practice: Set_Venn diagram [165 marks]

[1 mark]1a.

Music lessons in Piano (P), Violin (V) and Flute (F) are offered to students at a school.

The Venn diagram shows the number of students who learn each kind of instrument.

Write down the total number of students in the school.

Markscheme145 (A1) (C1)

[1 mark]

1b. [3 marks]Write down the number of students who

(i) learn violin only;

(ii) learn piano or flute or both;

(iii) do not learn flute.

Markscheme(i) 56 (A1)

(ii) 85 (A1)

(iii) 89 (A1) (C3)

[3 marks]

[2 marks]1c. Explain, in words, the meaning of the part of the diagram that represents the set .P ∩ F ′

MarkschemeThe students who learn the piano and do not learn the flute. (A1)(A1) (C2)

Notes: Award (A1) for students who learn piano, not flute, (A1) for and (accept but). Accept correct alternative statements. Accept“The number of students who learn the piano and do not learn the flute”.

[2 marks]

2a. [2 marks]

A and B are subsets of U such that A = {multiples of 3}, B = {factors of 24}.

List the elements of

(i) U ;

(ii) B .

Markscheme(i) 3, 4, 5, 6, 7, 8, 9 (A1)

(ii) 3, 4, 6, 8 (A1)(ft) (C2)

Notes: Follow through from part (a)(i).

[2 marks]

U = {x|x is an integer, 2 < x < 10}

2b. [3 marks]Write down the elements of U on the Venn diagram.

Markscheme

(A1)(ft) for their 3, 6(A1)(ft) for their 4, 8, 9(A1)(ft) for their 5, 7 (A1)(ft)(A1)(ft)(A1)(ft) (C3)

Note: Follow through from their universal set and set B in part (a).

[3 marks]

[1 mark]2c. Write down .

Markscheme2 (A1)(ft) (C1)

Note: Follow through from their Venn diagram.

[1 mark]

n(A ∩ B)

[4 marks]3a.

100 students at IB College were asked whether they study Music (M), Chemistry (C), or Economics (E) with the following results.

10 study all three

15 study Music and Chemistry

17 study Music and Economics

12 study Chemistry and Economics

11 study Music only

6 study Chemistry only

Draw a Venn diagram to represent the information above.

Markscheme

(A1) for rectangle and three labelled circles (U need not be seen)

(A1) for 10 in the correct region

(A1) for 2, 7 and 5 in the correct regions

(A1) for 6 and 11 in the correct regions (A4)

[1 mark]3b. Write down the number of students who study Music but not Economics.

Markscheme16 (A1)(ft)

Note: Follow through from their Venn diagram.

3c. [4 marks]There are 22 Economics students in total.

(i) Calculate the number of students who study Economics only.

(ii) Find the number of students who study none of these three subjects.

Markscheme(i) (M1)

Note: Award (M1) for summing their 10, 7 and 2.

(A1)(ft)(G2)

Note: Follow through from their diagram. Award (M1)(A1)(ft) for answers consistent with their diagram irrespective of whetherworking seen. Award a maximum of (M1)(A0) for a negative answer.

(ii) (M1)

Note: Award (M1) for summing 22, and their 11, 5 and 6.

(A1)(ft)(G2)

Note: Follow through from their diagram. Award (M1)(A1)(ft) for answers consistent with their diagram and the use of 22irrespective of whether working seen. If negative values are used or implied award (M0)(A0).

10 + 7 + 2

22 − 19

= 3

22 + 11 + 5 + 6

100 − 44

= 56

3d. [7 marks]A student is chosen at random from the 100 that were asked above.

Find the probability that this student

(i) studies Economics;

(ii) studies Music and Chemistry but not Economics;

(iii) does not study either Music or Economics;

(iv) does not study Music given that the student does not study Economics.

Markscheme(i) (A1)(G1)

(ii) (A1)(ft)(A1)(G2)

Note: Award (A1)(ft) for their 5 in numerator, (A1) for denominator.

Follow through from their diagram.

(iii) (A1)(ft)(A1)(G2)

Note: Award (A1)(ft) for , (A1) for denominator.

