EXM.1.AHL.TZ0.28

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

Consider the matrix A $= \left( {\begin{array}{*{20}{c}} {{{\text{e}}^x}}&{{{\text{e}}^{ - x}}} \\ {2 + {{\text{e}}^x}}&1 \end{array}} \right)$ , where $x \in \mathbb{R}$ .

Find the value of $x$ for which A is singular.

Markscheme / solution

finding det A = ${{\text{e}}^x} - {{\text{e}}^{ - x}}\left( {2 + {{\text{e}}^x}} \right)$ or equivalent A1

A is singular ⇒ det A = 0 (R1)

${{\text{e}}^x} - {{\text{e}}^{ - x}}\left( {2 + {{\text{e}}^x}} \right) = 0$

${{\text{e}}^{2x}} - {{\text{e}}^x} - 2 = 0$ A1

solving for ${{\text{e}}^x}$ (M1)

${{\text{e}}^x}$ > 0 (or equivalent explanation) (R1)

${{\text{e}}^x} = 2$

$x =$ ln 2 (only) A1 N0

[6 marks]

Examiners’ report

[N/A]