| Date | May 2017 | Marks available | 3 | Reference code | 17M.1.hl.TZ0.3 |
| Level | HL only | Paper | 1 | Time zone | TZ0 |
| Command term | Solve | Question number | 3 | Adapted from | N/A |
Question
Consider the system of equations
\[\left[ {\begin{array}{*{20}{l}} 1&2&1&3 \\ 2&1&3&1 \\ 5&1&8&0 \\ 3&3&4&4 \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {{x_1}} \\ {{x_2}} \\ {{x_3}} \\ {{x_4}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 2 \\ 3 \\ \lambda \\ \mu \end{array}} \right]\]
Determine the value of \(\lambda \) and the value of \(\mu \) for which the equations are consistent.
For these values of \(\lambda \) and \(\mu \), solve the equations.
State the rank of the matrix of coefficients, justifying your answer.
Markscheme
using row operations on \(4 \times 5\) matrix, M1
\(\left[ {\begin{array}{*{20}{c}} 1&2&1&3 \\ 0&{ - 3}&1&{ - 5} \\ 0&{ - 9}&3&{ - 15} \\ 0&{ - 3}&1&{ - 5} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} 2 \\ { - 1} \\ {\lambda - 10} \\ {\mu - 6} \end{array}} \right]\,\,\,\begin{array}{*{20}{c}} {} \\ {{\text{row}}2 - 2 \times {\text{row1}}} \\ {{\text{row}}3 - 5 \times {\text{row1}}} \\ {{\text{row}}4 - 3 \times {\text{row1}}} \end{array}\) A2
or any alternative correct row reductions
Note: Award A1 for two correct row reductions.
\(\lambda = 7\) A1
\(\mu = 5\) A1
[5 marks]
let \({x_3} = \alpha ,{\text{ }}{x_4} = \beta \) M1
\({x_2} = \frac{{1 + \alpha - 5\beta }}{3}\) A1
\({x_1} = \frac{{4 - 5\alpha + \beta }}{3}\) A1
Note: Alternative solutions are available.
[3 marks]
the rank is 2 A1
because the matrix has 2 independent rows or a correct comment based on the use of rref R1
[2 marks]
