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.

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]