1. 1
   A student is practising set notation using the universal set $\mathcal{E} = \{1,2,3,4,5,6,7,8,9,10\}$ with subsets $A = \{2,4,6,8,10\}$ and $B = \{3,6,9\}$.

   (a)

   The universal set is $\mathcal{E} = \{1,2,3,4,5,6,7,8,9,10\}$, $A = \{2,4,6,8,10\}$ and $B = \{3,6,9\}$. Find $A \cap B$.

   [1 mark]

   • A$\{2,3,4,6,8,9,10\}$
   • B$\{6\}$
   • C$\{3,9\}$
   • D$\varnothing$

   (b)

   Find $n(A \cup B)$ for $A = \{2,4,6,8,10\}$ and $B = \{3,6,9\}$.

   [1 mark]

   • A$5$
   • B$8$
   • C$7$
   • D$3$

   (c)

   Find $A'$ (the complement of $A$) where $\mathcal{E} = \{1,2,3,4,5,6,7,8,9,10\}$ and $A = \{2,4,6,8,10\}$.

   [1 mark]

   • A$\mathcal{E}$
   • B$\{2,4,6,8,10\}$
   • C$\varnothing$
   • D$\{1,3,5,7,9\}$

   Total for question 1: 3 marks

2. 2
   A class of 30 students studies French and/or Spanish (or neither), and their teacher is analysing overlaps using set notation.

   (a)

   In a class of 30 students, 18 study French, 14 study Spanish, and 6 study neither. Let $F$ be the set of students studying French and $S$ the set studying Spanish. Using $n(F \cup S) = n(F) + n(S) - n(F \cap S)$, find $n(F \cap S)$, the number of students who study both languages.

   [2 marks]

   (b)

   Using your answer to part (a), find the number of students who study French only (not Spanish).

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A market researcher surveys 60 people about laptop and tablet ownership and needs to use set notation and Venn diagram reasoning to find probabilities.

   (a)

   In a survey of 60 people, $P$ is the set of people who own a laptop and $T$ is the set who own a tablet. $n(P) = 38$, $n(T) = 27$, and $n(P \cap T) = 15$. Draw up the information as a Venn diagram calculation (without drawing) by finding the number of people who own a laptop only, a tablet only, and neither.

   [3 marks]

   (b)

   One person is selected at random from the 60 people surveyed. Using your working from part (a), find the probability that this person owns a tablet only, giving your answer as a fraction in its simplest form.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A school records how many students take part in music, sport and art, some in more than one activity, and wants to analyse the overlaps using three-set Venn diagram reasoning.

   (a)

   In a school of 120 students, sets are defined as $M$ = students who play music, $S$ = students who play sport, and $A$ = students who paint art. It is known that $n(\mathcal{E}) = 120$, $n(M) = 55$, $n(S) = 48$, $n(A) = 40$, $n(M \cap S) = 20$, $n(M \cap A) = 15$, $n(S \cap A) = 12$, and $n(M \cap S \cap A) = 5$. Find the number of students who play music only (not sport, not art).

   [4 marks]

   (b)

   Find the number of students who take part in none of the three activities.

   [4 marks]

   (c)

   A student is selected at random from the school. Using your answers to parts (a) and (b), find the probability that the student plays exactly one of the three activities (music only, sport only, or art only), giving your answer as a fraction in its simplest form. (Sport only $= 48 - 20 - 12 + 5 = 21$; art only $= 40 - 15 - 12 + 5 = 18$.)

   [5 marks]

   Total for question 4: 13 marks

End of questions