3.4

1a. [2 marks]

Markscheme

¬q ¬⇒ p      (A1)(A1) (C2)

Note: Award (A1) for two negations seen, (A1) for correct antecedent and consequent on either side of

an implication.

 

[2 marks]

Examiners report

[N/A]

1b. [2 marks]

Markscheme

if it can go wrong then it does go wrong    (A1)(A1) (C2)

Note: Award (A1) for “if…then” seen, (A1) for correct antecedent and consequent.

 

[2 marks]

Examiners report

[N/A]

1c. [2 marks]

Markscheme

if it cannot go wrong then it does not go wrong   (A1)(A1)(ft) (C2)

Note: Award (A1) for “if…then” seen, (A1)(ft) for their correct antecedent and consequent. Follow

through from part (b).

 

1

[2 marks]

Examiners report

[N/A]

2a. [3 marks]

Markscheme

if the car is less than 2 years old or the car has not been driven more than , then the car is

under warranty     (A1)(A1)(A1)     (C3)

 

Note:     Award (A1) for if …, then …, (A1) for “or”, (A1) for correct statements in correct order. Accept

“If the car has not been driven more than or the car is less than 2 years old, then the car is

under warranty”. Accept logical equivalent wording for each proposition, eg “less than ”.

 

[3 marks]

Examiners report

[N/A]

2b. [2 marks]

Markscheme

     (A1)(A1)(ft)     (C2)

2

 

 

Note:     Award (A1) for column correct and (A1)(ft) for column correct. Follow

through from their column.

 

[2 marks]

Examiners report

[N/A]

2c. [1 mark]

Markscheme

contrapositive     (A1)     (C1)

[1 mark]

Examiners report

[N/A]

3a. [1 mark]

Markscheme

I was not paid     (A1)     (C1)

[1 mark]

Examiners report

[N/A]

3b. [1 mark]

Markscheme

    (A1)     (C1)

[1 mark]

3

Examiners report

[N/A]

3c. [2 marks]

Markscheme

     (A1)(A1)     (C2)

 

Note:     Award (A1) for each correct column.

 

[2 marks]

Examiners report

[N/A]

3d. [2 marks]

Markscheme

yes     (A1)(ft)

as the last two columns of the truth table are the same     (R1)(ft)     (C2)

 

Note:     Do not award (A1)(R0). Follow through from part (c)(i).

 

[2 marks]

Examiners report

4

[N/A]

4a. [3 marks]

Markscheme

If Sandi gets up before eight o’clock then Sandi (either) goes for a run or goes for a swim, but not both.  

(A1)(A1)(A1)     (C3)

 

Note:     Award (A1) for If …… then ……, (A1) for all propositions in the correct order, (A1) for “… or …

but not both” (do not accept “either” as a replacement for “but not both”).

 

[3 marks]

Examiners report

[N/A]

4b. [2 marks]

Markscheme

     (A1)(A1)(ft)     (C2)

 

Note:     Award (A1) for correct column, and (A1)(ft) for their correct  column.

Follow through from their column.

5

 

[2 marks]

Examiners report

[N/A]

4c. [1 mark]

Markscheme

tautology     (A1)(ft)     (C1)

 

Note:     Follow through from part (b).

 

[1 mark]

Examiners report

[N/A]

5a. [2 marks]

Markscheme

   (A1)(A1)

 

Note:     Award (A1) for either  OR seen. Award (A1) for two correct terms added together.

 

[2 marks]

Examiners report

[N/A]

5b. [1 mark]

Markscheme6

   (A1)

 

Notes:     Units not required.

 

[1 mark]

Examiners report

[N/A]

5c. [1 mark]

Markscheme

   (A1)(ft)

 

Notes:     Award (A1)(ft) for equating to their part (b).

Do not accept unless is explicitly defined as their part (b).

 

[1 mark]

Examiners report

[N/A]

5d. [2 marks]

Markscheme

   (A1)(ft)(M1)

 

Note:     Award (A1)(ft) for their seen.

Award (M1) for correctly substituting only into a correct part (a).

7

Award (A1)(ft)(M1) for rearranging part (c) to and substituting for  in expression

for .

