Sets and Venn Diagrams Quiz 2 [101 marks]

Transcription

1 Sets and Venn Diagrams Quiz 2 [101 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. 1a. Enter the above information on the Venn diagram below. (A2) (A1) only if 1 error (A0) otherwise (C2) Venn diagrams continue to be a problem area. Quite a good number of candidates managed to fill in the information on the Venn diagram accurately. However, finding the correct value for x and calculating the number of students in the school posed a big problem for many candidates.

2 1b. 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. x = 99 x = 31 (A1) (C1) Venn diagrams continue to be a problem area. Quite a good number of candidates managed to fill in the information on the Venn diagram accurately. However, finding the correct value for x and calculating the number of students in the school posed a big problem for many candidates. 1c. 99 of the students play basketball, 88 sing in the choir and 110 do drama. Calculate the total number of students in the school. [3 marks] Choir only = 88 ( ) = 12 (A1)(ft) Drama only = 110 ( ) = 18 (A1)(ft) Total = = 156 (A1)(ft) (C3) [3 marks] Venn diagrams continue to be a problem area. Quite a good number of candidates managed to fill in the information on the Venn diagram accurately. However, finding the correct value for x and calculating the number of students in the school posed a big problem for many candidates. 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. 2a. List the elements of the following sets: A, A = 8, 10, 12, 14, 16 (A1) (C1) Parts (a) and (b) were well done although some candidates added 1 as a multiple of 3.

3 2b. List the elements of the following sets: B, B = 3, 6, 9, 12, 15, 18 (A1) (C1) Parts (a) and (b) were well done although some candidates added 1 as a multiple of 3. List the elements of the following sets: 2c. A B, A B = 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) Part (c) was reasonably well attempted although some candidates found the intersection instead of the union. List the elements of the following sets: 2d. A B. B = 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20 (A1)(ft) A B = 8, 10, 14, 16 (A1)(ft) (C2) Part (d) was successfully completed by those candidates who managed to find the complement of B correctly. If they had not shown the set containing the complement of B in the working they could not be awarded the method mark.

4 3. The Venn diagram shows the number sets N, Z, Q and R. Place each of the following numbers in the appropriate region of the [6 marks] Venn diagram , 3, π, cos 120, , (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] About half of the students answered this question correctly. The placement of cos120 and π appeared to cause the most problems.

5 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. 4a. [5 marks] (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, 8 and (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] Most candidates began the paper well by correctly drawing the Venn diagram and answering parts (b) and (c) correctly. Find the number of families who own no pets. 4b (M1) 19 (A1)(ft)(G2) Note: Award (M1) for subtracting their total from 100. Most candidates began the paper well by correctly drawing the Venn diagram and answering parts (b) and (c) correctly.

6 Find the percentage of families that own exactly one pet. 4c. [3 marks] (M1) Note: Award (M1) for adding their values from (a). 50 ( ) 100 % 100 (A1)(ft) 50 % (50) (A1)(ft)(G3) [3 marks] Most candidates began the paper well by correctly drawing the Venn diagram and answering parts (b) and (c) correctly. The Venn diagram shows the numbers of pupils in a school according to whether they study the sciences Physics ( P), Chemistry ( C), Biology ( B). Write down the number of pupils that study Chemistry only. 5a. 9 (A1) (C1) This question was well attempted by the majority. The major error was the omission of the 6 in the candidates calculations. Perhaps better positioning would have helped in this regard. 5b. Write down the number of pupils that study exactly two sciences. 12

7 This question was well attempted by the majority. The major error was the omission of the 6 in the candidates calculations. Perhaps better positioning would have helped in this regard. Write down the number of pupils that do not study Physics. 5c (M1) = 26 (A1) (C2) Note: Award (A1) for 20 seen if answer is incorrect. This question was well attempted by the majority. The major error was the omission of the 6 in the candidates calculations. Perhaps better positioning would have helped in this regard. 5d. Find n[(p B) C] (M1) = 10 (A1) (C2) Note: Award (A1) for 29 or 19 seen if answer is incorrect. This question was well attempted by the majority. The major error was the omission of the 6 in the candidates calculations. Perhaps better positioning would have helped in this regard. Shade (A B) C on the diagram below. 6a.

