Question 10:
(a)(i) State whether the following compound statement is true or false.
(ii) Write down the converse of the following implication.
(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, …
(a)(i) State whether the following compound statement is true or false.
| All straight lines have positive gradients. |
| If x = 5, then x2 = 25. |
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.
(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, .....