 

   (AG)

 

Notes:     The conclusion, , must be consistent with their working seen for the (A1)

to be awarded.

Accept as equivalent to .

 

[2 marks]

Examiners report

[N/A]

5e. [3 marks]

Markscheme

   (A1)(A1)(A1)

 

Note:     Award (A1) for , (A1) for or , (A1) for .

 

[3 marks]

Examiners report

[N/A]

5f. [3 marks]

Markscheme

8

   (M1)

 

Note:     Award (M1) for equating their part (e) to zero.

 

OR      (M1)

 

Note:     Award (M1) for isolating .

 

OR

sketch of derivative function     (M1)

with its zero indicated     (M1)

   (A1)(ft)(G2)

[3 marks]

Examiners report

[N/A]

5g. [2 marks]

Markscheme

   (M1)

 

Note:     Award (M1) for correct substitution of their part (f) into the given equation.

 

   (A1)(ft)(G2)

[2 marks]9

Examiners report

[N/A]

5h. [3 marks]

Markscheme

   (M1)

 

Note:     Award (M1) for dividing their part (g) by 2000.

 

   (A1)(ft)

 

Notes:     Follow through from part (g).

 

14 (cans)     (A1)(ft)(G3)

 

Notes:     Final (A1) awarded for rounding up their to the next integer.

 

[3 marks]

Examiners report

[N/A]

6a. [1 mark]

Markscheme

is not a multiple of      (A1)     (C1)

Examiners report

[N/A] 10

6b. [2 marks]

Markscheme

    (A1)(A1)(C2)

Note: Award (A1) for , (A1) for and in the correct order.

Accept .

Examiners report

[N/A]

6c. [1 mark]

Markscheme

Converse     (A1) (C1)

Examiners report

[N/A]

6d. [2 marks]

Markscheme

not valid     (A1)

for example  is a multiple of  and not a multiple of      (R1)     (C2)

 

Notes: Do not award (A1)(R0). Any multiple of 6 that is not a multiple of  can be accepted as a

counterexample.

Examiners report

[N/A]

7a. [3 marks]

Markscheme

    (A1)(A1)(A1)

11

 

Notes: Award (A1) for conjunction seen, award (A1) for implication seen, award (A1) for correct

simple propositions in correct order (the parentheses are required). Accept .

Examiners report

Forming the statement in part (a) was attainable by the great majority, although the lack of parentheses

was a common fault.

7b. [2 marks]

Markscheme

     (A1)(ft)(A1)(ft)

 

Notes: Award (A1)(ft) for each correct column, follow through to the final column from their

column. For the second (A1)(ft) to be awarded there must be an implication in part (a).

Follow through from part (a).

Examiners report

The truth table in part (b) saw less success and it was clear that some centres simply had not prepared

their candidates in this area of the course.

7c. [2 marks]

Markscheme

The argument is not valid since not all entries in the final column are T.     (A1)(ft)(R1)

 

Notes: Do not award (A1)(ft)(R0). Follow through from part (b).

12

Accept “The argument is not valid since is not a tautology”.

Examiners report

Where the truth table was correctly constructed many candidates were not aware of the conditions

required for an argument to be valid and in part (d) the converse and the inverse were often confused.

7d. [4 marks]

Markscheme

(i)         (A1)(ft)(A1)(ft)

 

OR

    (A1)(ft)(A1)(ft)

Notes: Award (A1)(ft) for the negation of their antecedent and the negation of their consequent, (A1)

(ft) for their fully correct answer.

Follow through from part (a). Accept or . Follow through from part

(a).

 

(ii)     if it is not the case that the land has been purchased and the building permit has been obtained

then the land can not be used for residential purposes.     (A1)(A1)(ft)

 

OR

if (either) the land has not been purchased or the building permit has not been obtained then the land

can not be used for residential purposes.     (A1)(A1)(ft)

Notes: Award (A1) for “if… then…” seen, (A1)(ft) for correct statements in correct order. Follow

through from part (d)(i).

Examiners report

Where the truth table was correctly constructed many candidates were not aware of the conditions

required for an argument to be valid and in part (d) the converse and the inverse were often confused.

