Mathematics questions, chapter 3

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Logical Reasoning

44 marks. Use Practice to check each answer as you go, or Exam to answer everything first. Show your working for Section B.

Section A: objective questions

12 marks

Choose the correct answer. Each question carries 1 mark.

Question 1

Which of the following is a statement?

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Is 15 a multiple of 5?
7 + 2 = 10
x + 2 = 9

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A statement is true or false but not both. 7 + 2 = 10 is false, but it is still a statement. A command and a question are not statements, and x + 2 = 9 is true only for some x.

For the teacher: Remembering, low difficulty, standard 3.1.1

Textbook p. 56

Question 2

Which statement is false?

Some prime numbers are even numbers.
All polygons have diagonals.
All rectangles have four right angles.
Some straight lines intersect the y-axis.

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A triangle is a polygon with no diagonal, so it is a counter-example to 'All polygons have diagonals'. The other three statements are true (2 is an even prime).

For the teacher: Understanding, low difficulty, standard 3.1.1

Textbook p. 58

Question 3

What is the negation of 'All multiples of 5 are multiples of 10'?

No multiples of 5 are multiples of 10.
All multiples of 5 are not multiples of 10.
Some multiples of 10 are multiples of 5.
Not all multiples of 5 are multiples of 10.

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To negate 'all', say 'not all'. One multiple of 5 that is not a multiple of 10 (such as 15) is enough. The first two options are too strong.

For the teacher: Understanding, medium difficulty, standard 3.1.2

Textbook p. 59

Question 4

Which compound statement is true?

6 is a multiple of 3 or 6 is a multiple of 4.
6 is a multiple of 3 and 6 is a multiple of 4.
√16 = 8 or 2³ = 6.
5 is an even number and 5 is a prime number.

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An 'or' statement is true if at least one part is true: 6 is a multiple of 3. The 'and' statements need both parts true. In C both parts are false.

For the teacher: Applying, low difficulty, standard 3.1.3

Textbook p. 62

Question 5

p: 8 is a perfect cube. q: 8 is a perfect square. Which is correct?

p and q is true; p or q is true
p and q is true; p or q is false
p and q is false; p or q is true
p and q is false; p or q is false

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p is true (2³ = 8) and q is false (8 is not the square of a whole number). 'and' needs both true, so it is false. 'or' needs only one true, so it is true.

For the teacher: Applying, low difficulty, standard 3.1.3

Textbook p. 61

Question 6

What is the antecedent of 'If x is a multiple of 6, then x is an even number'?

x is an even number
x is a multiple of 6
If x is a multiple of 6
then x is an even number

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The antecedent is the statement p after 'if', without the word 'if'. The consequent is 'x is an even number'.

For the teacher: Remembering, low difficulty, standard 3.1.4

Textbook p. 64

Question 7

What is the converse of 'If x = 4, then x² = 16'?

If x ≠ 4, then x² ≠ 16.
If x² ≠ 16, then x ≠ 4.
If x = 4, then x² ≠ 16.
If x² = 16, then x = 4.

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The converse swaps the antecedent and consequent. A is the inverse and B is the contrapositive.

For the teacher: Understanding, low difficulty, standard 3.1.5

Textbook p. 66

Question 8

What is the contrapositive of 'If a polygon is a square, then it has four sides'?

If a polygon does not have four sides, then it is not a square.
If a polygon has four sides, then it is a square.
If a polygon is not a square, then it does not have four sides.
If a polygon is a square, then it does not have four sides.

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The contrapositive is 'If ~q, then ~p': swap the two parts and negate both. B is the converse and C is the inverse.

For the teacher: Understanding, low difficulty, standard 3.1.5

Textbook p. 66

Question 9

Which implication is false?

If 5 is an even number, then 5 + 1 is an odd number.
If 9 is a multiple of 3, then 9 is a multiple of 6.
If 2 + 2 = 5, then 3 is an even number.
If 12 is divisible by 6, then 12 is divisible by 3.

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'If p, then q' is false only when p is true and q is false. In B, 9 is a multiple of 3 (true) but not of 6 (false). In A and C the antecedent is false, so the implication is true.

For the teacher: Analysing, medium difficulty, standard 3.1.5

Textbook p. 68

Question 10

Which argument is valid?

All squares are rhombuses. PQRS is a rhombus. So PQRS is a square.
If x is even, then x² is even. x² is even. So x is even.
If n is divisible by 6, then n is divisible by 3. n is not divisible by 6. So n is not divisible by 3.
If n is divisible by 6, then n is divisible by 3. 14 is not divisible by 3. So 14 is not divisible by 6.

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D follows Form III: If p, then q; not q; so not p. A, B and C reverse or deny the parts of the implication, so they match none of the three valid forms.

For the teacher: Analysing, high difficulty, standard 3.2.2

Textbook p. 75

Question 11

'All prime numbers greater than 2 are odd. 13 is a prime number greater than 2. So 13 is odd.' This argument is

deductive, because it goes from a general premise to a specific conclusion
deductive, because it goes from specific premises to a general conclusion
inductive, because it has two premises
inductive, because its conclusion is true

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The first premise is general and the conclusion is about the single number 13, which is the direction of a deductive argument.

For the teacher: Understanding, low difficulty, standard 3.2.1

Textbook p. 72

Question 12

'The sum of the exterior angles of a triangle is 360°. So is that of a quadrilateral, and so is that of a pentagon. So the sum of the exterior angles of every polygon is 360°.' This inductive argument is

weak and not cogent
strong but not cogent
strong and cogent
weak but cogent

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The premises are true and make the conclusion very probable, so it is strong. All premises are true, so it is also cogent. A weak argument can never be cogent.

