Worksheet Ch 3: Mathematical Reasoning

Section A • Bahagian A [12 Marks]

Question 1 [4 Marks]

(a) State whether each of the following is a statement or a non-statement:

(i) $3x + 4 = 19$.

(ii) 49 is a perfect square.

(b) Determine the truth value of the compound statement: "$2^3 = 6$ or $5 \times 0 = 0$." [2 marks]

(a)(i) Non-statement (Bukan pernyataan)

(a)(ii) Statement (Pernyataan)

P1 P1

(b) $2^3 = 6$ is False, but $5 \times 0 = 0$ is True. "OR" statement is True if at least one component is True.

Truth Value: True (Benar)

K1 N1

Question 2 [4 Marks]

Write down two implications based on the following compound statement:

"$x$ is an odd number if and only if $x + 1$ is an even number."

Implication 1:

If $x$ is an odd number, then $x + 1$ is an even number.

Implication 2:

If $x + 1$ is an even number, then $x$ is an odd number.

Question 3 [4 Marks]

Given the implication: "If $x > 7$, then $x^2 > 49$."

(a) State the converse and determine its truth value. [2 marks]
(b) State the contrapositive and determine its truth value. [2 marks]

(a) Converse: If $x^2 > 49$, then $x > 7$.

Truth Value: False (Counterexample: $x = -8 \implies (-8)^2 = 64 > 49$, but $-8 \ngtr 7$).

N1 P1

(b) Contrapositive: If $x^2 \le 49$, then $x \le 7$.

Truth Value: True.

N1 P1

Section B • Bahagian B (HOTS / KBAT) [18 Marks]

Question 4 (Deductive Arguments) [9 Marks]

Complete the premises or conclusions for the following deductive arguments and state whether each argument is valid and sound:

(a) [3 marks]
Premise 1: All regular pentagons have 5 axes of symmetry.
Premise 2: Polygon $P$ is a regular pentagon.
Conclusion: ___________________________________________________________

(b) [3 marks]
Premise 1: If $k$ is a multiple of 6, then $k$ is divisible by 3.
Premise 2: ___________________________________________________________
Conclusion: 14 is not a multiple of 6.

(c) [3 marks]
Premise 1: If angle $\theta$ is an acute angle, then $0^\circ < \theta < 90^\circ$.
Premise 2: $\theta = 45^\circ$ is an acute angle.
Conclusion: ___________________________________________________________

(a) Form I: Conclusion: Polygon $P$ has 5 axes of symmetry.

Valid and sound (Sah dan munasabah).

(b) Form III (Modus Tollens): Premise 2: 14 is not divisible by 3.

Valid and sound.

Question 5 (Inductive Generalisation) [9 Marks]

The table below shows the pattern of total tiles used to pave geometric hexagonal pathways:

Pathway 1: $5 = 3(1) + 2$

Pathway 2: $8 = 3(2) + 2$

Pathway 3: $11 = 3(3) + 2$

Pathway 4: $14 = 3(4) + 2$

(a) Form a strong inductive conclusion for the number of tiles in Pathway $n$. [3 marks]
(b) Find the number of tiles needed for Pathway 25. [2 marks]
(c) A contractor has a crate of 500 tiles. Determine the largest pathway number $n$ that can be paved completely and calculate the remaining unused tiles. [4 marks]

(a) Inductive formula with sequence definition

$$3n + 2, \quad n = 1, 2, 3, 4, \dots$$

K1 N2

(b) Substitute $n = 25$

$$\text{Tiles} = 3(25) + 2 = 75 + 2 = 77\text{ tiles}$$

Largest pathway $n = 166$

$$\text{Tiles used} = 3(166) + 2 = 498 + 2 = 500 \implies \mathbf{0} \text{ unused tiles!}$$

K2 N2