8 not shading C or shading A B (A1) correct shading (A1) (C2) This question proved to be one of the easier questions with a number of candidates able to shade in the required region and finding values in a set. They still had problems with part (b). 6b. In the Venn diagram below, the number of elements in each region is given. Find n((p Q) R). Identifying the correct 5 numbers 3, 4, 5, 6, 9 (A1) 27 (A1) (C2) This question proved to be one of the easier questions with a number of candidates able to shade in the required region and finding values in a set. They still had problems with part (b). U is the set of positive integers, Z. 6c. + E is the set of even numbers. M is the set of multiples of 3. (i) List the first six elements of the set M. (ii) List the first six elements of the set E M.

9 (i) M = {3, 6, 9, 12, 15, 18} brackets not required. (A1) (ii) E M = {3, 9, 15, 21, 27, 33} (ft) from (i). (A1)(ft) (C2) This question proved to be one of the easier questions with a number of candidates able to shade in the required region and finding values in a set. They still had problems with part (b). 7. The sets P, Q and U are defined as U = {Real Numbers}, P = {Positive Numbers} and Q = {Rational Numbers}. [6 marks] Write down in the correct region on the Venn diagram the numbers, , sin( 60 ), 0, 3 8, π (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] 22 Very few candidates gained full marks in this question. A common error turned out to be that and were not considered 7 rational numbers. Also, 0 and sin( 60 ) were often placed incorrectly. However, it was encouraging that very few candidates placed values in more than one region.

10 A fitness club has 60 members. 35 of the members attend the club s aerobics course (A) and 28 members attend the club s yoga course (Y). 17 members attend both courses. A Venn diagram is used to illustrate this situation. Write down the value of q. 8a. 17 (A1) (C1) This was probably the question that most candidates found the easiest. Nearly all candidates gained either 5 or 6 marks with the mark lost in shading the region on the Venn diagram. Find the value of p. 8b (M1) = 18 (A1) (C2) Note: Award (A1) for correct answer only. This was probably the question that most candidates found the easiest. Nearly all candidates gained either 5 or 6 marks with the mark lost in shading the region on the Venn diagram. Calculate the number of members of the fitness club who attend neither the aerobics course (A) nor the yoga course (Y). 8c.

11 60 (35 17) (28 17) 17 = 14 (A1)(ft) (C2) (M1) Note: Follow through from (a) and (b). This was probably the question that most candidates found the easiest. Nearly all candidates gained either 5 or 6 marks with the mark lost in shading the region on the Venn diagram. Shade, on your Venn diagram, A Y. 8d. (A1) (C1) This was probably the question that most candidates found the easiest. Nearly all candidates gained either 5 or 6 marks with the mark lost in shading the region on the Venn diagram. A group of 33 people was asked about the passports they have. 21 have Australian passports, 15 have British passports and 3 have neither. 9a. Find the number that have both Australian and British passports or equivalent (M1) Note: Award (M1) for correct use of all four numbers. = 6 (A1) (C2) Much good work was seen in parts (a) and (b). However, there was much confusion in candidates responses to part (c) as many could not determine the required answer where a union was involved with a complement. The result was that either candidates simply ignored n[(a B ) ] and evaluated n(a) = 21 or ignored n[(a B)] and evaluated n( B ) = 18. Irrespective of ability, the modal mark for this question was four with very few candidates achieving more than this mark.

