A2 June 2022 Paper 2 Q7

OCR ACurrent spec13 marksMatrices

7 You are given that \(a\) is a parameter which can take only real values.

The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} 2 & 4 & -6 \\ -3 & 10 - 4a & 9 \\ 7 & 4 & 4 \end{pmatrix}\).

(a) Find an expression for the determinant of \(\mathbf{A}\) in terms of \(a\). [2]

You are given the following system of equations in \(x\), \(y\) and \(z\).

\[\begin{aligned} 2x + 4y - 6z &= 6 \\ -3x + (10 - 4a)y + 9z &= -9 \\ 7x + 4y + 4z &= 11 \end{aligned}\]

The system can be written in the form \(\mathbf{A}\begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 6 \\ -9 \\ 11 \end{pmatrix}\).

(b)
(i) In the case where \(\mathbf{A}\) is not singular, solve the given system of equations by using \(\mathbf{A}^{-1}\). [5]
(ii) In the case where \(\mathbf{A}\) is singular describe the configuration of the planes whose equations are the three equations of the system. [3]

The transformation represented by \(\mathbf{A}\) is denoted by T.
A 3-D object of volume \(|5a - 20|\) is transformed by T to a 3-D image.

(c)
(i) Determine the range of values of \(a\) for which the orientation of the image is the reverse of the orientation of the object. [1]
(ii) Determine the range of values of \(a\) for which the volume of the image is less than the volume of the object. [2]
‹ Previous109 of 176 in MatricesNext ›