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\(\pi\)
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\(\frac{a}{b}\)
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Let \(T\) be a linear transformation defined by the matrix \(\mathbf{A} = \begin{pmatrix} 3 & x \\ 1 & 2 \end{pmatrix}\), where \(x\) is a constant.
A parallelogram with area \(5\) cm\(^2\) is transformed by \(T\) into an image with area \(30\) cm\(^2\).
Write down an expression for the determinant of \(\mathbf{A}\) in terms of \(x\).
Find the two possible values of \(x\).
Given that the transformation reverses the orientation of the shape (i.e., the determinant is negative), find the value of \(x\).
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