The cuboid $O A B C D E F G$ shown in the diagram has $\overrightarrow { O A } = 4 \mathbf { i } , \overrightarrow { O C } = 2 \mathbf { j } , \overrightarrow { O D } = 3 \mathbf { k }$, and $M$ is the mid-point of $G F$.\\
(i) Find the equation of the plane $A C G E$, giving your answer in the form $\mathbf{r} \cdot \mathbf{n} = p$.\\
(ii) The plane $O E F C$ has equation $\mathbf { r } \cdot ( 3 \mathbf { i } - 4 \mathbf { k } ) = 0$. Find the acute angle between the planes $O E F C$ and $A C G E$.\\
(iii) The line $A M$ meets the plane $O E F C$ at the point $W$. Find the ratio $A W : W M$.
\hfill \mbox{\textit{OCR FP3 2009 Q6 [13]}}