| Date | May 2011 | Marks available | 4 | Reference code | 11M.1.hl.TZ1.11 |
| Level | HL only | Paper | 1 | Time zone | TZ1 |
| Command term | Find | Question number | 11 | Adapted from | N/A |
Question
The points A(1, 2, 1) , B(−3, 1, 4) , C(5, −1, 2) and D(5, 3, 7) are the vertices of a tetrahedron.
Find the vectors \(\overrightarrow {{\text{AB}}} \) and \(\overrightarrow {{\text{AC}}} \).
Find the Cartesian equation of the plane \(\prod \) that contains the face ABC.
Markscheme
\(\overrightarrow {{\text{AB}}} = \left( {\begin{array}{*{20}{c}}
{ - 4} \\
{ - 1} \\
3
\end{array}} \right)\), \(\overrightarrow {{\text{AC}}} = \left( {\begin{array}{*{20}{c}}
4 \\
{ - 3} \\
1
\end{array}} \right)\) A1A1
Note: Accept row vectors.
[2 marks]
\(\overrightarrow {{\text{AB}}} \times \overrightarrow {{\text{AC}}} = \left| {\begin{array}{*{20}{c}}
{\boldsymbol{i}}&{\boldsymbol{j}}&{\boldsymbol{k}} \\
{ - 4}&{ - 1}&3 \\
4&{ - 3}&1
\end{array}} \right| = \left( {\begin{array}{*{20}{c}}
8 \\
{16} \\
{16}
\end{array}} \right)\) M1A1
normal \({\boldsymbol{n}} = \left( {\begin{array}{*{20}{c}}
1 \\
2 \\
2
\end{array}} \right)\) so \({\boldsymbol{r}} \cdot \left( {\begin{array}{*{20}{c}}
1 \\
2 \\
2
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
1 \\
2 \\
1
\end{array}} \right) \cdot \left( {\begin{array}{*{20}{c}}
1 \\
2 \\
2
\end{array}} \right)\) (M1)
\(x + 2y + 2z = 7\) A1
Note: If attempt to solve by a system of equations:
Award A1 for 3 correct equations, A1 for eliminating a variable and A2 for the correct answer.
[4 marks]
Examiners report
Most candidates attempted this question and scored at least a few marks in (a) and (b). Part (c) was more challenging to many candidates who were unsure how to find the required distance. Part (d) was attempted by many candidates some of whom benefited from follow through marks due to errors in previous parts. However, many candidates failed to give the correct answer to this question due to the use of the simplified vector found in (b) showing little understanding of the role of the magnitude of this vector. Part (e) was poorly answered. Overall, this question was not answered to the expected level, showing that many candidates have difficulties with vectors and are unable to answer even standard questions on this topic.
Most candidates attempted this question and scored at least a few marks in (a) and (b). Part (c) was more challenging to many candidates who were unsure how to find the required distance. Part (d) was attempted by many candidates some of whom benefited from follow through marks due to errors in previous parts. However, many candidates failed to give the correct answer to this question due to the use of the simplified vector found in (b) showing little understanding of the role of the magnitude of this vector. Part (e) was poorly answered. Overall, this question was not answered to the expected level, showing that many candidates have difficulties with vectors and are unable to answer even standard questions on this topic.