Follow through from their diagram.

(iv) (0.794871...) (A1)(ft)(A1)(G2)

Note: Award (A1)(ft) for numerator, (A1) for denominator. Follow

through from part (d)(iii) for numerator.

( ,0.22,22%)22100

1150

( ,0.05,5%)5100

120

( ,0.62,62%)62100

3150

100 − (22 + 11 + their 5)

( ,0.795,79.5%)6278

3139

4a. [1 mark]

The following Venn diagram shows the relationship between the sets of numbers

The number –3 belongs to the set of and , but not , and is placed in the appropriate position on the Venn diagram as an

example.

Write down the following numbers in the appropriate place in the Venn diagram.

4

Markscheme

(A1)(A1)(A1)(A1)(A1)(A1) (C6)

Note: Award (A1) for each number correctly placed.

Award (A0) for any entry in more than one region.

[1 mark]

N, Z, Q and R.

Z, Q R N

[1 mark]4b. 13

Markscheme

(A1)(A1)(A1)(A1)(A1)(A1) (C6)

Note: Award (A1) for each number correctly placed.

Award (A0) for any entry in more than one region.

[1 mark]

3

[1 mark]4c.

Markscheme

(A1)(A1)(A1)(A1)(A1)(A1) (C6)

Note: Award (A1) for each number correctly placed.

Award (A0) for any entry in more than one region.

[1 mark]

π

[1 mark]4d. 0.38

Markscheme

(A1)(A1)(A1)(A1)(A1)(A1) (C6)

Note: Award (A1) for each number correctly placed.

Award (A0) for any entry in more than one region.

[1 mark]

[1 mark]4e.

Markscheme

(A1)(A1)(A1)(A1)(A1)(A1) (C6)

Note: Award (A1) for each number correctly placed.

Award (A0) for any entry in more than one region.

[1 mark]

5√

[1 mark]4f. −0.25

Markscheme

(A1)(A1)(A1)(A1)(A1)(A1) (C6)

Note: Award (A1) for each number correctly placed.

Award (A0) for any entry in more than one region.

[1 mark]

[1 mark]5a.

is the set of positive integers less than or equal to .

, and are subsets of .

List the elements of .

Markscheme (A1) (C1)

Note: Do not penalize the use of .

[1 mark]

U 10

A B C U

A = {even integers}

B = {multiples of 3}

C = {6, 7, 8, 9}

A

2,4,6,8,10

{ }

[1 mark]5b. List the elements of .

Markscheme (A1) (C1)

Note: Do not penalize the use of .

Follow through from part (a) only if their is listed.

[1 mark]

B

3,6,9

{ }

U

5c. [4 marks]Complete the Venn diagram with all the elements of .

Markscheme

(A1)(ft)(A1)(ft)(A1)(ft)(A1)(ft) (C4)

Notes: Award (A1)(ft) for the correct placement of .

Award (A1)(ft) for the correct placement of and and the empty region.

Award (A1)(ft) for the correct placement of , , , , and .

Award (A1)(ft) for the correct placement of and .

If an element is in more than one region, award (A0) for that element.

Follow through from their answers to parts (a) and (b).

[4 marks]

U

6

8 9

2 4 3 7 10

1 5

[5 marks]6a.

A group of women in the USA were asked whether they had visited the continents of Europe ( ) or South America ( ) or Asia (

). had visited all three continents

had visited Europe only

had visited South America only

had visited Asia only

had visited Europe and South America but had not visited Asia

had visited South America and Asia but had not visited Europe

had visited Europe and Asia but had not visited South America

had not visited any of these continents

Draw a Venn diagram, using sets labelled , and , to show this information.

120 E S

A

7

28

22

16

15

x

2x

20

E S A

Markscheme

(A1)(A1)(A1)(A1)(A1)

Notes: Award (A1) for rectangle and three labelled intersecting circles.

Award (A1) for in correct place.

Award (A1) for , and in the correct places.

Award (A1) for , and in the correct places.

Award (A1) for in the correct place.

Accept and instead of and .

Do not penalize if is omitted from the diagram.

