The position vectors of points \(A\) and \(B\) are \(\begin{pmatrix}k\\6\end{pmatrix}\) and \(\begin{pmatrix}2\\-2\end{pmatrix}\) respectively. Given that \(AB=10\), find the negative value of \(k\).[3]
\(C\) is the point \((5,-6)\). Using the value of \(k\) found in part (a)(i), show that \(A\), \(B\) and \(C\) are collinear.[3]
The diagram shows a parallelogram \(PQRS\) and a triangle \(QXR\). \(QS\) is a diagonal of the parallelogram and \(Y\) is a point on \(RS\) such that \(OS:QS=3:10\). It is given that \(\overrightarrow{OP}=\mathbf p\) and \(\overrightarrow{OQ}=\mathbf q\).
Express \(\overrightarrow{QR}\), as simply as possible, in terms of \(\mathbf p\) and \(\mathbf q\).[3]
If \(\overrightarrow{QX}=m\overrightarrow{SQ}\) and \(\overrightarrow{RX}=n\left(\dfrac15\mathbf p+\dfrac17\mathbf q\right)\), where \(m\) and \(n\) are rational numbers, find the value of \(m\) and of \(n\).[4]
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