A2 June 2019 Paper 1 Q14
14 Three planes have equations
\[\begin{aligned} -x + ay \phantom{{}+ z} &= 2 \\ 2x + 3y + z &= -3 \\ x + by + z &= c \end{aligned}\]where \(a\), \(b\) and \(c\) are constants.
(a) In the case where the planes do not intersect at a unique point,
(i) find \(b\) in terms of \(a\), [4]
(ii) find the value of \(c\) for which the planes form a sheaf. [3]
(b) In the case where \(b = a\) and \(c = 1\), find the coordinates of the point of intersection of the planes in terms of \(a\). [6]
| Scheme | Marks | AO |
|---|---|---|
| (i) let \(\mathbf{M} = \begin{pmatrix} -1 & a & 0 \\ 2 & 3 & 1 \\ 1 & b & 1 \end{pmatrix}\) | M1 | 3.1a |
| \(\det\mathbf{M} = b - 3 - a\) | B1 | 1.1b |
| \(\det\mathbf{M} = 0\) | M1 | 1.1b |
| \(\Rightarrow b = a + 3\) | A1 | 3.2a |
| [4] | ||
| (ii) \(x = ay - 2 \Rightarrow (2a + 3)y + z = 1\) \((a + b)y + z = c + 2,\ b = a + 3\) | M1 | 3.1a |
| \(\Rightarrow (2a + 3)y + z = c + 2\) | M1 | 3.1a |
| \(\Rightarrow c = -1\) | A1cao | 3.2a |
| [3] |
Notes
(a)(i)
M1: finding matrix of coefficients
(a)(ii)
M1: reduce system to 2 equations in 2 variables, one including \(c\)
M1: use \(b = a + 3\) to find value of \(c\) for consistency
A1cao: or other valid method
| Scheme | Marks | AO |
|---|---|---|
| \(\mathbf{M}^{-1} = -\dfrac{1}{3}\begin{pmatrix} 3 - a & -a & a \\ -1 & -1 & 1 \\ 2a - 3 & 2a & -3 - 2a \end{pmatrix}\) | M1 A2 M1 | 3.1a 1.1b 1.1b |
| \(\mathbf{M}^{-1}\begin{pmatrix} 2 \\ -3 \\ 1 \end{pmatrix} = -\dfrac{1}{3}\begin{pmatrix} 6 + 2a \\ 2 \\ -4a - 9 \end{pmatrix}\) | M1 | 1.1b |
| coordinates are \(\left(-\dfrac{6 + 2a}{3}, -\dfrac{2}{3}, \dfrac{4a + 9}{3}\right)\) | A1cao | 3.2a |
| [6] |
Notes
M1: attempt to find \(\mathbf{M}^{-1}\)
A2: A1 any 6 entries correct
M1: \(\times\) 1/their det
M1: pre-multiplying by their \(\mathbf{M}^{-1}\)
A1cao: accept in vector form
Alternative solution
| Scheme | Marks |
|---|---|
| \(-x + ay = 2,\ x + ay + z = 1\) \(\Rightarrow 2x + z = -1,\ z = -2x - 1\) | M1 |
| \(-x + ay = 2 \Rightarrow y = (2 + x)/a\) | M1 |
| \(\Rightarrow 2x + \dfrac{3x + 6}{a} - 1 - 2x = -3\) | M1 |
| \(\Rightarrow x = -\dfrac{2a + 6}{3},\ y = -\dfrac{2}{3},\ z = \dfrac{4a + 9}{3}\) | A3 |
| [6] |
M1: eliminate one variable from 2 equations
M1: eliminate another variable
M1: substitute into 3rd eqn to get eqn in one unknown