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Let \(T\) be a linear transformation defined by the matrix \(\mathbf{M} = \begin{pmatrix} k & 2 \\ 1 & 4 \end{pmatrix}\), where \(k\) is a constant.
A triangle with area \(10\) cm\(^2\) is transformed by \(T\) into an image triangle with area \(180\) cm\(^2\).
Write down an expression for the determinant of \(\mathbf{M}\) in terms of \(k\).
Find the two possible values of \(k\).
Given that the transformation preserves the orientation of the shape (i.e., the determinant is positive), find the value of \(k\).
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