4.03r Solve simultaneous equations: using inverse matrix

128 questions

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Pre-U Pre-U 9795/1 2013 November Q9
10 marks Challenging +1.2
  1. Show that there is exactly one value of \(k\) for which the system of equations \begin{align} kx + 2y + kz &= 4
    3x + 10y + 2z &= m
    (k - 1)x - 4y + z &= k \end{align} does not have a unique solution. [4]
  2. Given that the system of equations is consistent for this value of \(k\), find the value of \(m\). [4]
  3. Explain the geometrical significance of a non-unique solution to a \(3 \times 3\) system of linear equations. [2]
Pre-U Pre-U 9795 Specimen Q10
10 marks Standard +0.3
  1. Find the inverse of the matrix \(\begin{pmatrix} 1 & 3 & 4 \\ 2 & 5 & -1 \\ 3 & 8 & 2 \end{pmatrix}\), and hence solve the set of equations \begin{align} x + 3y + 4z &= -5,
    2x + 5y - z &= 10,
    3x + 8y + 2z &= 8. \end{align} [5]
  2. Find the value of \(k\) for which the set of equations \begin{align} x + 3y + 4z &= -5,
    2x + 5y - z &= 15,
    3x + 8y + 3z &= k, \end{align} is consistent. Find the solution in this case and interpret it geometrically. [5]
OCR MEI FP2 2013 June Q1
Standard +0.3
1 & - 3 & - 2 \end{array} \right) \left( \begin{array} { l } x
y
z \end{array} \right) = \left( \begin{array} { c } p