[5 marks]

7

28 22 16

15 x 2x

20

4 8 x 2x

U

[2 marks]6b. Calculate the value of .

Markscheme (M1)

Note: Award (M1) for setting up a correct equation involving , the and values from their diagram.

(A1)(ft)(G2)

Note: Follow through from part (a). For the follow through to be awarded must be a positive integer.

[2 marks]

x

3x = 120 − (20 + 28 + 15 + 22 + 7 + 16)

x 120

x = 4

x

[2 marks]6c. Explain, in words, the meaning of .

Markscheme(Women who had visited) Europe or South America and (but had) not (visited) Asia (A1)(A1) Notes: Award (A1) for “(visited) Europe or South America” (or both).

Award (A1) for “and (but) had not visited Asia”.

(urope) union (outh America) intersected with not (sia) earns no marks, (A0). [2 marks]

(E ∪ S) ∩ A′

E S A

[1 mark]6d. Write down .n ((E ∪ S ∪ A ))′

Markscheme (A1)

Note: Award (A0) for the embedded answer of .

[1 mark]

20

n(20)

[2 marks]6e. Find the probability that a woman selected at random from the group had visited Europe.

Markscheme (A1)(ft)(A1)(G2)

Note: Award (A1)(ft) for numerator, follow through from their value of , or their diagram, (A1) for denominator.

[2 marks]

( , 0.483, 48.3% ) (0.48333 … )58120

2960

x

[2 marks]6f. Find the probability that a woman selected at random from the group had visited Europe, given that she had visited Asia.

Markscheme (A1)(ft)(A1)(ft)(G2)

Note: Award (A1)(ft) for numerator, (A1)(ft) for denominator, follow through from their value of or their diagram.

[2 marks]

( , 0.429, 42.9% ) (0.428571 … )1535

37

x

6g. [3 marks]Two women from the group are selected at random.

Find the probability that both women selected had visited South America.

Markscheme (A1)(ft)(M1)

Notes: Award (A1)(ft) for two correct fractions, follow through from their denominator in part (e), follow through the numerator from

their answer to part (b) or from their diagram, (M1) for multiplication of their two fractions.

(A1)(ft)(G2)

Notes: Award (A1)(M1)(A1) for correct fractions, correctly multiplied together with an answer of .

Award (A0)(M1)(A0) for .

Award (G1) for an answer of with no working seen.

[3 marks]

×48120

47119

= ( , 0.158, 15,8% ) (0.157983 … )225614 280

94595

0.16

× = 0.1648120

48120

0.16

[4 marks]7a.

A group of tourists went on safari to a game reserve. The game warden wanted to know how many of the tourists saw Leopard ( ),

Cheetah ( ) or Rhino ( ). The results are given as follows.

5 of the tourists saw all three

7 saw Leopard and Rhino

1 saw Cheetah and Leopard but not Rhino

4 saw Leopard only 3 saw Cheetah only 9 saw Rhino only

Draw a Venn diagram to show this information.

Markscheme

(A1)(A1)(A1)(A1)

Note: Award (A1) for rectangle and three labelled intersecting circles (the rectangle need not be labelled), (A1) for 5, (A1) for 2 and

1, (A1) for 4, 3 and 9.

[4 marks]

L

C R

7b. [2 marks]There were 25 tourists in the group and every tourist saw at least one of the three types of animal.

Find the number of tourists that saw Cheetah and Rhino but not Leopard.

Markscheme (M1)

Notes: Award (M1) for their seen even if total is greater than .

Do not award (A1)(ft) if their total is greater than .

(A1)(ft)(G2)

[2 marks]

25 − (5 + 2 + 1 + 4 + 3 + 9)

5 + 2 + 1 + 4 + 3 + 9 25

25

= 1

7c. [6 marks]There were 25 tourists in the group and every tourist saw at least one of the three types of animal.

Calculate the probability that a tourist chosen at random from the group

(i) saw Leopard;

(ii) saw only one of the three types of animal;

(iii) saw only Leopard, given that he saw only one of the three types of animal.

Markscheme(i) (A1)(ft)(A1)(G2)

Notes: Award (A1)(ft) for numerator, (A1) for denominator.

