CAIE A-Level Further Math A2 2.2.2 Determinants and Inverses Questions

Practise calculating determinants, using inverse matrices to solve equations and interpreting singular systems with Further Mathematics Paper 2 questions.

Exam points

Calculate determinants and use inverse matrices to solve equations and interpret singular systems.

[Maximum number: 4]

Show that the system of equations

\begin{aligned} 14 x-4 y+6 z & =5 \\ x+y+k z & =3 \\ -21 x+6 y-9 z & =14 \end{aligned}

where k is a constant, does not have a unique solution and interpret this situation geometrically.

$\begin{aligned} \left|\begin{array}{ccc} 14 -4 1 1 k -21 6 -9 \end{array}\right|=14\left|\begin{array}{cc} 1 k 6 -9 \end{array}\right|+4\left|\begin{array}{cc} 1 k -21 -9 \end{array}\right|+6\left|\begin{array}{cc} 1 -21 6 \end{array}\right| =14(-9-6 k)+4(-9+21 k)+6(6+21)=0 \end{aligned}$
Evaluates determinant. Can expand along any row e.g. -(36-36)+(-126+126)+k(84-84).

If using row operations, they must show an inconsistent system for M1. All their row operations must be correct for A1.

Two parallel planes, not identical.
Other plane not parallel.
4

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CAIE A-Level Further Math A2 2.2.2 Determinants and Inverses Questions question 1