Question 13 (5 marks):
(a) State whether the following statements are true statement or false statement.
$\begin{array}{l}\left(\text{i}\right)\text{ { }}\subset \{S,E,T\}\\ \left(\text{ii}\right)\text{ { 1 }}\subset \{1,2,3\}=\{1,2,3\}\end{array}$

(b) Diagram 7 shows the first three patterns of a sequence of patterns.

Diagram 7

It is given that the diameter of each circle is 20 cm.
(i) Make a general conclusion by induction for the area of the unshaded region.

(ii) Hence, calculate the area of the unshaded region for the 5th pattern.


Solution:
(a)(i) True

(a)(ii) False

(b)(i)
Area of unshaded region (first)
= (20 × 20) – π(10)²
= 400 – 100π
= 100 (4 – π)

Area of unshaded region (second)
= 4 × 100 (4 – π)
= 400 (4 – π)

Area of unshaded region (third)
= 9 × 100 (4 – π)
= 900 (4 – π)
100 (4 – π), 400 (4 – π), 900 (4 – π), …
10² (4 – π), 20² (4 – π), 30² (4 – π), …

General conclusion = n² (4 – π)
n = 10, 20, 30, …


(b)(ii)
Area of unshaded region (5th)
= 50² (4 – π)
= 2500 (4 – π)

Question 11 (5 marks):
(a) State whether the following compound statement is true or false.
2 > 3 and (–2)³ = –8

(b) Write down two implications based on the following statement:
a > b if and only if a – b > 0

(c) Table 1 shows the number of sides and the number of axes of symmetry for some regular polygons.

Table 1

Make a conclusion by induction by completing the statement in the answer space:
The number of axes of symmetry for a regular polygon with n sides is __________.


Solution:
(a)
Palsu. Nilai bagi 2 adalah kurang daripada 3, 2 < 3.

(b)
Implication 1: Jika a > b, maka a – b > 0
Implication 2: Jika a – b > 0, maka a – b > 0

(c)
The number of axes of symmetry for a regular polygon with n sides is
Number of sides of regular polygon = Number of axes of symmetry of regular polygon

Question 12 (5 marks):
(a) State whether the following statement is true or false.


(b) Write down the converse of the following implication:
 

(c) Write down Premise 2 to complete the following argument:
Premise 1   :  If x is an odd number, then x is not divisible by 2.
Premise 2   : ………………………………………………………
Conclusion : 24 is not an odd number.

(d) Based on the information below, make one conclusion by deduction for the surface area of sphere with radius 9 cm.



Solution:
(a)
True

(b)
If an angle is an acute angle, then the angle lies between 0^o and 90^o.

(c)
24 is divisible by 2.

(d)
4π(9)² = 324π
Surface area of the sphere is 324π.

Question 9:
(a) State whether the following statements are true statement or false statement.
$\begin{array}{l}\left(i\right)\left\{\right\}\subset \left\{\text{H,O,T}\right\}\\ \left(ii\right)\left\{2\right\}\subset \left\{2,3,4\right\}=\left\{2,3,4\right\}\end{array}$

(b) Complete the following statement to form a true statement by using the quantifier ‘all’ or ‘some’.
   ……… factor of 24 are factor of 40

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

The perimeter of square ABCD is 60 cm if and only if the side of square ABCD is 15 cm.

(d) It is given that the volume of the sphere is $\frac{4}{3}\pi r^3$ where r is the radius.
Make one conclusion by deduction for the volume of the sphere with radius 3 cm.

Solution:
(a)(i) True

(a)(ii) False

(b)    …Some… factor of 24 are factor of 40

(c)(i) If the perimeter of square ABCD is 60 cm, then the side of square ABCD is 15 cm.

(c)(ii) If the side of square ABCD is 15 cm, then the perimeter of square ABCD is 60 cm.

(d)
$\begin{array}{l}\frac{4}{3}\pi \times 3^3=36\pi \\ \text{Volume of the sphere is }36\pi .\end{array}$    

Question 10:
(a)(i) State whether the following compound statement is true or false.

All straight lines have positive gradients.

(ii) Write down the converse of the following implication.

If x = 5, then x2 = 25.

(b) Write down Premise 2 to complete the following argument:
Premise 1: If Q is an odd number, then 2 × Q is an even number.
Premise 2: _____________________
Conclusion: 2 × 3 is an even number.

(c) Make a general conclusion by induction for the sequence of numbers 4, 18, 48, 100, … which follows the following pattern.
4 = 1 (2)²
18 = 2 (3)²
48 = 3 (4)²
100 = 4 (5)²
  .
  .
  .

Solution:
(a)(i) False

(a)(ii) If x² = 25, then x = 5

(b) Premise 2: 3 is an odd number

(c) n (n + 1)², where n = 1, 2, 3, …

Example 1:

Premise 1: All regular polygons have equal sides. Premise 2: ABCD is a regular polygon. Conclusion:

Solution:



Example 2:

Premise 1: If m > 4, then 2m > 8. Premise 2: 2m < 8 Conclusion:

Solution:

Premise 1: Premise 2: m ×n is not an even number. Conclusion: m and n are not even numbers.

Premise 1: If x = 3, then x2 = 9. Premise 2: Conclusion: x ≠ 3

Example 1:

Solution:



Example 2:

Solution:



Example 3:

Solution: