FP1 January 2012 Q4

EdexcelOld spec11 marksMatrices

4. A right angled triangle \(T\) has vertices \(A(1,\ 1)\), \(B(2,\ 1)\) and \(C(2,\ 4)\). When \(T\) is transformed by the matrix \(\mathbf{P} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\), the image is \(T'\).

(a) Find the coordinates of the vertices of \(T'\). (2)

(b) Describe fully the transformation represented by \(\mathbf{P}\). (2)

The matrices \(\mathbf{Q} = \begin{pmatrix} 4 & -2 \\ 3 & -1 \end{pmatrix}\) and \(\mathbf{R} = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\) represent two transformations. When \(T\) is transformed by the matrix \(\mathbf{QR}\), the image is \(T''\).

(d) Find the determinant of \(\mathbf{QR}\). (2)

(e) Using your answer to part (d), find the area of \(T''\). (3)

Mark scheme (a)

Scheme	Marks
\(\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 1 & 2 & 2 \\ 1 & 1 & 4 \end{pmatrix} = \begin{pmatrix} 1 & 1 & 4 \\ 1 & 2 & 2 \end{pmatrix}\) Attempt to multiply the right way round with at least 4 correct elements	M1
\(T'\) has coordinates (1,1), (1,2) and (4,2) or \(\begin{pmatrix} 1 \\ 1 \end{pmatrix},\begin{pmatrix} 1 \\ 2 \end{pmatrix},\begin{pmatrix} 4 \\ 2 \end{pmatrix}\) NOT just \(\begin{pmatrix} 1 & 1 & 4 \\ 1 & 2 & 2 \end{pmatrix}\) Correct coordinates or vectors	A1
	(2)

Mark scheme (b)

Scheme	Marks
Reflection in the line \(y = x\) Reflection \(y = x\)	B1 B1
	(2)

Notes

Allow ‘in the axis’ ‘about the line’ \(y = x\) etc. Provided both features are mentioned ignore any reference to the origin unless there is a clear contradiction.

Mark scheme (c)

Scheme	Marks
\(\mathbf{QR} = \begin{pmatrix} 4 & -2 \\ 3 & -1 \end{pmatrix}\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} -2 & 0 \\ 0 & 2 \end{pmatrix}\) 2 correct elements Correct matrix	M1 A1
	(2)

Notes

Note that \(\mathbf{RQ} = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\begin{pmatrix} 4 & -2 \\ 3 & -1 \end{pmatrix} = \begin{pmatrix} 10 & -4 \\ 24 & -10 \end{pmatrix}\) scores M0A0 in (c) but allow all the marks in (d) and (e)

Scheme	Marks
\(\det(\mathbf{QR}) = -2 \times 2 - 0 = -4\) “\(-2\)”x”2” – “0”x”0” \(-4\)	M1 A1
	(2)

Mark scheme (e)

Scheme	Marks
Area of \(T = \dfrac{1}{2} \times 1 \times 3 = \dfrac{3}{2}\) Correct area for T	B1
Area of \(T'' = \dfrac{3}{2} \times 4 = 6\) Attempt at “\(\tfrac{3}{2}\)”\(\times \pm\)“4” 6 or follow through their det(QR) x Their triangle area provided area \(> 0\)	M1 A1ft
	(3)
	(11 marks)