Follow through from Venn diagram.

(ii) (A1)(A1)(G2)

Notes: Award (A1) for numerator, (A1) for denominator.

There is no follow through; all information is given.

(iii) ) (A1)(A1)(ft)(G2)

Notes: Award (A1) for numerator, (A1)(ft) for denominator.

Follow through from part (c)(ii) only.

[6 marks]

(0.48, 48%)1225

(0.64, 64%)1625

(0.25, 25%)416

7d. [2 marks]There were 25 tourists in the group and every tourist saw at least one of the three types of animal.

If a tourist chosen at random from the group saw Leopard, find the probability that he also saw Cheetah.

Markscheme (A1)(A1)(ft)(G2)

Notes: Award (A1) for numerator, (A1)(ft) for denominator.

Follow through from Venn diagram.

[2 marks]

(0.5, 50%)612

8. [6 marks]The sets , and are defined as

U = {Real Numbers} , P = {Positive Numbers} and Q = {Rational Numbers}.

Write down in the correct region on the Venn diagram the numbers

, , , , , .

P Q U

227

5 × 10−2 sin( )60∘ 0 −8−−−√3 −π

Markscheme

(A1)(A1)(A1)(A1)(A1)(A1) (C6)

Note: Award (A1) for each number placed once in the correct region. Accept equivalent forms for numbers.

[6 marks]

[1 mark]9a.

A fitness club has 60 members. 35 of the members attend the club’s aerobics course (A) and 28 members attend the club’s yogacourse (Y). 17 members attend both courses. A Venn diagram is used to illustrate this situation.

Write down the value of q.

Markscheme17 (A1) (C1)

[1 mark]

[2 marks]9b. Find the value of p.

Markscheme35 – 17 (M1)

= 18 (A1) (C2)

Note: Award (A1) for correct answer only.

[2 marks]

[2 marks]9c. Calculate the number of members of the fitness club who attend neither the aerobics course (A) nor the yoga course (Y).

Markscheme (M1)

= 14 (A1)(ft) (C2)

Note: Follow through from (a) and (b).

[2 marks]

60 − (35 − 17) − (28 − 17) − 17

[1 mark]9d. Shade, on your Venn diagram, .

Markscheme

(A1) (C1)

[1 mark]

∩ YA′

[2 marks]10a.

A group of 33 people was asked about the passports they have. 21 have Australian passports, 15 have British passports and 3 haveneither.

Find the number that have both Australian and British passports.

Markscheme or equivalent (M1)

Note: Award (M1) for correct use of all four numbers.

(A1) (C2)

[2 marks]

21 + 15 + 3 − 33

= 6

10b. [2 marks]In the Venn diagram below, set A represents the people in the group with Australian passports and set B those with Britishpassports.

Write down the value of

(i) q ;

(ii) p and of r .

Markscheme(i) q = 6 (A1)(ft)

(ii) p =15, r = 9 (A1)(ft) (C2)

Note: Follow through from their answer to part (a).

[2 marks]

10c. [2 marks]

A group of 33 people was asked about the passports they have. 21 have Australian passports, 15 have British passports and 3 haveneither.

In the Venn diagram below, set A represents the people in the group with Australian passports and set B those with Britishpassports.

Find .

Markscheme15 + 6 + 3 (M1)

Note: Award (M1) for their figures seen in a correct calculation:

15 + 6 + 3 or 21 + 3 or 33 − 9

= 24 (A1)(ft) (C2)

Note: Follow through from parts (a) and (b) or from values shown on Venn diagram.

[2 marks]

n(A ∪ )B′

N

11a. [1 mark]

Consider the universal set , and the subsets and .

List the elements of the following set.

A

Markscheme6, 9, 12 (A1) (C1)

[1 mark]

U = {x ∈ N|3 < x < 13} A = {multiples of 3} B = {4, 6, 12}

11b. [1 mark]List the elements of the following set.

Markscheme9 (A1)(ft) (C1)

Note: Follow through from their part (a)(i).

[1 mark]

A ∩ B′

[2 marks]11c. Write down one element of .

