EXM.1.AHL.TZ0.54

pestleMathematicsAIHLPaper 1EXM· ahl-1-14-introduction-to-matricessource ↗

Consider the system of equations A $\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 2 \\ { - 3} \end{array}} \right)$ where A $= \left( {\begin{array}{*{20}{c}} {k + 1}&{ - k} \\ 2&{k - 1} \end{array}} \right)$ and $k \in \mathbb{R}$ .

Find det A.

[2]

Find the set of values of $k$ for which the system has a unique solution.

[4]

Markscheme / solution

Attempting to find det A (M1)

det A $= {k^2} + 2k - 1$ A1 N2

[2 marks]

System has a unique solution provided det A ≠ 0 (R1)

${k^2} + 2k - 1 \ne 0$ (A1)

Solving ${k^2} + 2k - 1 \ne 0$ or equivalent for $k$ M1

$k \in \mathbb{R}\,{\text{\ }}\left\{ { - 1 \pm \,\sqrt 2 } \right\}\,\,\left( {{\text{accept}}\,\,k \ne - 1 \pm \,\sqrt 2 {\text{,}}\,\,\,k \ne - 2.41{\text{,}}\,\,0.414} \right)$ A1 N3

[4 marks]

Examiners’ report

[N/A]