A2 June 2022 Paper 2 Q10
10 The coordinates of the points \(A\) and \(B\) are \((3, -2, -1)\) and \((13, 10, 9)\) respectively.
- The plane \(\Pi_A\) contains \(A\) and the plane \(\Pi_B\) contains \(B\).
- The planes \(\Pi_A\) and \(\Pi_B\) are parallel.
- The \(x\) and \(y\) components of any normal to plane \(\Pi_A\) are equal.
- The shortest distance between \(\Pi_A\) and \(\Pi_B\) is 2.
There are two possible solution planes for \(\Pi_A\) which satisfy the above conditions.
Determine the acute angle between these two possible solution planes. [8]
| Scheme | Marks | AO |
|---|---|---|
| \(\overrightarrow{AB} = \begin{pmatrix} 10 \\ 12 \\ 10 \end{pmatrix}\) | *M1 | |
| \(\pm 2 = AB\cos\theta = \dfrac{|\overrightarrow{AB}.\mathbf{n}|}{|\mathbf{n}|}\) with \(\overrightarrow{AB}\) from above used | dep*M1 | |
| \(\mathbf{n} = \begin{pmatrix} a \\ a \\ c \end{pmatrix}\) or \(\mathbf{n} = \begin{pmatrix} 1 \\ 1 \\ s \end{pmatrix}\) oe | *M1 | |
| \(119a^2 + 110ac + 24c^2 = 0\) | A1 | |
| \((7a + 4c)(17a + 6c) = 0\) e.g. \(a = 4,\ c = -7 \Rightarrow \mathbf{n}_1 = \begin{pmatrix} 4 \\ 4 \\ -7 \end{pmatrix}\) | dep*M1 | |
| e.g. \(a = 6,\ c = -17 \Rightarrow \mathbf{n}_2 = \begin{pmatrix} 6 \\ 6 \\ -17 \end{pmatrix}\) | A1 | |
| \(\begin{pmatrix} 4 \\ 4 \\ -7 \end{pmatrix} . \begin{pmatrix} 6 \\ 6 \\ -17 \end{pmatrix} = 167\) | dep*M1 | |
| \(\cos\theta = \dfrac{167}{9 \times 19} = \dfrac{167}{171}\) \(\Rightarrow\) acute angle is awrt \(12.4^\circ\) | A1 | |
| [8] |
Notes
*M1: Finding \(\overrightarrow{AB}\) or an expression for \(p_B - p_A\) or \(d_B - d_A\)
from \(\mathbf{r}.\mathbf{n} = p_A = 3a - 2a - c\) and \(\mathbf{r}.\mathbf{n} = p_B = 13a + 10a + 9c\) or \(\mathbf{r}.\hat{\mathbf{n}} = d_A\) and \(\mathbf{r}.\hat{\mathbf{n}} = d_B\)
ie \(22a + 10c\)
dep*M1: Use of dot product to express correct shortest distance in terms of \(\overrightarrow{AB}\) and the normal to the planes or seeing an appropriate difference between plane constants
or \(d_B - d_A = \pm 2\) or \(\dfrac{p_B}{\sqrt{a^2 + b^2 + c^2}} - \dfrac{p_A}{\sqrt{a^2 + b^2 + c^2}} = \pm 2\)
allow omission of \(\pm\)
*M1: \(\begin{pmatrix} a \\ a \\ c \end{pmatrix}\) used consistently
A1: Quadratic formed
Or \(171a^2 \mp 22a - 24 = 0\)
\(171c^2 \mp 20c - 119 = 0\)
\(24s^2 + 110s + 119 = 0\)
dep*M1: Using one solution of quadratic to obtain a normal of one of the solution planes.
Dependent on all previous M marks
A1: Both correct.
dep*M1: Finding the dot product of their solution normals.
Dependent on all previous M marks
A1: or awrt 0.217 rads