Markschemeany element from {5, 7, 8, 10, 11} (A1)(A1)(ft) (C2)

Note: Award (A1)(ft) for finding , follow through from their A.

Award full marks if all correct elements of are listed.

[2 marks]

(A ∪ B)′

(A ∪ B)

(A ∪ B)′

11d. [2 marks]One of the statements in the table below is false. Indicate with an X which statement is false. Give a reason for your answer.

Markscheme

(R1)(A1) (C2)

Notes: Accept correct reason in words.

If the reason is incorrect, both marks are lost.

Do not award (R0)(A1).

[2 marks]

15 ∉ U

12a. [4 marks]

Beartown has three local newspapers: The Art Journal, The Beartown News, and The Currier.

A survey shows that

32 % of the town’s population read The Art Journal,

46 % read The Beartown News,

54 % read The Currier,

3 % read The Art Journal and The Beartown News only,

8 % read The Art Journal and The Currier only,

12 % read The Beartown News and The Currier only, and

5 % of the population reads all three newspapers.

Draw a Venn diagram to represent this information. Label A the set that represents The Art Journal readers, B the set thatrepresents The Beartown News readers, and C the set that represents The Currier readers.

Markscheme

(A1) for three circles and a rectangle (U need not be seen)

(A1) for 5

(A1) for 3, 8 and 12

(A1) for 16, 26 and 29 OR 32, 46, 54 placed outside the circles. (A4)

Note: Accept answers given as decimals or fractions.

[4 marks]

[2 marks]12b. Find the percentage of the population that does not read any of the three newspapers.

Markscheme100 – (16 + 26 + 29) – (8 + 5 + 3 + 12) (M1)

100 – 71 – 28

Note: Award (M1) for correct expression. Accept equivalent expressions, for example 100 – 71 – 28 or 100 – (71 + 28).

= 1 (A1)(ft)(G2)

Note: Follow through from their Venn diagram but only if working is seen.

[2 marks]

[2 marks]12c. Find the percentage of the population that reads exactly one newspaper.

Markscheme16 + 26 + 29 (M1)

Note: Award (M1) for 16, 26, 29 seen.

= 71 (A1)(ft)(G2)

Note: Follow through from their Venn diagram but only if working is seen.

[2 marks]

[2 marks]12d. Find the percentage of the population that reads The Art Journal or The Beartown News but not The Currier.

Markscheme16 + 3 + 26 (M1)

Note: Award (M1) for their 16, 3, 26 seen.

= 45 (A1)(ft)(G2)

Note: Follow through from their Venn diagram but only if working is seen.

[2 marks]

12e. [2 marks]A local radio station states that 83 % of the population reads either The Beartown News or The Currier.

Use your Venn diagram to decide whether the statement is true. Justify your answer.

MarkschemeTrue (A1)(ft)

100 – (1 –16) = 83 (R1)(ft)

OR

46 + 54 – 17 = 83 (R1)(ft)

Note: Do not award (A1)(R0). Follow through from their Venn diagram.

[2 marks]

12f. [4 marks]The population of Beartown is 120 000. The local radio station claimed that 34 000 of the town’s citizens read at least two ofthe local newspapers.

Find the percentage error in this claim.

Markscheme28% of 120000 (M1)

= 33600 (A1)

(M1)

Note: Award (M1) for 28 seen (may be implied by 33600 seen), award (M1) for correct substitution of their 33600 in the percentageerror formula. If an error is made in calculating 33600 award a maximum of (M1)(A0)(M1)(A0), the final accuracy mark is lost.

OR

(M1)

= 28.3(28.3333…) (A1)

(M1)

= 1.19% (1.19047...) (A1)(ft)(G3)

Note: % sign not required. Accept 1.07 (1.0714…) with use of 28.3. 1.18 with use of 28.33 and 1.19 with use of 28.333. Award(G3) for 1.07, 1.18 or 1.19 seen without working.

[4 marks]

% error = × 100(34000−33600)33600

× 10034000120000

% error = × 100(28.3333...−28)28

[1 mark]13a.

is the set of all the positive integers less than or equal to . , and are subsets of .

Write down the number of elements in .

