Date | May 2012 | Marks available | 5 | Reference code | 12M.3srg.hl.TZ0.2 |
Level | HL only | Paper | Paper 3 Sets, relations and groups | Time zone | TZ0 |
Command term | Find | Question number | 2 | Adapted from | N/A |
Question
The elements of sets P and Q are taken from the universal set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. P = {1, 2, 3} and Q = {2, 4, 6, 8, 10}.
Given that \(R = (P \cap Q')'\) , list the elements of R .
For a set S , let \({S^ * }\) denote the set of all subsets of S ,
(i) find \({P^ * }\) ;
(ii) find \(n({R^ * })\) .
Markscheme
\(P = \{ 1,{\text{ }}2,{\text{ }}3\} \)
\(Q' = \{ 1,{\text{ }}3,{\text{ }}5,{\text{ }}7,{\text{ }}9\} \)
so \(P \cap Q' = \{ 1,{\text{ }}3\} \) (M1)(A1)
so \((P \cap Q')' = \{ 2,{\text{ }}4,{\text{ }}5,{\text{ }}6,{\text{ }}7,{\text{ }}8,{\text{ }}9,{\text{ }}10\} \) A1
[3 marks]
(i) \({P^ * } = \left\{ {\{ 1\} ,{\text{ }}\{ 2\} ,{\text{ }}\{ 3\} ,{\text{ }}\{ 1,{\text{ }}2\} ,{\text{ }}\{ 2,{\text{ }}3\} ,{\text{ }}\{ 3,{\text{ }}1\} ,{\text{ }}\{ 1,{\text{ }}2,{\text{ }}3),{\text{ }}\emptyset } \right\}\) A2
Note: Award A1 if one error, A0 for two or more.
(ii) \({R^ * }\) contains: the empty set (1 element); sets containing one element (8 elements); sets containing two elements (M1)
\( = \left( {\begin{array}{*{20}{c}}
8 \\
0
\end{array}} \right) + \left( {\begin{array}{*{20}{c}}
8 \\
1
\end{array}} \right) + \left( {\begin{array}{*{20}{c}}
8 \\
2
\end{array}} \right) + ...\left( {\begin{array}{*{20}{c}}
8 \\
8
\end{array}} \right)\) (A1)
\( = {2^8}{\text{ }}( = 256)\) A1
Note: FT in (ii) applies if no empty set included in (i) and (ii).
[5 marks]
Examiners report
This was also a well answered question with many candidates obtaining full marks on both parts of the problem. Some candidates attempted to use a factorial rather than a sum of combinations to solve part (b) (ii) and this led to incorrect answers.
This was also a well answered question with many candidates obtaining full marks on both parts of the problem. Some candidates attempted to use a factorial rather than a sum of combinations to solve part (b) (ii) and this led to incorrect answers.