3.2

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. [4 marks]

Markscheme

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

 

Note: Award (A1) for each correct column followed through from the respective columns.

 

[4 marks]

Examiners report

[N/A]

2b. [2 marks]

Markscheme

neither      (A1)(ft)

since the entries in the final column are not all true and not all false        (R1)  (C2)

 

Note: Do not award (A1)(R0). Follow through from an incorrect truth table but only if their reasoning

is consistent with the final column.

Award (R1) only if the final column is clearly identified in the justification.

2

 

[2 marks]

Examiners report

[N/A]

3a. [2 marks]

Markscheme

If my Mathematical Studies homework is not due in tomorrow then today is Wednesday.     (A1)(A1) 

(C2)

Note: Award (A1) for If… then…

Award (A1) for correct propositions, my Mathematical Studies homework is not due in tomorrow and

today is Wednesday, in the correct order.

Award (A1)(A0) for “If ¬p then q”.

[2 marks]

Examiners report

[N/A]

3b. [3 marks]

Markscheme

  (A1)(A1)(A1)

(ft)  (C3)

 

[3 marks]

3

 

Examiners report

[N/A]

3c. [1 mark]

Markscheme

neither      (A1)(ft) (C1)

Note: Follow through from the final column of their truth table.

[1 mark]

Examiners report

[N/A]

4a. [3 marks]

Markscheme

if the baby is happy and wants to play then the baby does not cry     (A1)(A1)(A1) (C3)

Note: Award (A1) for “If… then…”; (A1) for “the baby is happy and wants to play”, (A1) for “the baby

does not cry”. Crying must be negated.

[3 marks]

Examiners report

[N/A]

4b. [2 marks]

Markscheme

4

    (A1)(A1) (C2)

Note: Award (A1) for each correct column.

[2 marks]

Examiners report

[N/A]

4c. [1 mark]

Markscheme

Neither     (A1)(ft)   (C1)

Note: Follow through from the last column in their part (b).

[1 mark]

Examiners report

[N/A]

5a. [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)

5

 

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]

5b. [2 marks]

Markscheme

     (A1)(A1)(ft)     (C2)

 

 

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

through from their column.

 

[2 marks]

6

Examiners report

[N/A]

5c. [1 mark]

Markscheme

contrapositive     (A1)     (C1)

[1 mark]

Examiners report

[N/A]

6a. [1 mark]

Markscheme

I was not paid     (A1)     (C1)

[1 mark]

Examiners report

[N/A]

6b. [1 mark]

Markscheme

    (A1)     (C1)

[1 mark]

Examiners report

[N/A]

6c. [2 marks]

Markscheme

7

     (A1)(A1)     (C2)

 

Note:     Award (A1) for each correct column.

 

[2 marks]

Examiners report

[N/A]

6d. [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

[N/A]

7a. [3 marks]

Markscheme

    (A1)(A1)(A1)

8

 

Note:     Award (A1) for “ ”.

Award (A1) for “ ” or “ ” (or “ ”or “ ”)

Award (A1) for “ ”.

Award at most (A1)(A1)(A0) if parentheses are missing for .

Award (A0)(A0)(A1) for .

 

[3 marks]

Examiners report

[N/A]

7b. [3 marks]

Markscheme

is not a multiple of 5 and ( ) is (either) a factor of 60 or ( ) is a multiple of 4, but not both     (A1)

(A1)(A1)

 

Note:     Award (A1) for “ is not a multiple of 5”, (A1) for “( ) is a factor of 60 or ( ) is a multiple of 4

but not both”, (A1) for “and” in the correct position. Accept only “but not both” in the second (A1).

Award at most (A1)(A1)(A0) for using extra statements such as “If ...then”, “if and only if” etc.

 

[3 marks]

Examiners report

[N/A]

7c. [3 marks]

Markscheme

9

     (A1)(A1)(A1)(ft)

 

Note:     Award (A1) for each correct column. Last column follows through from previous two.

