Sets and Venn Diagrams Quiz 2 - Uplift Education...Sets and Venn Diagrams Quiz 2 [101 marks] 1a. [2 marks] A school offers three activities, basketball (B), choir (C) and drama (D)

Sets and Venn Diagrams Quiz 2 [101 marks]

1a. [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]

Examiners reportVenn diagrams continue to be a problem area. Quite a good number of candidates managed to fill in the information on the Venndiagram accurately. However, finding the correct value for x and calculating the number of students in the school posed a big problemfor many candidates.

1b. [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.

Markscheme

(A1) (C1)

[1 mark]


x + 16 + 18 + 34 = 99

x = 31

1c. [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

2a. [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]

Examiners reportParts (a) and (b) were well done although some candidates added 1 as a multiple of 3.

2b. [1 mark]List the elements of the following sets:

B,

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

[1 mark]

Examiners reportParts (a) and (b) were well done although some candidates added 1 as a multiple of 3.

2c. [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]

Examiners reportPart (c) was reasonably well attempted although some candidates found the intersection instead of the union.

A ∪ B

A ∪ B

2d. [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]

Examiners reportPart (d) was successfully completed by those candidates who managed to find the complement of B correctly. If they had not shownthe set containing the complement of B in the working they could not be awarded the method mark.

A ∩ B′

B′

A ∩ B′

3. [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

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]

Examiners reportAbout half of the students answered this question correctly. The placement of cos120 and π appeared to cause the most problems.

N Z Q R

14

3 –2

[5 marks]4a.

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]

Examiners reportMost candidates began the paper well by correctly drawing the Venn diagram and answering parts (b) and (c) correctly.

[2 marks]4b. 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]4c. 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

[1 mark]5a.

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]

Examiners reportThis question was well attempted by the majority. The major error was the omission of the “ ” in the candidates’ calculations.Perhaps better positioning would have helped in this regard.

P C

B

9

6

[1 mark]5b. Write down the number of pupils that study exactly two sciences.

12


6

[2 marks]5c. 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

6

[2 marks]5d. 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

6

6a. [2 marks]Shade on the diagram below.(A ∪ B) ∩ C ′

Markscheme

not shading or shading (A1)

correct shading (A1) (C2)

[2 marks]

Examiners reportThis question proved to be one of the easier questions with a number of candidates able to shade in the required region and findingvalues in a set. They still had problems with part (b).

C A ∪ B

6b. [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

6c. [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 ′

7. [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

, , , , , .

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]

Examiners reportVery few candidates gained full marks in this question. A common error turned out to be that and were not consideredrational numbers. Also, and were often placed incorrectly. However, it was encouraging that very few candidates placedvalues in more than one region.

P Q U

227

5 × 10−2 sin( )60∘ 0 −8−−−√3 −π

227

5 × 10−2

0 sin( )60∘

[1 mark]8a.

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]

Examiners reportThis was probably the question that most candidates found the easiest. Nearly all candidates gained either 5 or 6 marks with the marklost in shading the region on the Venn diagram.

[2 marks]8b. Find the value of p.

Markscheme35 – 17 (M1)

= 18 (A1) (C2)

Note: Award (A1) for correct answer only.

[2 marks]


[2 marks]8c. 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]8d. Shade, on your Venn diagram, .

Markscheme

(A1) (C1)

[1 mark]


∩ YA′

[2 marks]9a.

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]

Examiners reportMuch good work was seen in parts (a) and (b). However, there was much confusion in candidates’ responses to part (c) as manycould not determine the required answer where a union was involved with a complement. The result was that either candidates simplyignored and evaluated or ignored and evaluated . Irrespective of ability, the modalmark for this question was four with very few candidates achieving more than this mark.

21 + 15 + 3 − 33

= 6

n[(A ∪ B ])′ n(A) = 21 n[(A ∩ B)] n( ) = 18B′

9b. [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]


n[(A ∪ B ])′ n(A) = 21 n[(A ∩ B)] n( ) = 18B′

9c. [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 .n(A ∪ )B′

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(A) = 21 n[(A ∩ B)] n( ) = 18B′

10a. [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]

Examiners reportThe question was not well answered by the majority of the candidates. Many did not identify the universal set correctly and so took 3to be a member of this set. This affected their answers in a)(i) and a)(ii).

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

10b. [1 mark]List the elements of the following set.

Markscheme9 (A1)(ft) (C1)

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

[1 mark]

Examiners reportThe question was not well answered by the majority of the candidates. Many did not identify the universal set correctly and so took 3to be a member of this set. This affected their answers in a)(i) and a)(ii).

A ∩ B′

[2 marks]10c. Write down one element of .(A ∪ B)′

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]

Examiners reportNot many students answered (b) correctly. Some listed all correct elements of the given set instead of just one, which shows that theydid not read the question carefully.

(A ∪ B)

(A ∪ B)′

10d. [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]

Examiners reportAlthough many candidates could indicate which statement in the table in c) was false, often they were unable either to identify orarticulate a correct reason for it.

15 ∉ U

[1 mark]11a.

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

Write down the number of elements in .

