Higher November 2021 Paper 1 Q16

EdexcelHigherCurrent spec5 marksSets

16 There are 32 students in a class.
In one term these 32 students each took a test in Maths (\(M\)), in English (\(E\)) and in French (\(F\)).

25 students passed the test in Maths.
20 students passed the test in English.
14 students passed the test in French.
18 students passed the tests in Maths and English.
11 students passed the tests in Maths and French.
4 students failed all three tests.
\(x\) students passed all three tests.

The incomplete Venn diagram gives some more information about the results of the 32 students.

Venn diagram of three overlapping circles M, E and F inside a universal set; 3 is written in the region for M only and 2 in the region for E and F but not M

(a) Use all the given information about the results of students who passed the test in Maths to find the value of \(x\). (2)

(b) Use your value of \(x\) to complete the Venn diagram to show the number of students in each subset. (2)

A student who passed the test in Maths is chosen at random.

Mark scheme (a)

Scheme	Marks
\((11 - x) + (x) + (18 - x) + 3 = 25\) oe or \((11 - x) + (x) + (18 - x) + 3 + 7 = 25 + 7\) oe or \(x + y + z = 25 - 3\) and \(x + z = 11\) and \(x + y = 18\) oe where \(y = M \cap E \cap F'\) and \(z = M \cap F \cap E'\)	M1
Correct answer scores full marks (unless from obvious incorrect working) Answer: 7	A1
	(2)

Notes

M1: for setting up a correct equation

A1: (allow 7 in the Venn diagram if no answer is given in (a))

Completed Venn diagram: M only (3), M and E only 11, E only 0, all three 7, M and F only 4, E and F only (2), F only 1, outside the circles 4 — Mark scheme (b)

Scheme	Marks
M only (3), \(M \cap E\) only 11, \(E\) only 0, all three 7, \(M \cap F\) only 4, \(E \cap F\) only (2), \(F\) only 1, outside 4	B2
	(2)

Mark scheme (c)

Scheme	Marks
\(\dfrac{3 + \text{``}{11}\text{''}}{25}\) or \(\dfrac{3 + (18 - \text{``}{7}\text{''})}{25}\) or 0.56 oe Answer: \(\dfrac{14}{25}\)	B1ft
	(1)
	(5 marks)

Notes

B1ft: for \(18 - x\), \(x\) and \(11 - x\) oe