13

8a. [2 marks]

Markscheme

If I do not break my arm, then it will not hurt     (A1)(A1)     (C2)

Note: Award (A1) for “if… then…”

For Spanish candidates, only accept “Si” and “entonces”.

Award (A1) for “not break my arm” and “not hurt” in correct order.

Examiners report

[N/A]

8b. [2 marks]

Markscheme

      (A1)(A1)     (C2)

Notes: Award (A1) for each correct column.

Examiners report

[N/A]

8c. [2 marks]

Markscheme

logically equivalent     (A1)(ft)

last two columns of the truth table are identical     (R1)(ft)     (C2)

Notes: Do not award (A1)(ft)(R0).

Follow through from the last two columns of the table in part (a).

14

Examiners report

[N/A]

9a. [2 marks]

Markscheme

    (A1)(A1)     (C2)

 

Note: Award the first (A1) for seeing the implication sign, the second (A1) is for a correct answer only.

Not using the implication earns no marks.

 

[2 marks]

Examiners report

[N/A]

9b. [1 mark]

Markscheme

    (A1)(ft)     (C1)

 

Note: Award (A1)(ft) where the propositions in the implication in part (a) are exchanged.

 

[1 mark]

Examiners report

[N/A]

9c. [2 marks]

Markscheme

15

Not equivalent; a kite or an isosceles trapezium (for example) can have diagonals that are equal in

length.     (A1)(R1)     (C2)

 

Notes: Accept a valid sketch as reasoning.

     If the reason given is that a square has diagonals of equal length, but is not a rectangle, then award

(R1)(A0).

     Do not award (A1)(R0).

     Do not accept solutions based on truth tables.

 

[2 marks]

Examiners report

[N/A]

9d. [1 mark]

Markscheme

Inverse     (A1)     (C1)

 

Note: Do not accept symbolic notation.

 

[1 mark]

Examiners report

[N/A]

10a. [2 marks]

Markscheme

If Eva is losing weight then Eva is on a diet     (A1)(A1)     (C2)

 

16

Notes: Award (A1) for If… then…

     For Spanish candidates, only accept “Si” and “entonces”.

     For French candidates, only accept “Si” and “alors”.

     For all 3 languages these words are from the subject guide.

     Award (A1) for correct propositions in correct order.

 

[2 marks]

Examiners report

[N/A]

10b. [2 marks]

Markscheme

If Eva is not on a diet then she is not losing weight     (A1)(A1)     (C2)

 

Notes: Award (A1) for “not on a diet” and “not losing weight” seen, (A1) for complete correct answer.

     No follow through from part (a).

 

[2 marks]

Examiners report

[N/A]

10c. [2 marks]

Markscheme

The statements are logically equivalent     (A1)(ft)

The contrapositive is always logically equivalent to the original statement     (R1)(ft)

OR

17

A correct truth table showing the equivalence     (R1)(ft)     (C2)

 

Note: Follow through from their answers to part (a) and part (b).

 

[2 marks]

Examiners report

[N/A]

11a. [2 marks]

Markscheme

If I do not have a bowl of soup then I have an ice cream.     (A1)(A1)     (C2)

 

Notes: Award (A1) for If… then…

     Award (A1) for correct statements in correct order.

 

[2 marks]

Examiners report

Most candidates were able to write the compound proposition in words, however many were not able

to write the converse in symbolic form. While they were able to fill in the third column of the truth

table, many were unable to complete the fourth column correctly.

11b. [2 marks]

Markscheme

18

     (A1)(A1)(ft)     (C2)

 

Note: Follow through from third column to fourth column.

 

[2 marks]

Examiners report

Most candidates were able to write the compound proposition in words, however many were not able

to write the converse in symbolic form. While they were able to fill in the third column of the truth

table, many were unable to complete the fourth column correctly.

11c. [2 marks]

Markscheme

    (A1)(A1)     (C2)

 

Notes: Award (A1) for .

     Award (A1) for and in correct order.

     Accept .

