A2 June 2024 Paper 1 Q5
5 The points \(A\), \(B\) and \(C\) have coordinates \(A(5, 3, 4)\), \(B(8, -1, 9)\) and \(C(12, 5, 10)\)
The points \(A\), \(B\) and \(C\) lie in the plane \(\Pi\)
(a) Find a vector that is normal to the plane \(\Pi\) [3 marks]
(b) Find a Cartesian equation of the plane \(\Pi\) [2 marks]
| Scheme | Marks | AO |
|---|---|---|
| Obtains two of the three vectors connecting \(A\), \(B\) and \(C\) \(\pm\begin{bmatrix}3 \\ -4 \\ 5\end{bmatrix},\ \pm\begin{bmatrix}7 \\ 2 \\ 6\end{bmatrix},\ \pm\begin{bmatrix}4 \\ 6 \\ 1\end{bmatrix}\) Condone one incorrect element. | M1 | 1.1a |
| Forms the vector product of two vectors. | M1 | 1.1a |
| Obtains \(k\begin{bmatrix}-2 \\ 1 \\ 2\end{bmatrix}\) | A1 | 1.1b |
| (3) |
Typical solution
\[\overrightarrow{AB} \times \overrightarrow{AC} = \begin{bmatrix}3 \\ -4 \\ 5\end{bmatrix} \times \begin{bmatrix}7 \\ 2 \\ 6\end{bmatrix} = 17\begin{bmatrix}-2 \\ 1 \\ 2\end{bmatrix}\]| Scheme | Marks | AO |
|---|---|---|
| Obtains their \(-2x + y + 2z = d\) where \(d\) is a number or Evaluates the scalar product of their normal vector and a point in the plane. | M1 | 1.1a |
| Obtains \(-2x + y + 2z = 1\) OE | A1 | 1.1b |
| (2) | ||
| (5 marks) |