 

[3 marks]

Examiners report

[N/A]

7d. [1 mark]

Markscheme

because not all the entries in the column are F     (R1)(ft)

 

Note:     If all entries in the last column of their truth table are T, award (R1)(ft) for an answer of “it is a

tautology”. Only award (R1)(ft) if the column is identified in the justification.

 

[1 mark]

Examiners report

[N/A]

10

7e. [1 mark]

Markscheme

accept one of: 1 OR 2 OR 3 OR 6     (A1)

 

Note:     Award (A1) for any one of the above answers.

 

[1 mark]

Examiners report

[N/A]

8a. [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]

8b. [2 marks]

Markscheme

11

     (A1)(A1)(ft)     (C2)

 

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

Follow through from their column.

 

[2 marks]

Examiners report

[N/A]

8c. [1 mark]

Markscheme

tautology     (A1)(ft)     (C1)

 

Note:     Follow through from part (b).

 

[1 mark]

Examiners report

[N/A]

9a. [2 marks] 12

Markscheme

   (A1)(A1)

 

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

 

[2 marks]

Examiners report

[N/A]

9b. [1 mark]

Markscheme

   (A1)

 

Notes:     Units not required.

 

[1 mark]

Examiners report

[N/A]

9c. [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).

13

 

[1 mark]

Examiners report

[N/A]

9d. [2 marks]

Markscheme

   (A1)(ft)(M1)

 

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

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

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]

9e. [3 marks] 14

Markscheme

   (A1)(A1)(A1)

 

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

 

[3 marks]

Examiners report

[N/A]

9f. [3 marks]

Markscheme

   (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)

15

[3 marks]

Examiners report

[N/A]

9g. [2 marks]

Markscheme

   (M1)

 

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

 

   (A1)(ft)(G2)

[2 marks]

Examiners report

[N/A]

9h. [3 marks]

Markscheme

   (M1)

 

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

 

   (A1)(ft)

 

Notes:     Follow through from part (g).

 

16

14 (cans)     (A1)(ft)(G3)

 

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

 

[3 marks]

Examiners report

[N/A]

10a. [2 marks]

Markscheme

i)      is not a rational number        (A1)

Note: Accept “ is an irrational number”.

 

ii)   any non-rational number (for example: , …)       (A1) (C2)

Examiners report

Question 5 Logic

In part (a), the majority of candidates were able to state the negation, but surprisingly many were

unable to give an example of a non-rational number.

10b. [3 marks]

Markscheme

      (A1)(A1)(A1) (C3)

Note: Award (A1) for “ ” seen, (A1) for “ ” as the consequent and (A1) for “ ” or “ ”

as the antecedent (the parentheses are required).

Examiners report

In part (b), a common error was the lack of parentheses in the antecedent. A further error was the use

of the “intersection” symbol rather than that for conjunction; care must be taken in this regard.

17

10c. [1 mark]

Markscheme

all integers are rational numbers (and therefore  cannot be an integer if it is not a rational number)    

(R1)

Note: Accept equivalent expressions.

OR

if is an integer, then  is a rational number, therefore if  is not a rational number, then  is not an

integer (contrapositive)   (R1) (C1)

Note: Accept “If  is not in , then  is not in ” with a Venn diagram showing ,  and  correctly.

Examiners report

Part (c) proved problematic for all but the best candidates.

11a. [3 marks]

Markscheme

if the teacher is absent then the lesson is cancelled and the students are in the library        (A1)(A1)(A1)

(C3)

Note: Award (A1) for If…then.

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

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

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

Award (A1) for “and”,

Award (A1) for correct propositions in correct order.

Examiners report

Question 4: Logic.

All candidates recognized that to fill in a truth table the answer is either true or false. However, given

that there are truth tables in the formula booklet it was surprising that some candidates made mistakes

when negating a given column of the truth table. Most candidates recognized that in a tautology the

column is always true with a small minority confusing tautology and contradiction. Candidates were

able to write a compound proposition in words.

18

11b. [2 marks]

Markscheme

(A1)(A1)(ft)    (C2)

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

Award (A0)(A1)(ft) for a  column that correctly follows from an incorrect column.

