4.6 SPM Practice (Long Questions)

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)