Standard +0.8 This question requires finding the determinant of a matrix with algebraic entries, recognizing that area scales by |det(T)|, setting up the optimization problem det(T) = (x²+1)(x²+5) - (-4)(3-2x²), expanding to get a quartic in x², and minimizing using calculus. While the individual steps are standard Further Maths techniques, the combination of matrix transformations, algebraic manipulation, and optimization makes this moderately challenging.
3 A transformation T is represented by the matrix \(\mathbf { T }\) where \(\mathbf { T } = \left( \begin{array} { c c } x ^ { 2 } + 1 & - 4 \\ 3 - 2 x ^ { 2 } & x ^ { 2 } + 5 \end{array} \right)\). A quadrilateral \(Q\), whose area is 12 units, is transformed by T to \(Q ^ { \prime }\).
Find the smallest possible value of the area of \(Q ^ { \prime }\).
3 A transformation T is represented by the matrix $\mathbf { T }$ where $\mathbf { T } = \left( \begin{array} { c c } x ^ { 2 } + 1 & - 4 \\ 3 - 2 x ^ { 2 } & x ^ { 2 } + 5 \end{array} \right)$. A quadrilateral $Q$, whose area is 12 units, is transformed by T to $Q ^ { \prime }$.
Find the smallest possible value of the area of $Q ^ { \prime }$.
\hfill \mbox{\textit{OCR FP1 AS 2021 Q3 [5]}}