Sure! Let's go through each part of the problem step by step.
### Part (a)
To find the value of [tex]\( m \)[/tex] in the given equation [tex]\( 2 \times 4 = \frac{1}{2} \times 8m \)[/tex], we can follow these steps:
1. Start by simplifying the left-hand side:
[tex]\( 2 \times 4 = 8 \)[/tex].
2. Now the equation looks like:
[tex]\( 8 = \frac{1}{2} \times 8m \)[/tex].
3. Simplify the right-hand side:
[tex]\( \frac{1}{2} \times 8m = 4m \)[/tex].
4. So, we have:
[tex]\( 8 = 4m \)[/tex].
5. Solve for [tex]\( m \)[/tex] by dividing both sides by 4:
[tex]\( m = \frac{8}{4} \)[/tex].
6. This gives us:
[tex]\( m = 2 \)[/tex].
So, the value of [tex]\( m \)[/tex] is [tex]\( 2 \)[/tex].
### Part (b)
To find the smallest angle in a triangle with side lengths in the ratio [tex]\( 2:3:4 \)[/tex], we use the fact that the smallest angle is opposite the smallest side.
1. The sides are proportional to 2, 3, and 4.
2. The smallest side length is 2.
3. By the properties of triangles, the angles are proportional to the sides opposite them. Therefore, the smallest angle is opposite to the smallest side.
As a result, the smallest angle is the one opposite the side with length 2.
### Part (c.i)
We want to factorize the expression [tex]\( 2xy - 6y + 7x - 21 \)[/tex].
We can use factorization by grouping:
1. Group the terms in pairs:
[tex]\( (2xy - 6y) + (7x - 21) \)[/tex].
2. Factor out the common factors from each group:
[tex]\( 2y(x - 3) + 7(x - 3) \)[/tex].
3. Notice that [tex]\( (x - 3) \)[/tex] is common in both terms, so factor it out:
[tex]\( (x - 3)(2y + 7) \)[/tex].
Thus, the factorized form of [tex]\( 2xy - 6y + 7x - 21 \)[/tex] is:
[tex]\[ (x - 3)(2y + 7) \][/tex]
### Part (c.ii)
Now, we need to evaluate the expression [tex]\( (x - 3)(2y + 7) \)[/tex] for [tex]\( x = 2 \)[/tex] and [tex]\( y = -1 \)[/tex].
1. Substitute [tex]\( x = 2 \)[/tex] and [tex]\( y = -1 \)[/tex] into the factorized expression:
[tex]\[ (2 - 3)(2(-1) + 7) \][/tex]
2. Simplify inside the parentheses first:
[tex]\[ (2 - 3)(-2 + 7) \][/tex]
3. This gives:
[tex]\[ (-1)(5) \][/tex]
4. Finally, multiply the results:
[tex]\[ -1 \times 5 = -5 \][/tex]
So, the evaluated expression for [tex]\( x = 2 \)[/tex] and [tex]\( y = -1 \)[/tex] is [tex]\( -5 \)[/tex].
### Summary of Answers:
1. a. The value of [tex]\( m \)[/tex] is [tex]\( 2 \)[/tex].
2. b. The smallest angle is the one opposite the side of length 2.
3. c. i. The factorized form of [tex]\( 2xy - 6y + 7x - 21 \)[/tex] is [tex]\( (x - 3)(2y + 7) \)[/tex].
ii. The evaluated expression for [tex]\( x = 2 \)[/tex] and [tex]\( y = -1 \)[/tex] is [tex]\( -5 \)[/tex].
