4.6 SPM Practice (Long Questions)

Posted on April 23, 2020 by user

Question 1:
(a) State if each of the following statements is true or false.
(i) $(i) 2^{4} = 16 and 12 \div \sqrt[3]{27} = 3.$
(ii) 17 is a prime number or an even number.

(b) Complete the statement, in the answer space, to form a true statement by using the quantifier ‘all’ or ‘some’.

(c) Write down two implications based on the following statement:

A number is a prime number if and only if it is only divisible by 1 and itself.

Solution:
(a)(i) False
(a)(ii) True

(b) Some multiples of 3 are multiples of 6.

(c) Implication 1: If a number is a prime number, then it is only divisible by 1 and itself.
Implication 2: If a number is only divisible by 1 and itself, then it is a prime
number.

Question 2:
(a) State if each of the following statements is true or false.
(i) 2 × 3 = 6 or 2 + 3 = 6
(ii) 2 is a prime number and 5 is an even number.

(b) Write down the converse of the following implication.
Hence, state whether the converse is true or false.

If x is a multiple of 12,

then x is a multiple of 3.

(c) Complete the premise in the following argument:

Premise 1: All hexagons have six sides

Premise 2: _____________________

Conclusion: ABCDEF has six sides.

Solution:
(a)(i) True
(a)(ii) False

(b) Converse: If x is a multiple of 3, then x is a multiple of 12.
The converse is false.

(c) Premise 2: ABCDEF is a hexagon.

4.3 Operations on Statements (Sample Questions 1)

Posted on April 23, 2020 by user

Example 1:

Form a compound statement by combining two given statements using the word ‘and’.

(a) 3 × 12 = 36

7 × 5 = 35

(b) 5 is a prime number.

5 is an odd number.

(c) Rectangles have 4 sides.

Rectangles have 4 vertices.

Solution:

(a) 3 × 12 = 36 and 7 × 5 = 35

(b) 5 is a prime number and an odd number.

(c) Rectangles have 4 sides and 4 vertices.

Example 2:

Form a compound statement by combining two given statements using the word ‘or’.

(a) 16 is a perfect square. 16 is an even number.

(b) 4 > 3. -5 < -1

Solution:

(a) 16 is a perfect square or an even number.

(b) 4 > 3 or -5 < -1.

4.1 Statements (Sample Questions)

Posted on April 23, 2020 by user

Example 1:

Determine whether the following sentences are statements or not. Give a reason for your answer.

(a) 3 + 3 = 8

(b) 9 – 4 = 5

(d) 4 is a prime number.

(e) Is 40 divisible by 3?

(f) Find the perimeter of a square with each side of 4 cm.

(g) Help!

Answer:

(a) Statement; it is a false statement.

(b) Statement; it is a true statement.

(d) Statement; it is a false statement.

(e) Not a statement; it is a question.

(f) Not a statement; it is an instruction.

(g) Not a statement; it is an exclamation.

Deduction and Induction

Posted on April 23, 2020 by user

4.6 Deduction and Induction

(A) Reasoning by Deduction and Induction

1. Reasoning by deduction is a process of making a conclusion for a specific case based on a given general statement.

2. Reasoning by induction is a process of making a generalization based on specific cases.

Maths Tip

1. General statement → Special conclusion → Deduction

2. Specific cases → General conclusion → Induction

Example:

Determine whether the following conclusion is made based on a deductive reasoning or inductive reasoning.

(a) Area of triangle = ½ × Base × Height

(i)

Area of ∆ ABC

= ½ × 7cm × 5cm

= 17.5 cm²

(ii)

Area of ∆ DEF

= ½ × 7cm × 4cm

= 14 cm²

(b)

1 = 7 (1)² – 6

22 = 7 (2)² – 6

57 = 7 (3)² – 6

106 = 7 (4)² – 6

7n² – 6, n = 1, 2, 3, 4…

Solution:

(a)

The specific conclusion is made based on a general statement ~ Area of triangle = ½ × Base × Height. Therefore, the conclusion is made based on deductive reasoning.