12 In the Venn diagram below, set A represents the people in the group with Australian passports and set B those with British 9b. passports. Write down the value of (i) q ; (ii) p and of r. (i) q = 6 (A1)(ft) (ii) p =15, r = 9 (A1)(ft) (C2) Note: Follow through from their answer to part (a). Much good work was seen in parts (a) and (b). However, there was much confusion in candidates responses to part (c) as many could not determine the required answer where a union was involved with a complement. The result was that either candidates simply ignored n[(a B ) ] and evaluated n(a) = 21 or ignored n[(a B)] and evaluated n( B ) = 18. Irrespective of ability, the modal mark for this question was four with very few candidates achieving more than this mark. A group of 33 people was asked about the passports they have. 21 have Australian passports, 15 have British passports and 3 have neither. 9c. In the Venn diagram below, set A represents the people in the group with Australian passports and set B those with British passports. Find n(a B ).

13 (M1) Note: Award (M1) for their figures seen in a correct calculation: or or 33 9 = 24 (A1)(ft) (C2) Note: Follow through from parts (a) and (b) or from values shown on Venn diagram. Much good work was seen in parts (a) and (b). However, there was much confusion in candidates responses to part (c) as many could not determine the required answer where a union was involved with a complement. The result was that either candidates simply ignored n[(a B ) ] and evaluated n(a) = 21 or ignored n[(a B)] and evaluated n( B ) = 18. Irrespective of ability, the modal mark for this question was four with very few candidates achieving more than this mark. Consider the universal set U = {x N 3 < x < 13}, and the subsets A = {multiples of 3} and B = {4, 6, 12}. 10a. List the elements of the following set. A 6, 9, 12 (A1) (C1) The question was not well answered by the majority of the candidates. Many did not identify the universal set correctly and so took 3 to be a member of this set. This affected their answers in a)(i) and a)(ii). List the elements of the following set. 10b. A B 9 (A1)(ft) (C1) Note: Follow through from their part (a)(i). The question was not well answered by the majority of the candidates. Many did not identify the universal set correctly and so took 3 to be a member of this set. This affected their answers in a)(i) and a)(ii). 10c. Write down one element of (A B).

14 any element from {5, 7, 8, 10, 11} (A1)(A1)(ft) (C2) Note: Award (A1)(ft) for finding (A B), follow through from their A. Award full marks if all correct elements of (A B) are listed. Not many students answered (b) correctly. Some listed all correct elements of the given set instead of just one, which shows that they did not read the question carefully. 10d. One of the statements in the table below is false. Indicate with an X which statement is false. Give a reason for your answer. 15 U (R1)(A1) (C2) Notes: Accept correct reason in words. If the reason is incorrect, both marks are lost. Do not award (R0)(A1). Although many candidates could indicate which statement in the table in c) was false, often they were unable either to identify or articulate a correct reason for it. U is the set of all the positive integers less than or equal to 12. A, B and C are subsets of U. A = {1, 2, 3, 4, 6, 12} B = {odd integers} C = {5, 6, 8} Write down the number of elements in A C. 11a.

15 1 (one) (A1) (C1) Note: 6, {6} or {1} earns no marks. There was much confusion amongst candidates as to the understanding of the words number of elements. Many candidates simply wrote down 6 or {6} and consequently lost the first mark. List the elements of B. 11b. 1, 3, 5, 7, 9, 11 (A1) (C1) Note: Do not penalise if braces, parentheses or brackets are seen. There was much confusion amongst candidates as to the understanding of the words number of elements. Many candidates simply wrote down 6 or {6} and consequently lost the first mark. Part (b) was done well and many successful attempts were made at completing the Venn diagram in part (c). The most common error in the last part of the question was the omission of the element c. Complete the following Venn diagram with all the elements of U. [4 marks]

16 (A1)(A1)(ft)(A1)(ft)(A1)(ft) (C4) Notes: Award (A1) for the empty set A B C. Award (A1)(ft) for the correct placement of 6, 5, 1 and 3. Award (A1)(ft) for the correct placement of 2, 4, 12, 7, 9, 11, 8. Award (A1)(ft) for the correct placement of 10. Follow through from part (b). [4 marks] Part (b) was done well and many successful attempts were made at completing the Venn diagram in part (c). The most common error in the last part of the question was the omission of the element 10. 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. 12a. Draw a Venn diagram to represent this information. Label A the set that represents The Art Journal readers, B the set that represents The Beartown News readers, and C the set that represents The Currier readers. [4 marks]