For the teacher: Evaluating, medium difficulty, standard 3.2.4

Textbook p. 81

Section B: structured questions

32 marks

Answer all questions. Show your working.

Question 13

State whether each sentence is a statement. If it is a statement, state whether it is true or false.

[1 mark] (a)Is 12 a factor of 36?

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Not a statement (it is a question) [1]1

Tick each point you got.

[1 mark] (b)7 × 8 = 54

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A statement; false because 7 × 8 = 56 [1]1

Tick each point you got.

[1 mark] (c)A cube has 12 edges.

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A statement; true [1]1

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For the teacher: Remembering, low difficulty, standard 3.1.1

Textbook p. 56

Question 14

Answer the following.

[2 marks] (a)Write the negation of 'A kite has two axes of symmetry' and state its truth value.

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A kite does not have two axes of symmetry [1]1
True, because a kite has only one axis of symmetry [1]1

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[2 marks] (b)'All multiples of 3 are odd numbers.' Is this statement true? If it is false, give a counter-example.

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False [1]1
Counter-example: 6 is a multiple of 3 but it is even [1]1

Tick each point you got.

For the teacher: Understanding, low difficulty, standard 3.1.2

Textbook p. 59

Question 15

p: √49 = 7. q: 2⁴ = 8.

[1 mark] (a)State the truth value of p and of q.

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p is true; q is false because 2⁴ = 16 [1]1

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[1 mark] (b)Write the compound statement 'p and q' and state its truth value.

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√49 = 7 and 2⁴ = 8; false [1]1

Tick each point you got.

[1 mark] (c)State the truth value of 'p or q'. Give a reason.

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True, because at least one part (p) is true [1]1

Tick each point you got.

For the teacher: Applying, low difficulty, standard 3.1.3

Textbook p. 61

Question 16

Answer the following.

[1 mark] (a)Write an implication 'if p, then q' with antecedent 'x = 5' and consequent 'x + 4 = 9'.

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If x = 5, then x + 4 = 9 [1]1

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[2 marks] (b)Write two implications from 'x² = 49 if and only if x = 7 or x = −7'.

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If x² = 49, then x = 7 or x = −7 [1]1
If x = 7 or x = −7, then x² = 49 [1]1

Tick each point you got.

For the teacher: Applying, low difficulty, standard 3.1.4

Textbook p. 64

Question 17

Implication: 'If x > 4, then x > 1.'

[3 marks] (a)Write the converse, inverse and contrapositive.

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Converse: If x > 1, then x > 4 [1]1
Inverse: If x ≤ 4, then x ≤ 1 [1]1
Contrapositive: If x ≤ 1, then x ≤ 4 [1]1

Tick each point you got.

[3 marks] (b)State the truth value of each of the three statements in (a). Give a counter-example for any false one.

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Converse: false, x = 2 is greater than 1 but not greater than 4 [1]1
Inverse: false, x = 2 is not greater than 4 but is greater than 1 [1]1
Contrapositive: true (same truth value as the implication) [1]1

Tick each point you got.

For the teacher: Analysing, medium difficulty, standard 3.1.5

Textbook p. 66

Question 18

Study the two arguments.

[2 marks] (a)'All multiples of 8 are multiples of 4. 56 is a multiple of 8. So 56 is a multiple of 4.' State the type of argument and whether it is valid.

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Deductive (general premise, specific conclusion) [1]1
Valid: Form I (All A are B; C is A; so C is B) [1]1

Tick each point you got.

[1 mark] (b)'1² + 1 = 2, 2² + 1 = 5, 3² + 1 = 10. So the sequence 2, 5, 10, … is n² + 1; n = 1, 2, 3, …' State the type of argument.

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Inductive (specific cases lead to a general conclusion) [1]1

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For the teacher: Analysing, low difficulty, standard 3.2.1

Textbook p. 73

Question 19

Premise 1: If a quadrilateral is a square, then it has four equal sides. Premise 2: Quadrilateral ABCD has four equal sides. Conclusion: ABCD is a square.

[2 marks] (a)Is this argument valid? Explain.

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Not valid [1]1
It does not follow any of the three forms; it affirms the consequent [1]1

Tick each point you got.

[1 mark] (b)Give a counter-example to show the conclusion may be false.

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A rhombus that is not a square has four equal sides [1]1

Tick each point you got.

For the teacher: Evaluating, high difficulty, standard 3.2.2

Textbook p. 76

Question 20

A culture of bacteria triples every hour. The number of new bacteria after t hours is N(t) = 3^t.

[1 mark] (a)Find the number of new bacteria after 5 hours.

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3⁵ = 243 [1]1

Tick each point you got.

[2 marks] (b)After how many hours is the number 729?

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3^t = 729 = 3⁶ [1]1
t = 6 hours [1]1

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[2 marks] (c)Another culture has 5, 8, 11, 14, … bacteria in the first four hours. By induction, form a general formula and find the 10th term.

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Common difference 3, so 3n + 2; n = 1, 2, 3, … [1]1
3(10) + 2 = 32 [1]1

Tick each point you got.

For the teacher: Applying, medium difficulty, standard 3.2.6

Textbook p. 84

Question 21

Premise 1: If a quadrilateral is a square, then it has four equal sides. Use this premise to form valid arguments.

[1 mark] (a)Write premise 2 and the conclusion for a valid argument of Form II. Use quadrilateral PQRS.

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Premise 2: PQRS is a square. Conclusion: PQRS has four equal sides [1]1

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[1 mark] (b)Write premise 2 and the conclusion for a valid argument of Form III. Use quadrilateral ABCD.

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Premise 2: ABCD does not have four equal sides. Conclusion: ABCD is not a square [1]1

Tick each point you got.

For the teacher: Creating, medium difficulty, standard 3.2.3

Textbook p. 78

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