Answer :
Certainly! Let's address both parts of the question using the given set of potential values for [tex]\( a \)[/tex]: [tex]\( \{1, -1, 2, -2, 3, -3\} \)[/tex].
### Part (i) [tex]\( \frac{1}{a} = 1 \)[/tex]?
First, let’s verify the expression [tex]\( \frac{1}{a} = 1 \)[/tex] for each value of [tex]\( a \)[/tex]:
1. For [tex]\( a = 1 \)[/tex]:
[tex]\[ \frac{1}{1} = 1 \][/tex]
This is true.
2. For [tex]\( a = -1 \)[/tex]:
[tex]\[ \frac{1}{-1} = -1 \][/tex]
This is not true.
3. For [tex]\( a = 2 \)[/tex]:
[tex]\[ \frac{1}{2} = 0.5 \][/tex]
This is not true.
4. For [tex]\( a = -2 \)[/tex]:
[tex]\[ \frac{1}{-2} = -0.5 \][/tex]
This is not true.
5. For [tex]\( a = 3 \)[/tex]:
[tex]\[ \frac{1}{3} = 0.3333\ldots \][/tex]
This is not true.
6. For [tex]\( a = -3 \)[/tex]:
[tex]\[ \frac{1}{-3} = -0.3333\ldots \][/tex]
This is not true.
From these computations, we conclude that [tex]\( \frac{1}{a} = 1 \)[/tex] only holds for [tex]\( a = 1 \)[/tex] and does not hold for other given values of [tex]\( a \)[/tex].
### Part (ii) [tex]\( \frac{a}{-1} = -a \)[/tex]?
Now, let’s check the expression [tex]\( \frac{a}{-1} = -a \)[/tex] for each value of [tex]\( a \)[/tex]:
1. For [tex]\( a = 1 \)[/tex]:
[tex]\[ \frac{1}{-1} = -1 \][/tex]
This is true.
2. For [tex]\( a = -1 \)[/tex]:
[tex]\[ \frac{-1}{-1} = 1 \][/tex]
This is true.
3. For [tex]\( a = 2 \)[/tex]:
[tex]\[ \frac{2}{-1} = -2 \][/tex]
This is true.
4. For [tex]\( a = -2 \)[/tex]:
[tex]\[ \frac{-2}{-1} = 2 \][/tex]
This is true.
5. For [tex]\( a = 3 \)[/tex]:
[tex]\[ \frac{3}{-1} = -3 \][/tex]
This is true.
6. For [tex]\( a = -3 \)[/tex]:
[tex]\[ \frac{-3}{-1} = 3 \][/tex]
This is true.
From these computations, we confirm that [tex]\( \frac{a}{-1} = -a \)[/tex] holds true for all the given values of [tex]\( a \)[/tex].
### Conclusion
- For part (i) [tex]\( \frac{1}{a} = 1 \)[/tex]: This holds true only when [tex]\( a = 1 \)[/tex].
- For part (ii) [tex]\( \frac{a}{-1} = -a \)[/tex]: This holds true for all the given values of [tex]\( a \)[/tex].
### Part (i) [tex]\( \frac{1}{a} = 1 \)[/tex]?
First, let’s verify the expression [tex]\( \frac{1}{a} = 1 \)[/tex] for each value of [tex]\( a \)[/tex]:
1. For [tex]\( a = 1 \)[/tex]:
[tex]\[ \frac{1}{1} = 1 \][/tex]
This is true.
2. For [tex]\( a = -1 \)[/tex]:
[tex]\[ \frac{1}{-1} = -1 \][/tex]
This is not true.
3. For [tex]\( a = 2 \)[/tex]:
[tex]\[ \frac{1}{2} = 0.5 \][/tex]
This is not true.
4. For [tex]\( a = -2 \)[/tex]:
[tex]\[ \frac{1}{-2} = -0.5 \][/tex]
This is not true.
5. For [tex]\( a = 3 \)[/tex]:
[tex]\[ \frac{1}{3} = 0.3333\ldots \][/tex]
This is not true.
6. For [tex]\( a = -3 \)[/tex]:
[tex]\[ \frac{1}{-3} = -0.3333\ldots \][/tex]
This is not true.
From these computations, we conclude that [tex]\( \frac{1}{a} = 1 \)[/tex] only holds for [tex]\( a = 1 \)[/tex] and does not hold for other given values of [tex]\( a \)[/tex].
### Part (ii) [tex]\( \frac{a}{-1} = -a \)[/tex]?
Now, let’s check the expression [tex]\( \frac{a}{-1} = -a \)[/tex] for each value of [tex]\( a \)[/tex]:
1. For [tex]\( a = 1 \)[/tex]:
[tex]\[ \frac{1}{-1} = -1 \][/tex]
This is true.
2. For [tex]\( a = -1 \)[/tex]:
[tex]\[ \frac{-1}{-1} = 1 \][/tex]
This is true.
3. For [tex]\( a = 2 \)[/tex]:
[tex]\[ \frac{2}{-1} = -2 \][/tex]
This is true.
4. For [tex]\( a = -2 \)[/tex]:
[tex]\[ \frac{-2}{-1} = 2 \][/tex]
This is true.
5. For [tex]\( a = 3 \)[/tex]:
[tex]\[ \frac{3}{-1} = -3 \][/tex]
This is true.
6. For [tex]\( a = -3 \)[/tex]:
[tex]\[ \frac{-3}{-1} = 3 \][/tex]
This is true.
From these computations, we confirm that [tex]\( \frac{a}{-1} = -a \)[/tex] holds true for all the given values of [tex]\( a \)[/tex].
### Conclusion
- For part (i) [tex]\( \frac{1}{a} = 1 \)[/tex]: This holds true only when [tex]\( a = 1 \)[/tex].
- For part (ii) [tex]\( \frac{a}{-1} = -a \)[/tex]: This holds true for all the given values of [tex]\( a \)[/tex].