Markscheme (one) (A1) (C1)

Note: , or earns no marks.

[1 mark]

U 12A B C U

A = {1, 2, 3, 4, 6, 12}

B = {odd integers}

C = {5, 6, 8}

A ∩ C

1

6 {6} {1}

[1 mark]13b. List the elements of .

Markscheme, , , , , (A1) (C1)

Note: Do not penalise if braces, parentheses or brackets are seen.

[1 mark]

B

1 3 5 7 9 11

13c. [4 marks]Complete the following Venn diagram with all the elements of .

Markscheme

(A1)(A1)(ft)(A1)(ft)(A1)(ft) (C4)

Notes: Award (A1) for the empty set .

Award (A1)(ft) for the correct placement of , , and .

Award (A1)(ft) for the correct placement of , , , , , , .

Award (A1)(ft) for the correct placement of .

Follow through from part (b).

[4 marks]

U

A ∩ B ∩ C

6 5 1 3

2 4 12 7 9 11 8

10

[2 marks]14a.

Let , and .

Find .

Markscheme (M1)

(A1) (C2)

Note: Award (M1) for correct substitution, (A1) for correct answer.

[2 marks]

P(A) = 0.5 P(B) = 0.6 P(A ∪ B) = 0.8

P(A ∩ B)

0.8 = 0.5 + 0.6 − P(A ∩ B)P(A ∩ B) = 0.3

[2 marks]14b. Find .P(A|B)

Markscheme (M1)

= 0.5 (A1)(ft) (C2)

Note: Award (M1) for correct substitution in conditional probability formula. Follow through from their answer to part (a), providedprobability is not greater than one.

[2 marks]

P(A|B) = 0.30.6

[2 marks]14c. Decide whether A and B are independent events. Give a reason for your answer.

Markscheme or 0.3 = 0.5 × 0.6 (R1)

OR

(R1)

they are independent. (Yes) (A1)(ft) (C2)

Note: Follow through from their answers to parts (a) or (b).

Do not award (R0)(A1).

[2 marks]

P(A ∩ B) = P(A) × P(B)

P(A|B) = P(A)

15. [6 marks]The Venn diagram shows the number sets , , and . Place each of the following numbers in the appropriate region of theVenn diagram.

, −3, π, cos 120°, 2.7 × 10 , 3.4 × 10

N Z Q R

14

3 –2

Markscheme

(A1)(A1)(A1)(A1)(A1)(A1) (C6)

Note: Award (A1) for each number placed once in the correct section. Accept equivalent forms for numbers.

[6 marks]

[5 marks]16a.

A survey of 100 families was carried out, asking about the pets they own. The results are given below.

56 owned dogs (S)

38 owned cats (Q)

22 owned birds (R)

16 owned dogs and cats, but not birds

8 owned birds and cats, but not dogs

3 owned dogs and birds, but not cats

4 owned all three types of pets

Draw a Venn diagram to represent this information.

Markscheme

(A1)(A1)(A1)(A1)(A1)

Note: Award (A1) for rectangle (U not required), (A1) for 3 intersecting circles, (A1) for 4 in central intersection, (A1) for 16, 3, 8and (A1) for 33, 10, 7 (ft) if subtraction is carried out, or for S(56), Q(38) and R(22) seen by the circles.

[5 marks]

[2 marks]16b. Find the number of families who own no pets.

Markscheme100 − 81 (M1)

19 (A1)(ft)(G2)

Note: Award (M1) for subtracting their total from 100.

[2 marks]

[3 marks]16c. Find the percentage of families that own exactly one pet.

Markscheme (M1)

Note: Award (M1) for adding their values from (a).

(A1)(ft)

50 % (50) (A1)(ft)(G3)

[3 marks]

33 + 10 + 7

( ) × 100 %50100

[2 marks]16d. A family is chosen at random. Find the probability that they own a cat, given that they own a bird.

MarkschemeP (own a cat given they own a bird) (A1)(ft)(A1)(ft)

Note: Award (A1)(ft) for the numerator, (A1)(ft) for the denominator.

[2 marks]

= (0.545, )1222

611

17a. [1 mark]

The universal set U is the set of integers from 1 to 20 inclusive.