Examiners report

Question 4: Logic.

All candidates recognized that to fill in a truth table the answer is either true or false. However, given

that there are truth tables in the formula booklet it was surprising that some candidates made mistakes

when negating a given column of the truth table. Most candidates recognized that in a tautology the

column is always true with a small minority confusing tautology and contradiction. Candidates were

able to write a compound proposition in words.

11c. [1 mark]

Markscheme

not all of the entries are true (or equivalent)        (R1)     (C1)

Note: Accept “One entry is false”.

Examiners report

Question 4: Logic.

All candidates recognized that to fill in a truth table the answer is either true or false. However, given

that there are truth tables in the formula booklet it was surprising that some candidates made mistakes

when negating a given column of the truth table. Most candidates recognized that in a tautology the

19

column is always true with a small minority confusing tautology and contradiction. Candidates were

able to write a compound proposition in words.

12a. [1 mark]

Markscheme

is not a multiple of      (A1)     (C1)

Examiners report

[N/A]

12b. [2 marks]

Markscheme

    (A1)(A1)(C2)

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

Accept .

Examiners report

[N/A]

12c. [1 mark]

Markscheme

Converse     (A1) (C1)

Examiners report

[N/A]

12d. [2 marks]

Markscheme

not valid     (A1)

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

 

20

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]

13a. [4 marks]

Markscheme

   

(A1)(A1)(ft)(A1)(ft)(A1)     (C4)

 

Notes: Award (A1) for each correct column.

For the “ ” follow through from the “ ” column.

For the “ ” column, follow through from the preceding column.

Examiners report

[N/A]

13b. [2 marks]

Markscheme

tautology     (A1)(ft)

21

columns and are identical     (R1)(C2)

 

Notes: Do not award (R0)(A1)(ft). Follow through from their table in part (a).

Award the (R1) for an additional column representing that is

consistent with their table.

Examiners report

[N/A]

14a. [3 marks]

Markscheme

    (A1)(A1)(A1)

 

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.

14b. [2 marks]

Markscheme

     (A1)(ft)(A1)(ft)

 

22

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.

14c. [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).

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.

14d. [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).

23

 

(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.

15a. [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]

15b. [2 marks]

Markscheme

24

      (A1)(A1)     (C2)

Notes: Award (A1) for each correct column.

Examiners report

[N/A]

15c. [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).

Examiners report

[N/A]

16a. [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]

16b. [1 mark]

Markscheme

25

    (A1)(ft)     (C1)

 

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

 

[1 mark]

Examiners report

[N/A]

16c. [2 marks]

Markscheme

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]

16d. [1 mark]

Markscheme

Inverse     (A1)     (C1)

 

26

Note: Do not accept symbolic notation.

 

[1 mark]

Examiners report

[N/A]

17a. [3 marks]

Markscheme

If the food is well cooked and the drinks are chilled then dinner is not spoilt.     (A1)(A1)(A1)     (C3)

 

Note: Award (A1) for “If…then” (then must be seen), (A1) for the two correct propositions connected

with “and”, (A1) for “not spoilt”.

     Only award the final (A1) if correct statements are given in the correct order.

 

[3 marks]

Examiners report

[N/A]

17b. [3 marks]

Markscheme

27

     (A1)(A1)(A1)(ft)     (C3)

 

Notes: Award (A1) for each correct column.

     The final column must follow through from the previous two columns.

 

[3 marks]

Examiners report

[N/A]

18a. [2 marks]

Markscheme

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

 

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.

28

 

[2 marks]

Examiners report

[N/A]

18b. [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]

18c. [2 marks]

Markscheme

The statements are logically equivalent     (A1)(ft)

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

OR

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

 

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

 

[2 marks]

29

Examiners report

[N/A]

19a. [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.

19b. [2 marks]

Markscheme

     (A1)(A1)(ft)     (C2)

 

Note: Follow through from third column to fourth column.

 

30

[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.

19c. [2 marks]

Markscheme

    (A1)(A1)     (C2)

 

Notes: Award (A1) for .

