OCR MEI C4 (Core Mathematics 4)

2 In Fig. 6, \(\mathrm { ABC } , \mathrm { ACD }\) and AED are right-angled triangles and \(\mathrm { BC } = 1\) unit. Angles CAB and CAD are \(\theta\) and \(\phi\) respectively. \begin{figure}[h] \end{figure}

1. Find AC and AD in terms of \(\theta\) and \(\phi\).
2. Hence show that \(\mathrm { DE } = 1 + \frac { \tan \phi } { \tan \theta }\).

3 Verify that the vector \(\mathbf { d } - \mathbf { j } + 4 \mathbf { k }\) is perpendicular to the plane through the points \(\mathrm { A } ( 2,0,1 ) , \mathrm { B } ( 1,2,2 )\) and \(\mathrm { C } ( 0 , - 4,1 )\). Hence find the cartesian equation of the plane. [5]

4 Show that the straight lines with equations \(\mathbf { r } = \begin{array} { r r r } 2 & + \lambda & 0
4 & & 1 \end{array}\) and \(\mathbf { r } = \quad + \mu \quad\) meet.
Find their point of intersection.

5 The points A , B and C have coordinates \(( 2,0 , - 1 ) , ( 4,3 , - 6 )\) and \(( 9,3 , - 4 )\) respectively.

1. Show that AB is perpendicular to BC .
2. Find the area of triangle ABC .

6

1. Verify that the lines \(\left. \mathbf { r } = \begin{array} { r } - 5
   3
   4 \end{array} \right) + \lambda \left( \begin{array} { r } 3
   0
   - 1 \end{array} \right)\) and \(\left. \left. \mathbf { r } = \begin{array} { r } - 1
   4
   2 \end{array} \right) + \mu - \begin{array} { r } 2
   - 1
   0 \end{array} \right)\) meet at the point ( \(1,3,2\) ).
2. Find the acute angle between the lines.