 

[2 marks]

Examiners report

19

Most candidates were able to write the compound proposition in words, however many were not able

to write the converse in symbolic form. While they were able to fill in the third column of the truth

table, many were unable to complete the fourth column correctly.

12a. [3 marks]

Markscheme

(i)      (A1)(A1)

Note: Award (A1) for conjunction, (A1) for negation of q.

(ii) OR      (A1)     (C3)

Examiners report

Some candidates found the phrase “Yuiko is studying French but not Chinese” confusing as they did not

realize in this context the word “but” means “and”. Alternative but correct logic notation was accepted.

12b. [3 marks]

Markscheme

If Yuiko is not studying Chinese, (then) she is studying French.     (A1)(A1)(A1)     (C3)

Notes: Award (A1) for “if … (then)” seen, award (A1) for “not studying Chinese” seen, (A1) for correct

propositions in correct order.

Examiners report

[N/A]

13a. [2 marks]

Markscheme

20

    

(A1) for third column and (A1)(ft) for fourth column     (A1)(A1)(ft)     (C2)

Examiners report

This was provocative in the G2 and the comments indicate that candidates found the wording

confusing. Candidates were able to write in words the compound proposition and following

from their truth table the candidates could state if this was true or false.

13b. [2 marks]

Markscheme

is greater than or equal to (not less than) 10 or is greater than 100.     (A1)(A1)     (C2)

Note: Award (A1) for “greater than or equal to (not less than) 10”, (A1) for “or is greater than 100”.

Examiners report

This was provocative in the G2 and the comments indicate that candidates found the wording

confusing. Candidates were able to write in words the compound proposition and following

from their truth table the candidates could state if this was true or false. In part (c) many candidates

either stated the correct answer “true” or stated an answer consistent with their truth table and

received follow-through marks. Candidates had difficulty writing down a value of for which is

false.

13c. [1 mark]

Markscheme

True     (A1)(ft)     (C1)

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

21

Examiners report

This was provocative in the G2 and the comments indicate that candidates found the wording

confusing. Candidates were able to write in words the compound proposition and following

from their truth table the candidates could state if this was true or false. In part (c) many candidates

either stated the correct answer “true” or stated an answer consistent with their truth table and

received follow-through marks. Candidates had difficulty writing down a value of for which \(\neg

p \vee q\]) is false.

13d. [1 mark]

Markscheme

Any value of such that .     (A1)(ft)     (C1)

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

Examiners report

This was provocative in the G2 and the comments indicate that candidates found the wording

confusing. Candidates were able to write in words the compound proposition and following

from their truth table the candidates could state if this was true or false.

14a. [2 marks]

Markscheme

Carlos is not playing the guitar and he is studying for his IB exams.     (A1)(A1)     (C2)

Note: Award (A1) for “and”, (A1) for correct statements.

[2 marks]

Examiners report

In part (a) occasionally ‘if…then…’ was not seen but generally this was well done.

14b. [1 mark]

Markscheme

22

     (A1)     (C1)

[1 mark]

Examiners report

Part (b) was also well done despite the dearth of previous testing of the exclusive or statement.

14c. [3 marks]

Markscheme

    (A1)(A1)(A1)     (C3)

Notes: Award (A1) for implication, (A1) for the , (A1) for both and in the correct order. If

correct converse seen in words only award (A1)(A1)(A0). Accept . Accept for .

[3 marks]

Examiners report

Finding the converse of a statement in part (c) proved to be difficult for a significant number of

candidates and incorrect answers of the form were more frequently seen than the correct

answer. Such incorrect answers lost two marks.

15a. [2 marks]

Markscheme

If (both) the numbers x and y are even (then) the sum of x and y is an even number.     (A1)(A1)        (C2)

Note: Award (A1) for If…(then), (A1) for the correct statements in the correct order.

[2 marks]

Examiners report

Although a few candidates did not seem to understand the meaning of the symbol, many scored a

minimum of two marks on the first two parts of the question. Indeed, many correct statements were

seen in part (a). Many candidates however confused converse with inverse in part (b) resulting in the

incorrect statement "if the sum of x and y are both even then the numbers x and y are both even"

appearing on many scripts earning (M1)(A0). Despite this incorrect compound statement, many

23

candidates recovered with correct reasoning in part (c) from their correct (or incorrect) statement in

part (b). Candidate's responses to part (c) of the question should have been given in the context of the

question set and those that simply inferred their answer from truth tables only, earned no marks.