A and B are subsets of U where:

A is the set of even numbers between 7 and 17.

B is the set of multiples of 3.

List the elements of the following sets:

A,

MarkschemeA = 8, 10, 12, 14, 16 (A1) (C1)

[1 mark]

17b. [1 mark]List the elements of the following sets:

B,

MarkschemeB = 3, 6, 9, 12, 15, 18 (A1) (C1)

[1 mark]

17c. [2 marks]List the elements of the following sets:

,

Markscheme = 3, 6, 8, 9,10,12,14,15,16,18 (A2)(ft)

Award (A1) only if a single element is missing or a single extra element is present, (A0) otherwise. (C2)

[2 marks]

A ∪ B

A ∪ B

17d. [2 marks]List the elements of the following sets:

.

Markscheme = 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20 (A1)(ft)

= 8, 10, 14, 16 (A1)(ft) (C2)

[2 marks]

A ∩ B′

B′

A ∩ B′

[1 mark]18a.

The Venn diagram shows the numbers of pupils in a school according to whether they study the sciences Physics ( ), Chemistry ( ),Biology ( ).

Write down the number of pupils that study Chemistry only.

Markscheme (A1) (C1)

[1 mark]

P C

B

9

[1 mark]18b. Write down the number of pupils that study exactly two sciences.

Markscheme (A1) (C1)

[1 mark]

12

[2 marks]18c. Write down the number of pupils that do not study Physics.

Markscheme (M1)

(A1) (C2)

Note: Award (A1) for seen if answer is incorrect.

[2 marks]

8 + 3 + 9 + 6

= 26

20

[2 marks]18d. Find .

Markscheme (M1)

(A1) (C2)

Note: Award (A1) for or seen if answer is incorrect.

[2 marks]

n[(P ∪ B) ∩ C]

5 + 2 + 3

= 10

29 19

19a. [2 marks]Shade on the diagram below.

Markscheme

not shading or shading (A1)

correct shading (A1) (C2)

[2 marks]

(A ∪ B) ∩ C ′

C A ∪ B

19b. [2 marks]In the Venn diagram below, the number of elements in each region is given.

Find .

MarkschemeIdentifying the correct 5 numbers , , , , (A1)

(A1) (C2)

[2 marks]

n((P ∩ Q) ∪ R)

3 4 5 6 9

27

19c. [2 marks] is the set of positive integers, .

is the set of even numbers.

is the set of multiples of .

(i) List the first six elements of the set .

(ii) List the first six elements of the set .

U Z+

E

M 3

M

∩ ME ′

Markscheme(i) brackets not required. (A1)

(ii) (ft) from (i). (A1)(ft) (C2)

[2 marks]

M = {3, 6, 9, 12, 15, 18}

∩ M = {3, 9, 15, 21, 27, 33}E ′

20a. [2 marks]

A school offers three activities, basketball (B), choir (C) and drama (D). Every student must participate in at least one activity.

16 students play basketball only.

18 students play basketball and sing in the choir but do not do drama.

34 students play basketball and do drama but do not sing in the choir.

27 students are in the choir and do drama but do not play basketball.

Enter the above information on the Venn diagram below.

Markscheme

(A2)

(A1) only if 1 error

(A0) otherwise (C2)

[2 marks]

20b. [1 mark]99 of the students play basketball, 88 sing in the choir and 110 do drama.

Calculate the number of students x participating in all three activities.

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© International Baccalaureate Organization 2016 International Baccalaureate® - Baccalauréat International® - Bachillerato Internacional®

Markscheme

(A1) (C1)

[1 mark]

x + 16 + 18 + 34 = 99

x = 31

20c. [3 marks]99 of the students play basketball, 88 sing in the choir and 110 do drama.

Calculate the total number of students in the school.

MarkschemeChoir only (A1)(ft)

Drama only (A1)(ft)

Total (A1)(ft) (C3)

[3 marks]

= 88 − (18 + 27 + 31) = 12

= 110 − (27 + 34 + 31) = 18

= 16 + 34 + 18 + 31 + 12 + 27 + 18 = 156