AS June 2024 Paper 1 Q7
7 Three planes have equations
\[\begin{aligned} x + 2y - 3z &= 0, \\ -x + 3y - 2z &= 0, \\ x - 2y + kz &= k, \end{aligned}\]where \(k\) is a constant.
(a) For the case \(k = 0\), the origin lies on all three planes.
Use a determinant to explain whether there are any other points that lie on all three planes in this case. [2]
Use a determinant to explain whether there are any other points that lie on all three planes in this case. [2]
(b) You are now given that \(k = 1\).
(i) Show that there are no points that lie on all three planes. [3]
(ii) Describe the geometrical arrangement of the three planes. [1]
| Scheme | Marks | AO |
|---|---|---|
| \(\begin{vmatrix} 1 & 2 & -3 \\ -1 & 3 & -2 \\ 1 & -2 & 0 \end{vmatrix} = -5\) | B1 | 1.1 |
| Non-zero value means there are no other points that lie on all three planes | B1 | 2.4 |
| [2] |
Notes
B1: det \(= -5\)
B1: correct conclusion with reference to non-zero determinant
| Scheme | Marks | AO |
|---|---|---|
| (i) Eliminate one variable, e.g. \(x\) | M1 | 3.1a |
| \(5y - 5z = 0\), \(y - z = 1\) | A1 | 1.1 |
| Inconsistent equations, so planes have no common point | A1 | 2.2a |
| [3] | ||
| (ii) The planes form a prismatic intersection | B1 | 1.2 |
| [1] |
Notes
(b)(i)
M1: Can eliminate any of \(x\), \(y\), \(z\)
A1: oe
A1: from correct working
(b)(ii)
B1: accept ‘prism’, ‘triangular prism’. Accept a diagram