4.7 Mathematics Reasoning, SPM Paper 2 (Long Questions)


Question 8:
(a) State whether the following sentence is a statement or not a statement.
x + 7 = 9
(b) Complete the following compound statement by writing the word ‘or’ or ‘and’ to form a true statement.
23 = 6 …… 5 × 0 = 0

(c) Write down Premise 2 to complete the following argument:
Premise 1: All isosceles triangles have two equal sides.
Premise 2: _____________________
Conclusion: ABC has two equal sides.

(d) Make a general conclusion by induction for the sequence of numbers 1, 7, 17, 31, … which follows the following pattern.
1 = (2 × 1) – 1
7 = (2 × 4) – 1
17 = (2 × 9) – 1
31 = (2 × 16) – 1

Solution:
(a) Not a statement

(b) 23 = 6 …or… 5 × 0 = 0

(c) ABC is an isosceles triangle.

(d)
1 = (2 × 1) – 1 =  (2 × 12) – 1
7 = (2 × 4) – 1 =  (2 × 22) – 1
17 = (2 × 9) – 1 =  (2 × 32) – 1
31 = (2 × 16) – 1 =  (2 × 42) – 1
  =  2n2 – 1, n = 1, 2, 3, …


Question 9:
(a) State whether the following statements are true statement or false statement.
( i ) { }{ H,O,T } ( ii ) { 2 }{ 2,3,4 }={ 2,3,4 }

(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 4 3 π 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)
4 3 π× 3 3 =36π Volume of the sphere is 36π.    


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, …


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