# 3.7.5 Logical Reasoning, SPM Paper 2 (Long Questions)

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, …

Question 11:
(a) For each of the following statements, determine whether the statement is true or  false.
(i) 35 is multiple of  3 and 5
(ii)   7 is a factor of 42 or 16 is a multiple of 6.
(b)   Write down two implications based on the following sentence:
p3 = –8 if and only if p = –2
(c) Make a general conclusion by induction for a list of numbers 2, 11, 26, 47, …….   which  follows the following pattern.
2 = 3(1)2– 1
11 = 3(2)2– 1
26 = 3(3)2– 1
47 = 3(4)2– 1
…………… [6 marks]

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

(b)
If p3 = –8, then p = –2
If p = –2, then p3 = –8

(c)
3n2– 1, where n = 1, 2, 3, 4, .....