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3 marksMatrices and Systems of Linear Equations

CAPE Pure Mathematics Unit 2 · May/June 2008 (T&T) · Paper 2 · Question 6(b)(i)

The Popular Taxi Service in a certain city provides transportation for tours of the city using cars, coaches and buses. Selection of vehicles for tours of distances (in km) is as follows: x cars, 2y coaches and 3z buses cover 34\text{ km} tours. 2x cars, 3y coaches and 4z buses cover 49\text{ km} tours. 3x cars, 4y coaches and 6z buses cover 71\text{ km} tours.

Express the information above as a matrix equation AX = Y, where A is a 3 \times 3 matrix, and X and Y are 3 \times 1 matrices with X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}.

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Other parts of this question

  1. 6(a)Solve for x the equation \begin{vmatrix} 1 & 1 & 1 \\ x & 2 & 1 \\ x^3 & 8 & 1 \end{vmatrix} = 0.[12 marks]
  2. 6(b)(ii)a)Calculate AB.[3 marks]
  3. 6(b)(ii)b)Deduce the inverse A^{-1} of A.[3 marks]
  4. 6(b)(iii)Hence, or otherwise, determine the number of cars and buses used in the 34\text{ km} tours.[4 marks]

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