4.04a Line equations: 2D and 3D, cartesian and vector forms

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Pre-U Pre-U 9795/1 2015 June Q10
11 marks Standard +0.8
  1. Find a vector equation for the line of intersection of the planes with cartesian equations $$x + 7y - 6z = -10 \quad \text{and} \quad 3x - 5y + 8z = 48.$$ [5]
  2. Determine the value of \(k\) for which the system of equations \begin{align} x + 7y - 6z &= -10
    3x - 5y + 8z &= 48
    kx + 2y + 3z &= 16 \end{align} does not have a unique solution and show that, for this value of \(k\), the system of equations is inconsistent. [6]
Edexcel AEA 2011 June Q6
19 marks Hard +2.3
The line \(L\) has equation $$\mathbf{r} = \begin{pmatrix} 13 \\ -3 \\ -8 \end{pmatrix} + t \begin{pmatrix} -5 \\ 3 \\ 4 \end{pmatrix}$$ The point \(P\) has position vector \(\begin{pmatrix} -7 \\ 2 \\ 7 \end{pmatrix}\). The point \(P'\) is the reflection of \(P\) in \(L\).
  1. Find the position vector of \(P'\). [6]
  2. Show that the point \(A\) with position vector \(\begin{pmatrix} -7 \\ 9 \\ 8 \end{pmatrix}\) lies on \(L\). [1]
  3. Show that angle \(PAP' = 120°\). [3]
% Figure 3 shows kite APBP' with angle at A = 120° \includegraphics{figure_3} Figure 3 The point \(B\) lies on \(L\) and \(APBP'\) forms a kite as shown in Figure 3. The area of the kite is \(50\sqrt{3}\)
  1. Find the position vector of the point \(B\). [5]
  2. Show that angle \(BPA = 90°\). [2]
The circle \(C\) passes through the points \(A\), \(P\), \(P'\) and \(B\).
  1. Find the position vector of the centre of \(C\). [2]
[Total 19 marks]