     Award (A1) for and in correct order.

     Accept .

 

[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.

20a. [3 marks]

Markscheme

(i)      (A1)(A1)

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

(ii) OR      (A1)     (C3)

31

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.

20b. [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]

21a. [2 marks]

Markscheme

    

(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.

21b. [2 marks]

Markscheme

32

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.

21c. [1 mark]

Markscheme

True     (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. 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.

21d. [1 mark]

Markscheme

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

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

Examiners report

33

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.

22a. [2 marks]

Markscheme

    (A1)(A1)     (C2)

Note: Award (A1) for any 2 correct symbols seen in a statement, (A1) for all 3 correct symbols in

correct order.

Examiners report

[N/A]

22b. [3 marks]

Markscheme

     (A1)(A1)(ft)(A1)(ft)     (C3)

Note: Award (A1) for each correct column. 4th column is follow through from 3rd, 5th column is follow

through from 4th.

Examiners report

[N/A]

22c. [1 mark] 34

Markscheme

Not all of last column is F     (R1)(ft)     (C1)

Note: Award (R1)(ft) if final column does not lead to a contradiction.

Examiners report

[N/A]

23a. [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.

23b. [1 mark]

Markscheme

     (A1)     (C1)

[1 mark]

Examiners report

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

23c. [3 marks]

Markscheme

    (A1)(A1)(A1)     (C3)

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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.

24a. [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

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.

24b. [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”.

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[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.

24c. [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]

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.

25a. [2 marks]

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Markscheme

     (A1)(A1)(ft)     (C2)

Notes: Award (A1) for each correct column. Follow through in 4th column from their 3rd column.

[2 marks]

Examiners report

This question was well answered by most of the candidates who could complete the truth table, write

the proposition in symbolic form and write the given proposition in words, although the 'If ' was

sometimes omitted. Where marks were lost on Question 2, it was generally in the second column of the

truth table.

25b. [2 marks]

Markscheme

    (A1)(A1)     (C2)

Note: Award (A1) for and p in any order, (A1) for .

[2 marks]

Examiners report

This question was well answered by most of the candidates who could complete the truth table, write

the proposition in symbolic form and write the given proposition in words, although the 'If ' was

sometimes omitted. Where marks were lost on Question 2, it was generally in the second column of the

truth table.

25c. [2 marks]

Markscheme

38

If I can swim 50 metres (then) I do not take swimming lessons.     (A1)(A1)     (C2)

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

[2 marks]

Examiners report

This question was well answered by most of the candidates who could complete the truth table, write

the proposition in symbolic form and write the given proposition in words, although the 'If ' was

sometimes omitted. Where marks were lost on Question 2, it was generally in the second column of the

truth table.

26a. [4 marks]

Markscheme

     (A1)(A1)(A1)(ft)(A1)(ft)    

(C4)

Notes: Award (A1) for each correct column.

            Award first (A1)(ft) from their third column in the table.

            Award second (A1)(ft) from their fourth and fifth column in the table.

[4 marks]

Examiners report

Weaker candidates had some difficulty here with the majority scoring less than 2 marks on this

question. The more confident candidates were able to score well with most marks being lost only on

completing the truth table for . As a consequence, the final column entries of the table were

often incorrect but earned the (A1)(ft) mark. Many candidates went on to correctly identify the correct

(ft) response to (b)(i) and were able to support their answer with a correct reason.

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26b. [1 mark]

Markscheme

Tautology     (A1)(ft)     (C1)

Note: Answer must be consistent with last column in table.

[1 mark]

Examiners report

Weaker candidates had some difficulty here with the majority scoring less than 2 marks on this

question. The more confident candidates were able to score well with most marks being lost only on

completing the truth table for . As a consequence, the final column entries of the table were

often incorrect but earned the (A1)(ft) mark. Many candidates went on to correctly identify the correct

(ft) response to (b)(i) and were able to support their answer with a correct reason.

26c. [1 mark]

Markscheme

All entries (in the final column) are true.     (R1)(ft)     (C1) 

Note: Answer must be consistent with their answer to part (b)(i).

