4.6 SPM Practice (Long Questions)

Question 5:
(a) State if each of the following statements is true or false.
(b) Write down two implications based on the following statement:

$\boxed{\frac{x}{a}+\frac{y}{b}=1\text{ if and only if }bx+ay=ab.}$

(c) It is given that the interior angle of a regular polygon of n sides 
  is   $\left(1-\frac{2}{n}\right)\times {180}^{\circ }$ .
  Make one conclusion by deduction on the size of the interior angle of a regular hexagon.

Solution:
(a)(i) True

(a)(ii) False

(b) 
$\begin{array}{l}\text{Implication 1: }\underset{\_}{\text{ If }\frac{x}{a}+\frac{y}{b}=1,\text{ then }bx+ay=ab.}\\ \text{Implication 2: }\underset{\_}{\text{ If }bx+ay=ab,\text{ then }\frac{x}{a}+\frac{y}{b}=1.}\end{array}$

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

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

(b) Complete the conclusion in the following argument:
$$\boxed{\begin{array}{l}\text{ Premise 1 : If }{n}^{\frac{1}{2}}=\sqrt{n},\text{ then }{4}^{\frac{1}{2}}=\sqrt{4}=2.\\ \text{ Premise 2 : }{n}^{\frac{1}{2}}=\sqrt{n}\\ \text{ Conclusion : \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}\end{array}}$$
(c) Make a general conclusion by induction for the sequence of numbers 10, 35, 70, … which follows the following pattern.


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

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

(b)
$\text{Conclusion :}{4}^{\frac{1}{2}}=\sqrt{4}=2$

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