(b)

The general conclusion 7n² – 6, n = 1, 2, 3, 4… is made based on specific cases. Therefore, the conclusion is based on inductive reasoning.

4.4 Implication

Posted on April 23, 2020 by user

(A) Antecedent and Consequent of an Implication

For two statements p and q, the sentence ‘if p, then q’ is called an implication.
p is called the antecedent.
q is called the consequent.

Example:

Identify the antecedent and consequent of the following implications.

(a) If m = 2, then 2m² + m = 10

$(b) I f P \cup Q = P, t h e n Q \subset P$

Solution:

Antecedent: m = 2
Consequent: 2m² + m = 10
$\begin{array}{l} Antecedent : P \cup Q = P \\ Consequent : Q \subset P \end{array}$

(B) Implications of the Form ‘p if and only if q’

Two implications ‘if p, then q’ and ‘if q, then p’ can be written as ‘p if and only if q’.
Likewise, two statements can be written from a statement in the form ‘p if and only if q’ as follows:
Implication 1: If p, then q.
Implication 2: If q, then p.

Example 1:

Given that p: x + 1 = 8
q: x = 7
Construct a mathematical statement in the form of implication

If p, then q.
p if and only if q.

Solution:

If x + 1 = 8, then x = 7.
x + 1 = 8 if and only if x = 7.

Example 2:

Write down two implications based on the following sentence:
x³ = 64 if and only if x = 4.

Solution:

If x³ = 64, then x = 4.
If x = 4, then x³ = 64.

(C) Converse of an Implication

The converse of an implication ‘if p, then q’ is ‘if q, then p’.

Example:

State the converse of each of the following implications.

If x² + x – 2 = 0, then (x – 1)(x + 2) = 0.
If x = 7, then x + 2 = 9.

Solution:

If (x – 1)(x + 2) = 0, then x² + x – 2.
If x + 2 = 9, then x = 7.

4.6 SPM Practice (Long Questions)

Posted on April 23, 2020 by user

Question 7:
(a)(i) State whether the following compound statement is true or false.

3 + 3 = 9 or 3 × 3 = 9

(a)(ii) Determine whether the following converse is true or false.

If x > 3, then x > 7

(b) Write down Premise 2 to complete the following argument:
Premise 1: If y = mx + 5 is a linear equation, then m is a gradient of the straight line.
Premise 2: _____________________
Conclusion: 2 is the gradient of the straight line.

(c) Angle subtended at the centre of a regular polygon with n sides is

\frac{360^{o}}{n} .

Make one conclusion by deduction for the angle subtended at the centre of a regular polygon with 5 sides.

Solution:
(a)(i) True

(a)(ii) The converse is true

(b) Premise 2: y = 2x + 5 is a linear equation

(c)

\begin{array}{l} Angle subtended at the centre of a regular pentagon \\ = \frac{360^{o}}{n} \\ = \frac{360^{o}}{5} \\ = 72^{o} \end{array}

Question 8:
(a) State whether the following sentence is a statement or not a statement.

x + 7 = 9

(b) Complete the following compound statement by writing the word ‘or’ or ‘and’ to form a true statement.
2³ = 6 …… 5 × 0 = 0

(c) Write down Premise 2 to complete the following argument:
Premise 1: All isosceles triangles have two equal sides.
Premise 2: _____________________
Conclusion: ABC has two equal sides.

(d) Make a general conclusion by induction for the sequence of numbers 1, 7, 17, 31, … which follows the following pattern.
1 = (2 × 1) – 1
7 = (2 × 4) – 1
17 = (2 × 9) – 1
31 = (2 × 16) – 1

Solution:
(a) Not a statement

(b) 2³ = 6 …or… 5 × 0 = 0

(c) ABC is an isosceles triangle.