15b. [2 marks]

Markscheme

If (both) the numbers x and y are not even (then) the sum of x and y is not an even number.     (A1)

(A1)        (C2)

Notes: Award (A1) for If…(then), (A1) for the correct not p, and not q in the correct order. Accept the

word odd for the phrase “not even”.

[2 marks]

Examiners report

Although a few candidates did not seem to understand the meaning of the symbol, many scored a

minimum of two marks on the first two parts of the question. Indeed, many correct statements were

seen in part (a). Many candidates however confused converse with inverse in part (b) resulting in the

incorrect statement "if the sum of x and y are both even then the numbers x and y are both even"

appearing on many scripts earning (M1)(A0). Despite this incorrect compound statement, many

candidates recovered with correct reasoning in part (c) from their correct (or incorrect) statement in

part (b). Candidate's responses to part (c) of the question should have been given in the context of the

question set and those that simply inferred their answer from truth tables only, earned no marks.

15c. [2 marks]

Markscheme

The inverse of a statement is not (necessarily) true, because two odd (not even) numbers, always have

an even sum.     (A1)(R1)(ft)    (C2)

Notes: Award (A1)(R1) if a specific counter example given instead of a reason stated in general terms,

e.g. the inverse is not true because, 5 and 7 have an even sum. Do not award (A1)(R0). Follow through

from their statement in part (b).

[2 marks]

24

Examiners report

Although a few candidates did not seem to understand the meaning of the symbol, many scored a

minimum of two marks on the first two parts of the question. Indeed, many correct statements were

seen in part (a). Many candidates however confused converse with inverse in part (b) resulting in the

incorrect statement "if the sum of x and y are both even then the numbers x and y are both even"

appearing on many scripts earning (M1)(A0). Despite this incorrect compound statement, many

candidates recovered with correct reasoning in part (c) from their correct (or incorrect) statement in

part (b). Candidate's responses to part (c) of the question should have been given in the context of the

question set and those that simply inferred their answer from truth tables only, earned no marks.

16a. [2 marks]

Markscheme

     (A1)(A1)        (C2)

 

Note: Award (A1) for ¬q , (A1) for last column.

 

[2 marks]

Examiners report

This question was well answered with most candidates able to complete the truth table correctly in

part a) and write the correct compound proposition in symbolic form in part b). A significant number of

candidates could not write the correct contrapositive, although most were awarded one mark for

writing an implication.

16b. [2 marks]

Markscheme

25

    (A1)(A1)     (C2)

Note: Award (A1) for , (A1) for p and q in the correct order.

 

[2 marks]

Examiners report

This question was well answered with most candidates able to complete the truth table correctly in

part a) and write the correct compound proposition in symbolic form in part b). A significant number of

candidates could not write the correct contrapositive, although most were awarded one mark for

writing an implication.

16c. [2 marks]

Markscheme

If Cristina does not do well on the logic test then she does not understand logic.     (A1)(A1)    (C2)

Note: Award (A1) for If…(then), must be an implication, (A1) for the correct propositions in the correct

order.

 

[2 marks]

Examiners report

This question was well answered with most candidates able to complete the truth table correctly in

part a) and write the correct compound proposition in symbolic form in part b). A significant number of

candidates could not write the correct contrapositive, although most were awarded one mark for

writing an implication.

17a. [2 marks]

Markscheme

If a quadrilateral is not a square (then) the four sides of the quadrilateral are not equal.     (A1)(A1)    

(C2)

26

Note: Award (A1) for “if…(then)”, (A1) for the correct phrases in the correct order.

 

[2 marks]

Examiners report

There was confusion among some students about which was the inverse and converse of the given

statement.

17b. [2 marks]

Markscheme

If the four sides of the quadrilateral are equal (then) the quadrilateral is a square.     (A1)(A1)(ft)    

(C2)

Note: Award (A1) for “if…(then)”, (A1)(ft) for the correct phrases in the correct order.

