Volume/area scale factors

Questions involving the relationship between determinants and volume or area scale factors of transformations.

6 questions · Standard +1.0

4.03j Determinant 3x3: calculation
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Edexcel F3 2021 October Q4
11 marks Challenging +1.2
4. The matrix \(\mathbf { M }\) is given by $$\left( \begin{array} { r r r } 2 & 0 & - 1 \\ k & 3 & 2 \\ - 2 & 1 & k \end{array} \right)$$
  1. Show that \(\operatorname { det } \mathbf { M } = 5 k - 10\) Given that \(k \neq 2\)
  2. find \(\mathbf { M } ^ { - 1 }\) in terms of \(k\). The points \(O ( 0,0,0 ) , A ( 4 , - 8,3 ) , B ( - 2,5 , - 4 )\) and \(C ( 4 , - 6,8 )\) are the vertices of a tetrahedron \(T\). The transformation represented by matrix \(\mathbf { M }\) transforms \(T\) to a tetrahedron with volume 50
  3. Determine the possible values of \(k\).
OCR Further Pure Core 1 2024 June Q3
8 marks Standard +0.8
3 A transformation T is represented by the matrix \(\mathbf { N } = \left( \begin{array} { l l l } a & 4 & 2 \\ 5 & 1 & 0 \\ 3 & 6 & 3 \end{array} \right)\), where \(a\) is a constant.
  1. Find \(\mathbf { N } ^ { 2 }\) in terms of \(a\).
  2. Find det \(\mathbf { N }\) in terms of \(a\). The value of \(a\) is 13 to the nearest integer.
    A shape \(S _ { 1 }\) has volume 11.6 to 1 decimal place. Shape \(S _ { 1 }\) is mapped to shape \(S _ { 2 }\) by the transformation T . A student claims that the volume of \(S _ { 2 }\) is less than 400 .
  3. Comment on the student's claim.
OCR MEI Further Pure Core 2022 June Q4
7 marks Standard +0.8
4
  1. A transformation with associated matrix \(\left( \begin{array} { r r r } m & 2 & 1 \\ 0 & 1 & - 2 \\ 2 & 0 & 3 \end{array} \right)\), where \(m\) is a constant, maps the vertices of a cube to points that all lie in a plane. Find \(m\).
  2. The transformations S and T of the plane have associated matrices \(\mathbf { M }\) and \(\mathbf { N }\) respectively, where \(\mathbf { M } = \left( \begin{array} { r r } k & 1 \\ - 3 & 4 \end{array} \right)\) and the determinant of \(\mathbf { N }\) is \(3 k + 1\). The transformation \(U\) is equivalent to the combined transformation consisting of S followed by T . Given that U preserves orientation and has an area scale factor 2, find the possible values of \(k\).
OCR Further Pure Core AS 2019 June Q7
7 marks Standard +0.3
7 A transformation A is represented by the matrix \(\mathbf { A }\) where \(\mathbf { A } = \left( \begin{array} { c c c } - 1 & x & 2 \\ 7 - x & - 6 & 1 \\ 5 & - 5 x & 2 x \end{array} \right)\).
The tetrahedron \(H\) has vertices at \(O , P , Q\) and \(R\). The volume of \(H\) is 6 units. \(P ^ { \prime } , Q ^ { \prime } , R ^ { \prime }\) and \(H ^ { \prime }\) are the images of \(P , Q , R\) and \(H\) under A .
  1. In the case where \(x = 5\)
OCR FP1 AS 2021 June Q4
7 marks Standard +0.3
4 A transformation A is represented by the matrix \(\mathbf { A }\) where \(\mathbf { A } = \left( \begin{array} { c c c } - 1 & x & 2 \\ 7 - x & - 6 & 1 \\ 5 & - 5 x & 2 x \end{array} \right)\).
The tetrahedron \(H\) has vertices at \(O , P , Q\) and \(R\). The volume of \(H\) is 6 units. \(P ^ { \prime } , Q ^ { \prime } , R ^ { \prime }\) and \(H ^ { \prime }\) are the images of \(P , Q , R\) and \(H\) under A .
  1. In the case where \(x = 5\)
AQA Further Paper 2 2020 June Q8
9 marks Hard +2.3
  1. Factorise \(\begin{vmatrix} 2u + h + x & x + h & x^2 + h^2 \\ 0 & a & -a^2 \\ a + b & b & b^2 \end{vmatrix}\) as fully as possible. [6 marks]
  2. The matrix \(\mathbf{M}\) is defined by $$\mathbf{M} = \begin{bmatrix} 13 + x & x + 3 & x^2 + 9 \\ 0 & 5 & 25 \\ 8 & 3 & 9 \end{bmatrix}$$ Under the transformation represented by \(\mathbf{M}\), a solid of volume \(0.625 \text{m}^3\) becomes a solid of volume \(300 \text{m}^3\) Use your answer to part (a) to find the possible values of \(x\). [3 marks]