AS June 2025 Paper 1 Q10

10 The \(3 \times 3\) matrix \(\mathbf{M}\) represents a reflection in the plane \(z = 0\)

The point \(A\) has position vector \(\begin{bmatrix} 2 \\ 5 \\ 4 \end{bmatrix}\)

The point \(A^{\prime}\) is the image of \(A\) under the reflection represented by the matrix \(\mathbf{M}\)

Line \(l\) passes through \(A\) and \(A^{\prime}\)

(a) Write down the matrix \(\mathbf{M}\) [1 mark]

(b) Find the position vector of \(A^{\prime}\) [1 mark]

(d) The point \(B\) has coordinates \((3, 7, -2)\)

(i) Write down the coordinates of the point on \(l\) which is closest to \(B\) [1 mark]

(ii) Calculate the shortest distance between \(l\) and \(B\) [1 mark]

(iii) Calculate the area of the triangle \(ABA^{\prime}\) [2 marks]

Mark scheme (a)

Scheme	Marks	AO
Writes down \(\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix}\)	B1	1.2
	(1)

\[\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{bmatrix}\]

Scheme	Marks	AO
Obtains \(\begin{bmatrix} 2 \\ 5 \\ -4 \end{bmatrix}\)	B1	1.1b
	(1)

Mark scheme (c)

Scheme	Marks	AO
Finds a correct direction vector between \(A\) and their \(A^{\prime}\) Do not award this mark if their ‘direction’ vector is used as a position vector.	M1	2.2a
Writes a correct vector equation using accurate notation.	A1	2.5
	(2)

\[\mathbf{r} = \begin{bmatrix} 2 \\ 5 \\ 4 \end{bmatrix} + \lambda\begin{bmatrix} 0 \\ 0 \\ 8 \end{bmatrix}\]

Mark scheme (d)

Scheme	Marks	AO
(i) Writes down \((2, 5, -2)\) Condone the position vector.	B1	3.1a
	(1)
(ii) Obtains \(\sqrt{5}\)	B1	3.1a
	(1)
(iii) Correct method for the area of triangle \(ABA^{\prime}\) for their part (d)(ii) PI by AWRT 8.9	M1	3.1a
Obtains \(4\sqrt{5}\) Accept AWRT 8.94	A1	2.2a
	(2)
	(8 marks)

(i)

\((2, 5, -2)\)

(ii)

Shortest distance between \(l\) and \(B\) is

\[\begin{aligned} &= \sqrt{(2 - 3)^2 + (5 - 7)^2 + (0)^2} \\ &= \sqrt{5}\end{aligned}\]

(iii)

\[\begin{aligned}\text{Area} &= \frac{1}{2} \times \big(4 - (-4)\big) \times \sqrt{5} \\ &= 4\sqrt{5}\end{aligned}\]