(a) px + qy = z
\(p \begin{pmatrix} 2 \\ 3 \end{pmatrix} + q \begin{pmatrix} 5 \\ -2 \end{pmatrix} = \begin{pmatrix} -4 \\ 13 \end{pmatrix}\)
\(\begin{pmatrix} 2p + 5q \\ 3p - 2q \end{pmatrix} = \begin{pmatrix} -4 \\ 13 \end{pmatrix}\)
Thus,
\(2p + 5q = -4 .....(1)\)
\(3p - 2q = 13 ......(2)\)
Solving the equations, we have p = 3 and q = -2.
(b)(i)
(ii) See attached graph for the quadrilateral W(2, 3), X(4, -1), Y(-3, -4) and Z(-3,2).
(iii) See attached for the image for the quadrilateral WXYZ under an anticlockwise rotation of 90° is \(W_{1}(-3, 2), X_{1}(1, 4), Y_{1}(4, -3)\) and \(Z_{1}(-2, -3)\).