(d)
1 = (2 × 1) – 1 = (2 × 1²) – 1
7 = (2 × 4) – 1 = (2 × 2²) – 1
17 = (2 × 9) – 1 = (2 × 3²) – 1
31 = (2 × 16) – 1 = (2 × 4²) – 1
= 2n² – 1, n = 1, 2, 3, …

4.5 Arguments

Posted on April 23, 2020 by user

(A) Premises and Conclusions

An argument is a process of making conclusion based on several given statements.
The statements given are known as premises.
An argument consists of premises and a conclusion.

Example 1:

Identify the premises and conclusion of the following argument.

A pentagon has 5 sides. ABCDE is a pentagon. Therefore, ABCDE has 5 sides.

Solution:

Premise 1: A pentagon has 5 sides.
Premise 2: ABCDE is a pentagon.
Conclusion: ABCDE has 5 sides.

(B) Forms of Arguments

Based on two given premises, a conclusion can be made for three different forms of arguments.

Argument Form I

Premise 1: All A are B.
Premise 2: C is A.
Conclusion: C is B.

Example 2:

Make a conclusion based on the two premises given below.

Premise 1: All multiples of 5 are divisible by 5.
Premise 2: 45 is a multiple of 5.
Conclusion: _______________

Solution:

Conclusion: 45 is divisible by 5.

Argument Form II

Premise 1: If p, then q.
Premise 2: p is true.
Conclusion: q is true.

Example 3:

Make a conclusion based on the two premises given below.
Premise 1: If a number is a factor of 18, then the number is a factor of 54.
Premise 2: 3 is a factor of 18.
Conclusion: _______________

Solution:

Conclusion: 3 is a factor of 54.

Argument Form III

Premise 1: If p, then q.
Premise 2: Not q is true.
Conclusion: Not p is true.

Example 4:

Make a conclusion based on the two premises given below.
Premise 1: If P is a subset of Q, then P ∩ Q = P .
Premise 2: P ∩ Q ≠ P
Conclusion: _______________

Solution:

Conclusion: P is not the subset of Q.

4.6 SPM Practice (Long Questions)

Posted on April 23, 2020 by user

Question 5:
(a) State if each of the following statements is true or false.

(i) 2³= 8 or ⅓ = 1.33.

(ii) – 6 > – 8 and 6 > 8.

(b) Write down two implications based on the following statement:

\frac{x}{a} + \frac{y}{b} = 1 if and only if b x + a y = a b .

(c) It is given that the interior angle of a regular polygon of n sides
is

(1 - \frac{2}{n}) \times 180^{\circ}

.
Make one conclusion by deduction on the size of the interior angle of a regular hexagon.

Solution:
(a)(i) True

(a)(ii) False

(b)

\begin{array}{l} Implication 1: \underline{If \frac{x}{a} + \frac{y}{b} = 1, then b x + a y = a b .} \\ Implication 2: \underline{If b x + a y = a b, then \frac{x}{a} + \frac{y}{b} = 1.} \end{array}

(c)

\begin{array}{l} Size of an interior angle of a regular hexagon \\ = (1 - \frac{2}{6}) \times 180^{\circ} \\ = \frac{2}{3} \times 180^{\circ} \\ = 120^{\circ} \end{array}

Question 6:
(a) Complete the following mathematical sentence by writing the symbol > or <.

(i) 5³____ 20 is a false statement.

(ii) – 3 ____ – 10 is a true statement.

(b) Complete the conclusion in the following argument:

\begin{array}{l} Premise 1 : If n^{\frac{1}{2}} = \sqrt{n}, then 4^{\frac{1}{2}} = \sqrt{4} = 2. \\ Premise 2 : n^{\frac{1}{2}} = \sqrt{n} \\ Conclusion : _____________________ \end{array}

(c) Make a general conclusion by induction for the sequence of numbers 10, 35, 70, … which follows the following pattern.

