**Question 8**:

(a) State whether the following sentence is a statement or not a statement.

x + 7 = 9 |

**true**statement.

2

^{3}= 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:

Solution:

**(a)**Not a statement

**(b)**2

^{3}= 6 …or… 5 × 0 = 0

**(c)**

*ABC*is an isosceles triangle.

**(d)**

1 = (2 × 1) – 1 = (2 × 1

^{2}) – 1

7 = (2 × 4) – 1 = (2 × 2

^{2}) – 1

17 = (2 × 9) – 1 = (2 × 3

^{2}) – 1

31 = (2 × 16) – 1 = (2 × 4

^{2}) – 1

= 2

*n*

^{2}– 1,

*n*= 1, 2, 3, …

**Question 9**:

(a) State whether the following statements are true statement or false statement.

$\begin{array}{l}\left(i\right)\text{}\left\{\right\}\subset \left\{\text{H,O,T}\right\}\\ \left(ii\right)\text{}\left\{2\right\}\subset \left\{2,3,4\right\}=\left\{2,3,4\right\}\end{array}$

(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 $\frac{4}{3}\pi {r}^{3}$ where

*r*is the radius.

Make one conclusion by deduction for the volume of the sphere with radius 3 cm.

Solution:

Solution:

**(a)(i)**True

**False**

(a)(ii)

(a)(ii)

**(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.

**If the side of square**

(c)(ii)

(c)(ii)

*ABCD*is 15 cm, then the perimeter of square

*ABCD*is 60 cm.

**(d)**

$\begin{array}{l}\frac{4}{3}\pi \times {3}^{3}=36\pi \\ \text{Volumeofthesphereis}36\pi .\end{array}$

**Question 10**:

(a)(i) State whether the following compound statement is true or false.

All straight lines have positive gradients. |

If x = 5, then x^{2} = 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

**If**

(a)(ii)

(a)(ii)

*x*

^{2}= 25, then

*x*= 5

**(b)**Premise 2: 3 is an odd number

**(c)**

*n*(

*n*+ 1)

^{2}, where

*n*= 1, 2, 3, …