Note: Special case (A1)(R0) may be awarded.

[1 mark]

Examiners report

Weaker candidates had some difficulty here with the majority scoring less than 2 marks on this

question. The more confident candidates were able to score well with most marks being lost only on

completing the truth table for . As a consequence, the final column entries of the table were

often incorrect but earned the (A1)(ft) mark. Many candidates went on to correctly identify the correct

(ft) response to (b)(i) and were able to support their answer with a correct reason.

27a. [3 marks]

Markscheme

    (A1)(A1)(A1)        (C3)

40

Note: Award (A1) for two conjunctions, (A1) for negation seen on , (A1) for correct compound

statement.

[3 marks]

Examiners report

The logic question was clearly difficult for many students. Part a was very poorly done with the

majority of students not recognising that two conjunctions were required. Although candidates

performed better on part b, many omitted the 'if, (then)'. One of the most common errors in part b was

to translate the disjunction as 'and' rather than 'or'.

27b. [3 marks]

Markscheme

If I visited (either) Sarah’s Snackbar or Pete’s Eats (then) I did not visit Alan’s Diner.     (A1)(A1)(A1)       

(C3) 

Note: Award (A1) for If (then), (A1) for Sarah’s Snackbar or Pete’s Eats, (A1) for did not visit Alan’s

Diner.

[3 marks]

Examiners report

The logic question was clearly difficult for many students. Part a was very poorly done with the

majority of students not recognising that two conjunctions were required. Although candidates

performed better on part b, many omitted the 'if, (then)'. One of the most common errors in part b was

to translate the disjunction as 'and' rather than 'or'.

28a. [2 marks]

Markscheme

    (A2)

(A1) for

(A1) for

(A1)(A0) if brackets are missing.

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OR

    (A2)

(A1) for both brackets correct, (A1) for disjunctive “or” (A1)(A0) if brackets are missing.        (C2)

[2 marks]

Examiners report

(a) This caused problems for many candidates. They seem to expect to include the implication symbol

somewhere.

28b. [2 marks]

Markscheme

If you choose a salad then you do not choose a dessert.     (A2)

(A1) for “if …then…” (A1) for salad and no dessert in the correct order.

OR

If you choose a salad you do not choose a dessert.     (A2)     (C2)

[2 marks]

Examiners report

(b) Most candidates managed to write this correctly.

28c. [2 marks]

Markscheme

(A1) for each correct column         (A1)(A1)(ft)         (C2)

[2 marks]

Examiners report42

(c) Not all candidates could complete the truth table correctly. Many managed the first column but then

made mistakes in the last column.

29a. [4 marks]

Markscheme

(i)      (A1)

Brackets not required

(ii)      (A1)(ft)

Follow through only from incorrect B

(iii)      (A1)(ft)

     (A1)(ft)

Note: Award (A1) for correct  seen. The first (A1)(ft) in (iii) can be awarded only if C was listed

incorrectly and a mark was lost as a result in (a)(ii). If C was not listed and  is wrong, the first mark is

lost. The second mark can (ft) within part (iii) as  well as from (i).        (C4)

 

[4 marks]

 

Examiners report

a) Many candidates included 1 as a prime number for set . Most candidates were able to list the

intersection of and correctly with many receiving a follow through for their incorrect . Very few

candidates were able to list correctly with many listing the intersection. It was disappointing

that only a few candidates listed separately – those that did often received a mark for this working.

29b. [2 marks]

Markscheme43

“If x is not a positive integer between 1 and 7, then x is not a prime number less than 10.”     (A1)(A1)

Award (A1) for both (not) statements, (A1) for correct order.        (C2)

[2 marks]

Examiners report

b) The majority of candidates were able to write down the contrapositive correctly but many gave the

inverse or the converse instead.

30a. [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]

30b. [2 marks]

Markscheme

     (A1)(A1)     (C2)

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Examiners report

[N/A]

30c. [1 mark]

Markscheme

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

Examiners report

[N/A]

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