Challenging +1.2 This question requires knowing that area scales by |det(T)|, computing a determinant with parameters, then minimizing a quadratic expression. It combines matrix transformations with optimization, requiring multiple steps and conceptual understanding beyond routine calculation, but the algebra itself is straightforward and the techniques are standard A-level Further Maths content.
A transformation T is represented by the matrix \(\mathbf{T}\) where \(\mathbf{T} = \begin{pmatrix} x^2 + 1 & -4 \\ 3 - 2x^2 & x^2 + 5 \end{pmatrix}\).
A quadrilateral \(Q\), whose area is 12 units, is transformed by T to \(Q'\).
Find the smallest possible value of the area of \(Q'\). [5]
A transformation T is represented by the matrix $\mathbf{T}$ where $\mathbf{T} = \begin{pmatrix} x^2 + 1 & -4 \\ 3 - 2x^2 & x^2 + 5 \end{pmatrix}$.
A quadrilateral $Q$, whose area is 12 units, is transformed by T to $Q'$.
Find the smallest possible value of the area of $Q'$. [5]
\hfill \mbox{\textit{SPS SPS ASFM 2020 Q5 [5]}}