| Date | None Specimen | Marks available | 5 | Reference code | SPNone.2.sl.TZ0.4 |
| Level | SL only | Paper | 2 | Time zone | TZ0 |
| Command term | Find | Question number | 4 | Adapted from | N/A |
Question
Consider the lines \({L_1}\) , \({L_2}\) , \({L_2}\) , and \({L_4}\) , with respective equations.
\({L_1}\) : \(\left( {\begin{array}{*{20}{c}}
x\\
y\\
z
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
1\\
2\\
3
\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}
3\\
{ - 2}\\
1
\end{array}} \right)\)
\({L_2}\) : \(\left( \begin{array}{l}
x\\
y\\
z
\end{array} \right) = \left( \begin{array}{l}
1\\
2\\
3
\end{array} \right) + p\left( \begin{array}{l}
3\\
2\\
1
\end{array} \right)\)
\({L_3}\) : \(\left( {\begin{array}{*{20}{c}}
x\\
y\\
z
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
0\\
1\\
0
\end{array}} \right) + s\left( {\begin{array}{*{20}{c}}
{ - 1}\\
2\\
{ - a}
\end{array}} \right)\)
\({L_4}\) : \(\left( {\begin{array}{*{20}{c}}
x\\
y\\
z
\end{array}} \right) = q\left( {\begin{array}{*{20}{c}}
{ - 6}\\
4\\
{ - 2}
\end{array}} \right)\)
Write down the line that is parallel to \({L_4}\) .
Write down the position vector of the point of intersection of \({L_1}\) and \({L_2}\) .
Given that \({L_1}\) is perpendicular to \({L_3}\) , find the value of a .
Markscheme
\({L_1}\) A1 N1
[1 mark]
\(\left( \begin{array}{l}
1\\
2\\
3
\end{array} \right)\) A1 N1
[1 mark]
choosing correct direction vectors A1A1
e.g. \(\left( {\begin{array}{*{20}{c}}
3\\
{ - 2}\\
1
\end{array}} \right)\) , \(\left( {\begin{array}{*{20}{c}}
{ - 1}\\
2\\
{ - a}
\end{array}} \right)\)
recognizing that \({\boldsymbol{a}} \bullet {\boldsymbol{b}} = 0\) M1
correct substitution A1
e.g. \( - 3 - 4 - a = 0\)
\(a = - 7\) A1 N3
[5 marks]
