Edexcel A-Level Mathematics A2 Fp3 6 4 Evaluation of 3 3 Determinants Questions

Practise Edexcel IAL FP3.6.4 by evaluating parameterised determinants, setting det M to zero and applying determinant scaling to volumes.

Exam points

set det M = 0 to find parameter values where a matrix has no inverse
expand a parameterised 3×3 determinant correctly and simplify it
use |det M| as a volume scale factor to determine a parameter

[Maximum number: 7]

The matrix M is given by

\left(\begin{array}{rrr} 2 & 0 & -1 \\ k & 3 & 2 \\ -2 & 1 & k \end{array}\right)

(a)

Show that $\operatorname{det} \mathbf{M}=5 k-10$

Given that $k \neq 2$

[ 2 ]

$\det M= \begin{vmatrix} 2&0&-1\\ k&3&2\\ -2&1&k \end{vmatrix} =2\begin{vmatrix}3&2\\1&k\end{vmatrix} -0\begin{vmatrix}k&2\\-2&k\end{vmatrix} (-1)\begin{vmatrix}k&3\\-2&1\end{vmatrix}$ =2(3k-2)-(k+6)=6k-4-k-6=5k-10

M1 A1*

(b)

Determine the possible values of k.

[ 5 ]

$\text{Volume of }T=\frac16|\mathbf a\cdot(\mathbf b\times\mathbf c)| =\frac{20}{3}$ $\text{Volume image}=|\det M|\cdot\frac{20}{3} \Rightarrow \frac{20}{3}|5k-10|=50$ $k=\frac72\quad\text{or}\quad k=\frac12$

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Edexcel A-Level Mathematics A2 Fp3 6 4 Evaluation of 3 3 Determinants Questions question 1

Question (a)

Question (b)