U 12A B C U

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

B = {odd integers}

C = {5, 6, 8}

A ∩ C

Markscheme (one) (A1) (C1)

Note: , or earns no marks.

[1 mark]

Examiners reportThere was much confusion amongst candidates as to the understanding of the words number of elements. Many candidates simplywrote down or and consequently lost the first mark.

1

6 {6} {1}

6 {6}

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

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

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

[1 mark]

Examiners reportThere was much confusion amongst candidates as to the understanding of the words number of elements. Many candidates simplywrote down or and consequently lost the first mark. Part (b) was done well and many successful attempts were made atcompleting the Venn diagram in part (c). The most common error in the last part of the question was the omission of the element .

B

1 3 5 7 9 11

6 {6}10

11c. [4 marks]Complete the following Venn diagram with all the elements of .U

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]

Examiners reportPart (b) was done well and many successful attempts were made at completing the Venn diagram in part (c). The most common errorin the last part of the question was the omission of the element .

A ∩ B ∩ C

6 5 1 3

2 4 12 7 9 11 8

10

10

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]

Examiners reportThis question was accessible to the great majority of candidates. The common errors were:

the lack of a bounding rectangle in (a);the lack of subtraction for the entries in the disjoint regions of the type and the subsequent total exceeding 100%;the incorrect interpretation of “either ...or” as “exclusive or”. It is of the utmost importance to note that the ambiguity of the “or”statement will be removed and “exclusive or” signalled by the phrase “either ...or....but not both”. Otherwise, “inclusive or”must always be assumed.

A number of candidates were unable to interpret the percentage error question correctly and scored 0/4. This was somewhatdisappointing.

∩ ∩ CA′ B′

[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]




∩ ∩ CA′ B′

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


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

= 71 (A1)(ft)(G2)


[2 marks]


the lack of a bounding rectangle in (a);the lack of subtraction for the entries in the disjoint regions of the type and the subsequent total exceeding 100%;the incorrect interpretation of “either ...or” as “exclusive or”. It is of the utmost importance to note that the ambiguity of the “or”statement will be removed and“exclusive or” signalled by the phrase “either ...or....but not both”. Otherwise, “inclusive or” mustalways be assumed.


∩ ∩ CA′ B′

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


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

= 45 (A1)(ft)(G2)


[2 marks]




∩ ∩ CA′ B′

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]




∩ ∩ CA′ B′

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


the lack of a bounding rectangle in (a);the lack of subtraction for the entries in the disjoint regions of the type and the subsequent total exceeding 100%;the incorrect interpretation of “either ...or” as “exclusive or”. It is of the utmost importance to note that the ambiguity of the “or”statement will be removed and“exclusive or” signalled by the phrase “either ...or....but not both”. Otherwise, “inclusive or” mustalways be assumed.


∩ ∩ CA′ B′

13a. [4 marks]

In a college 450 students were surveyed with the following results

150 have a television

205 have a computer

220 have an iPhone

75 have an iPhone and a computer

60 have a television and a computer

70 have a television and an iPhone

40 have all three.

Draw a Venn diagram to show this information. Use T to represent the set of students who have a television, C the set ofstudents who have a computer and I the set of students who have an iPhone.

Markscheme

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

Notes: Award (A1) for labelled sets T, C, and I included inside an enclosed universal set. (Label U is not essential.) Award (A1) forcentral entry 40. (A1) for 20, 30 and 35 in the other intersecting regions. (A1) for 60, 110 and 115 or T(150), C(205), I(220).

[4 marks]

Examiners reportThe question was moderately well answered. The majority of candidates answered part (a) and at least parts of (b), and (d).

13b. [2 marks]Write down the number of students that

(i) have a computer only;

(ii) have an iPhone and a computer but no television.

MarkschemeIn parts (b), (c) and (d) follow through from their diagram.

(i) 110 (A1)(ft)

(ii) 35 (A1)(ft)

[2 marks]


[1 mark]13c. Write down .


60 (A1)(ft)

[2 marks]

Examiners reportThe question was moderately well answered. Part (c) proved to be difficult, as it required understanding and interpreting set notation.

n[T ∩ (C ∪ I ])′

[2 marks]13d. Calculate the number of students who have none of the three.


450 − (60 + 20 + 40 + 30 + 115 + 35 + 110) (M1)

Note: Award (M1) for subtracting all their values from 450.

= 40 (A1)(ft)(G2)

[2 marks]


[1 mark]14a.

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]

Examiners reportThe most common error in Question 4 was to omit counting the four non-music students. Explaining in words the meaning of the setnotation was difficult for some candidates.

14b. [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]14c. Explain, in words, the meaning of the part of the diagram that represents the set .P ∩ F ′

Printed for North hills Preparatory

© International Baccalaureate Organization 2015 International Baccalaureate® - Baccalauréat International® - Bachillerato Internacional®

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]


Documents

Sets and Venn Diagrams Quiz 2 - Uplift Education...Sets and Venn Diagrams Quiz 2 [101 marks] 1a. [2 marks] A school offers three activities, basketball (B), choir (C) and drama (D)