The universal set \(\xi\) represents the 60 members who visited a community library last month. Set \(F\) is the set of members who borrowed a fiction book,...
Question 1 Report
The universal set \(\xi\) represents the 60 members who visited a community library last month. Set \(F\) is the set of members who borrowed a fiction book, and set \(A\) is the set of members who borrowed an audiobook. The Venn diagram shows the number of members in each region, in terms of \(x\).
- Form an equation in \(x\) and solve it. (2)
- Write down \(n(A)\). (1)
Answer Details
Every one of the 60 members falls into exactly one region of the diagram, so adding all four regions (each written in terms of \(x\)) and setting the total to 60 solves for \(x\); substituting back into the audiobook regions then gives \(n(A)\).
(a) Adding the four regions: \((x+4)+x+(2x-5)+(x+1) = 5x = 60\), so \(x=12\), giving the completed regions 16, 12, 19 and 13 shown above. [2 marks]
(b) \(n(A)\) is every region inside the audiobook circle: \((2x-5)+x = 19+12 = 31\). [1 mark]
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