 

Note: Follow through in (b) if the inverse and converse in (a) and (b) are correct and reversed.

 

[2 marks]

Examiners report

There was confusion among some students about which was the inverse and converse of the given

statement.

17c. [2 marks]

Markscheme

The converse is not always true, for example a rhombus (diamond) is a quadrilateral with four equal

sides, but it is not a square.     (A1)(R1)     (C2)

Note: Do not award (A1)(R0).

27

 

[2 marks]

Examiners report

There was confusion among some students about which was the inverse and converse of the given

statement. Part (c) was poorly done with very few students able to provide an example that shows that

the converse is not always true.

18a. [2 marks]

Markscheme

If the sun is shining then I will go swimming.     (A1)(A1)     (C2)

 

Note: Award (A1) for “if…then” and (A1) for correct order.

 

[2 marks]

Examiners report

The most common error was poor use of the “If...then” connective.

18b. [2 marks]

Markscheme

Either the sun is not shining or I will go swimming.     (A1)(A1)     (C2)

Note: Award (A1) for both correct statements and (A1) for “either” “…or”.

 

[2 marks]

Examiners report

Confusion between “and” and “or” was rare, however, the use of implication in this part was a little too

common.

28

18c. [1 mark]

Markscheme

     (A1)     (C1)

[1 mark]

Examiners report

Precise, correct terminology was expected in this part.

18d. [1 mark]

Markscheme

They are (logically) equivalent.     (A1)     (C1)

Note: Do not accept any other answers.

 

[1 mark]

Examiners report

[N/A]

19a. [4 marks]

Markscheme

(i)

29

     (A3)

Note: Award (A1) for column correct, (A1)(ft) for column correct, (A1) for last column

correct.

 

(ii)    Yes.     (R1)(ft)     (C4)

Note: (ft) from their second and the last columns. Must be correct from their table.

[4 marks]

Examiners report

This question was well answered by many of the candidates. It is an area of the syllabus that is well

taught and many managed to get a follow through mark even though one of the columns in the table

might have been incorrect.

19b. [2 marks]

Markscheme

.     (A1)(A1)        (C2)

Note: Award (A1) for , (A1) for . Accept or .

[2 marks]

Examiners report

This question was well answered by many of the candidates. It is an area of the syllabus that is well

taught and many managed to get a follow through mark even though one of the columns in the table

might have been incorrect.

30

20a. [2 marks]

Markscheme

If ABCD is a square, then ABCD has four equal sides.     (A1)(A1)     (C2)

Note: Award (A1) for if… then, (A1) for propositions in the correct order.

Examiners report

[N/A]

20b. [2 marks]

Markscheme

If ABCD is not a square, then ABCD does not have four equal sides.     (A1)(A1)     (C2)

 

Note: Award (A1) for if… then, (A1) for propositions in the correct order.

Examiners report

[N/A]

20c. [2 marks]

Markscheme

Not a valid argument. ABCD may have 4 equal sides but will not necessarily be a square. (It may be a

rhombus)     (A1)(R1)     (C2)

 

Note: Award (R1) for correct reasoning, award (A1) for a consistent conclusion with their answer in

part (b).

It is therefore possible that (R1)(A0) may be awarded, but (R0)(A1) can never be awarded.

 

Note: Simple examples of determining the validity of an argument without the use of a truth table may

be tested.

31

Examiners report

[N/A]

21a. [3 marks]

Markscheme

If Alex does not play the flute then he is either a scientist or from Uruguay.     (A1)(A1)(A1)     (C3)

 

Note: Award (A1) if… then, correct (A1) antecedent, (A1) correct consequent.

Examiners report

[N/A]

21b. [2 marks]

Markscheme

     (A1)(A1)     (C2)

Examiners report

[N/A]

21c. [1 mark]

Markscheme

32

Not all entries in the final column are T.     (R1)     (C1)

Examiners report

[N/A]

Printed for International School of Europe

© International Baccalaureate Organization 2019

International Baccalaureate® - Baccalauréat International® - Bachillerato Internacional®

33