| Date | November 2014 | Marks available | 4 | Reference code | 14N.2.hl.TZ0.5 |
| Level | HL only | Paper | 2 | Time zone | TZ0 |
| Command term | Find | Question number | 5 | Adapted from | N/A |
Question
The lines \({l_1}\) and \({l_2}\) are defined as
\({l_1}:\frac{{x - 1}}{3} = \frac{{y - 5}}{2} = \frac{{z - 12}}{{ - 2}}\)
\({l_2}:\frac{{x - 1}}{8} = \frac{{y - 5}}{{11}} = \frac{{z - 12}}{6}\).
The plane \(\pi \) contains both \({l_1}\) and \({l_2}\).
Find the Cartesian equation of \(\pi \).
The line \({l_3}\) passing through the point \((4,{\text{ }}0,{\text{ }}8)\) is perpendicular to \(\pi \).
Find the coordinates of the point where \({l_3}\) meets \(\pi \).
Markscheme
attempting to find a normal to \(\pi {\text{ }}eg{\text{ }}\left( {\begin{array}{*{20}{c}} 3 \\ 2 \\ { - 2} \end{array}} \right) \times \left( {\begin{array}{*{20}{c}} 8 \\ {11} \\ 6 \end{array}} \right)\) (M1)
\(\left( {\begin{array}{*{20}{c}} 3 \\ 2 \\ { - 2} \end{array}} \right) \times \left( {\begin{array}{*{20}{c}} 8 \\ {11} \\ 6 \end{array}} \right) = 17\left( {\begin{array}{*{20}{c}} 2 \\ { - 2} \\ 1 \end{array}} \right)\) (A1)
\({{r}} \bullet \left( {\begin{array}{*{20}{c}} 2 \\ { - 2} \\ 1 \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 1 \\ 5 \\ {12} \end{array}} \right) \bullet \left( {\begin{array}{*{20}{c}} 2 \\ { - 2} \\ 1 \end{array}} \right)\) M1
\(2x - 2y + z = 4\) (or equivalent) A1
[4 marks]
\({l_3}:{{r}} = \left( {\begin{array}{*{20}{c}} 4 \\ 0 \\ 8 \end{array}} \right) + t\left( {\begin{array}{*{20}{c}} 2 \\ { - 2} \\ 1 \end{array}} \right),\;\;\;t \in \mathbb{R}\) (A1)
attempting to solve \(\left( {\begin{array}{*{20}{c}} {4 + 2t} \\ { - 2t} \\ {8 + t} \end{array}} \right) \bullet \left( {\begin{array}{*{20}{c}} 2 \\ { - 2} \\ 1 \end{array}} \right) = 4\;\;\;{\text{for }}t\;\;\;ie{\text{ }}9t + 16 = 4\;\;\;{\text{for }}t\) M1
\(t = - \frac{4}{3}\) A1
\(\left( {\frac{4}{3},{\text{ }}\frac{8}{3},{\text{ }}\frac{{20}}{3}} \right)\) A1
[4 marks]
Total [8 marks]
Examiners report
Part (a) was reasonably well done. Some candidates made numerical errors when attempting to find a normal to \(\pi \).
In part (b), a number of candidates were awarded follow through marks from numerical errors committed in part (a).
