AS June 2023 Paper 1 Q8
8 The equations of three planes are
\[\begin{aligned} 2x + y + 3z &= 3, \\ 3x - y - 2z &= 2, \\ -4x + 3y + 7z &= k, \end{aligned}\]where \(k\) is a constant.
(a) By considering a suitable determinant, show that the planes do not meet at a single point. [2]
(b) Given that the planes form a sheaf, determine the value of \(k\). [4]
| Scheme | Marks | AO |
|---|---|---|
| \(\begin{vmatrix} 2 & 1 & 3 \\ 3 & -1 & -2 \\ -4 & 3 & 7 \end{vmatrix}\ [= 0]\) | M1 | 1.1a |
| \(\det = 0 \Rightarrow\) planes do not meet at a point | A1 | 1.1 |
| [2] |
Notes
M1: finding determinant of correct matrix
| Scheme | Marks | AO |
|---|---|---|
| Method 1: use 2 equations to find e.g. \(z\) and \(y\) in terms of \(x\), and substitute in 3rd equation e.g. (1) + (2): \(5x + z = 5\), | M1 | 2.1 |
| \(\Rightarrow z = 5 - 5x,\ y = 13x - 12\) | M1 | 2.1 |
| substitute in (3): \(-4x + 39x - 36 + 35 - 35x = k\) | M1 | 2.1 |
| \(\Rightarrow k = -1\) | A1 | 2.2a |
| [4] |
Notes
M1: (1st) eliminating one variable
M1: (2nd) finding e.g. \(y\) and \(z\) in terms of \(x\)
M1: (3rd) substituting in 3rd equation
Method 2: eliminate one variable from 2 pairs of eqns, and compare eqns in 2 remaining v’bles, e.g.
| Scheme | Marks |
|---|---|
| from (1) and (2): \(5x + z = 5\) (4) | M1 |
| from (1) and (3): \(10x + 2z = 9 - k\) (5) | M1 |
| \(\Rightarrow 9 - k = 10\) | M1 |
| \(\Rightarrow k = -1\) | A1 |
M1: (1st) eliminating one variable using two equations
M1: (2nd) eliminating same variable using two other equations
M1: (3rd) eliminating both v’bles using (4) and (5) equations (coeffs in (4) and (5) must be correct)
Method 3: find linear combination by inspection
| Scheme | Marks |
|---|---|
| (3) = (1) − 2×(2) | M3 |
| \(\Rightarrow k = 3 - 2 \times 2 = -1\) | A1 |
M3: oe
A1: for \(k = -1\) unsupported, allow SCB3
Method 4: substitute a value of \(x\), \(y\) or \(z\), e.g.
| Scheme | Marks |
|---|---|
| substituting \(x = 0\) \(y + 3z = 3,\ -y - 2z = 2\) | M1 |
| \(\Rightarrow z = 5,\ y = -12\) | A1 |
| substitute in (3): \(-36 + 35 = k\) | M1 |
| \(\Rightarrow k = -1\) | A1 |
M1: (1st) any value for \(x\), \(y\) or \(z\) substituted
A1: (1st) solve 2 equations for \(y\), \(z\)
M1: (2nd) substitute values into third equation