10 = 5 (2)² – 10

35 = 5 (3)² – 10

70 = 5 (4)² – 10

…. = ………..

Solution:
(a)(i) 5³ < 20 is a false statement.

(a)(ii) – 3 > – 10 is a true statement.

(b)

Conclusion : 4^{\frac{1}{2}} = \sqrt{4} = 2

(c) 5 (n + 1)² – 10, where n = 1, 2, 3, …

4.4 Implications Short Notes

Posted on April 23, 2020 by user

4.4 Implications

(A) Antecedent and Consequent of an Implication

1. For two statements p and q, the sentence ‘if p, then q’ is called an implication.

2. p is called the antecedent.

q is called the consequent.

Example:

Identify the antecedent and consequent of the following implications.

(a) If m = 2, then 2m² + m = 10

(b) I f P \cup Q = P, t h e n Q \subset P

Solution:
(a) Antecedent: m = 2
Consequent:: 2m² + m = 10

\begin{array}{l} (b) Antecedent : P \cup Q = P \\ Consequent : Q \subset P \end{array}

(B) Implications of the Form ‘p if and only if q’

1. Two implications ‘if p, then q’ and ‘if q, then p’ can be written as ‘p if and only if q’.

2. Likewise, two statements can be written from a statement in the form ‘p if and only if q’ as follows:

Implication 1: If p, then q.

Implication 2: If q, then p.

Example 1:

Given that p: x + 1 = 8

q: x = 7

Construct a mathematical statement in the form of implication

(a) If p, then q.

(b) p if and only if q.

Solution:

(a) If x + 1 = 8, then x = 7.

(b) x + 1 = 8 if and only if x = 7.

Example 2:

Write down two implications based on the following sentence:

x³ = 64 if and only if x = 4.

Solution:

If x³ = 64, then x = 4.

If x = 4, then x³ = 64.

(C) Converse of an Implication

1. The converse of an implication ‘if p, then q’ is ‘if q, then p’.

Example:

State the converse of each of the following implications.

(a) If x² + x – 2 = 0, then (x - 1)(x + 2) = 0.

(b) If x = 7, then x + 2 = 9.

Solution:

(a) If (x - 1)(x + 2) = 0, then x² + x – 2.

(b) If x + 2 = 9, then x = 7.

4.6 SPM Practice (Long Questions)

Posted on April 23, 2020 by user

Question 3:
(a) Complete each of the following statements with the quantifier ‘all’ or ‘some’ so that it will become a true statement.
(i) ___________ of the prime numbers are odd numbers.
(ii) ___________ pentagons have five sides.

(b) Write down two implications based on the following statement:

A ∩ B = B if and only if A υ B = A.

(c) Complete the premise in the following argument:

Premise 1: If a number is a factor of 24, then it is a factor of 48.

Premise 2: 12 is a factor of 24.

Conclusion: _____________________

Solution:
(a)(i) Some of the prime numbers are odd numbers.
(a)(ii) All pentagons have five sides.

(b) Implication 1: If A ∩ B = B, then A υ B = A.
Implication 2: If A υ B = A, then A ∩ B = B.

(c) Conclusion: 12 is a factor of 48.

Question 4:
(a) Combine the following two statements to form one true statement.
Statement 1: (– 3)² = 9
Statement 2: –3 (3) = 19

(b) Complete the premise in the following argument:

Premise 1: _____________________

Premise 2: x is a multiple of 25.

Conclusion: x is a divisible of 5.

(c) Make a general conclusion by induction for the sequence of numbers 7, 14, 27, … which follows the following pattern.

7 = 3 (2)¹ + 1

14 = 3 (2)² + 2

27 = 3 (2)³ + 3

…. = ………..

Solution:
(a) (– 3)² = 9 or –3 (3) = 19.

(b) Premise 1: All multiples of 25 is divisible by 5.

(c) 3 (2)ⁿ + n, where n = 1, 2, 3, …