17 (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. Note: Accept answers given as decimals or fractions. [4 marks] (A4) This 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 A B C 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 somewhat disappointing. Find the percentage of the population that does not read any of the three newspapers. 12b. 100 ( ) ( ) (M1) Note: Award (M1) for correct expression. Accept equivalent expressions, for example or 100 ( ). = 1 (A1)(ft)(G2) Note: Follow through from their Venn diagram but only if working is seen. This 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 A B C 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 somewhat disappointing.

18 Find the percentage of the population that reads exactly one newspaper. 12c (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. This 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 A B C 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 somewhat disappointing. Find the percentage of the population that reads The Art Journal or The Beartown News but not The Currier. 12d (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. This 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 A B C 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 somewhat disappointing. 12e. 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.

19 True (A1)(ft) 100 (1 16) = 83 (R1)(ft) OR = 83 (R1)(ft) Note: Do not award (A1)(R0). Follow through from their Venn diagram. This 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 A B C 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 somewhat disappointing. 12f. The population of Beartown is The local radio station claimed that of the town s citizens read at least two of the local newspapers. Find the percentage error in this claim. [4 marks] 28% of (M1) = (A1) ( ) % error = (M1) Note: Award (M1) for 28 seen (may be implied by seen), award (M1) for correct substitution of their in the percentage error formula. If an error is made in calculating award a maximum of (M1)(A0)(M1)(A0), the final accuracy mark is lost. OR (M1) = 28.3( ) (A1) % error = 100 ( ) 28 = 1.19% ( ) (A1)(ft)(G3) (M1) Note: % sign not required. Accept 1.07 ( ) with use of with use of and 1.19 with use of Award (G3) for 1.07, 1.18 or 1.19 seen without working. [4 marks]

20 This 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 A B C 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 somewhat disappointing. 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. 13a. Draw a Venn diagram to show this information. Use T to represent the set of students who have a television, C the set of students who have a computer and I the set of students who have an iphone. [4 marks] (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) for central 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] The question was moderately well answered. The majority of candidates answered part (a) and at least parts of (b), and (d). Write down the number of students that 13b. (i) have a computer only; (ii) have an iphone and a computer but no television.

21 In parts (b), (c) and (d) follow through from their diagram. (i) 110 (A1)(ft) (ii) 35 (A1)(ft) The question was moderately well answered. The majority of candidates answered part (a) and at least parts of (b), and (d). Write down n[t (C I ) ]. 13c. In parts (b), (c) and (d) follow through from their diagram. 60 (A1)(ft) The question was moderately well answered. Part (c) proved to be difficult, as it required understanding and interpreting set notation. Calculate the number of students who have none of the three. 13d. In parts (b), (c) and (d) follow through from their diagram. 450 ( ) (M1) Note: Award (M1) for subtracting all their values from 450. = 40 (A1)(ft)(G2) The question was moderately well answered. The majority of candidates answered part (a) and at least parts of (b), and (d).

22 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. 14a. 145 (A1) (C1) The most common error in Question 4 was to omit counting the four non-music students. Explaining in words the meaning of the set notation was difficult for some candidates. Write down the number of students who 14b. (i) learn violin only; [3 marks] (ii) learn piano or flute or both; (iii) do not learn flute. (i) 56 (A1) (ii) 85 (A1) (iii) 89 (A1) (C3) [3 marks] The most common error in Question 4 was to omit counting the four non-music students. Explaining in words the meaning of the set notation was difficult for some candidates. 14c. Explain, in words, the meaning of the part of the diagram that represents the set P F.

23 The 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. The most common error in Question 4 was to omit counting the four non-music students. Explaining in words the meaning of the set notation was difficult for some candidates. International Baccalaureate Organization 2015 International Baccalaureate - Baccalauréat International - Bachillerato Internacional Printed for North hills Preparatory