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Sagot :
Sure, let's go through each question in a detailed and step-by-step manner.
### Section A: Multiple Choice Questions
Question 1:
Which of the following is the reciprocal of [tex]\(a\)[/tex]?
- (a) [tex]\(1 \rightarrow a\)[/tex]
- (b) [tex]\(a\)[/tex]
- (c) [tex]\(1 / 2\)[/tex]
- (d) [tex]\(1 / a\)[/tex]
The reciprocal of a number [tex]\(a\)[/tex] is a number which, when multiplied by [tex]\(a\)[/tex], gives the product as [tex]\(1\)[/tex]. Therefore, the reciprocal of [tex]\(a\)[/tex] is [tex]\(1/a\)[/tex].
Answer: (d) [tex]\(1 / a\)[/tex]
---
Question 2:
Which of the following lies between 0 and -1?
- (a) 0
- (b) [tex]\(1 / 3\)[/tex]
- (c) [tex]\(-2 / 3\)[/tex]
- (d) [tex]\(4 / 3\)[/tex]
We need to find a fraction that lies between 0 and -1. Among the options, only [tex]\(-2 / 3\)[/tex] fits this criterion.
Answer: (c) [tex]\(-2 / 3\)[/tex]
---
Question 3:
Which of the following is neither a positive nor a negative rational number?
- (a) 1
- (b) 0
- (c) Such a rational number does not exist
- (d) None of these
A number that is neither positive nor negative is [tex]\(0\)[/tex].
Answer: (b) 0
---
Question 4:
_____ is the identity for the addition of rational numbers.
- (a) 1
- (b) 0
- (c) [tex]\(1/2\)[/tex]
- (d) None of these
The identity element for addition is the number which, when added to any other number, does not change the value of that number. For rational numbers, this is [tex]\(0\)[/tex].
Answer: (b) 0
---
Question 5:
Which of the following can be expressed as terminating or non-terminating?
- (a) [tex]\(1 / 3\)[/tex]
- (b) [tex]\(-14 / 15\)[/tex]
- (c) [tex]\(-38 / 81\)[/tex]
A fraction can be expressed as a terminating or non-terminating decimal. Any fraction can either terminate or continue indefinitely as a repeating decimal. Thus, the fraction [tex]\(-14 / 15\)[/tex], like the others, can be expressed as either terminating or non-terminating.
Answer: (b) [tex]\(-14 / 15\)[/tex]
---
### Section B: Short Answer Questions
Question 6:
Multiply the negative of [tex]\(2 / 3\)[/tex] by the inverse of [tex]\(9 / 7\)[/tex].
First, find the negative of [tex]\(2 / 3\)[/tex]:
[tex]\[ - \frac{2}{3} \][/tex]
Next, find the inverse of [tex]\(9 / 7\)[/tex], which means flipping the numerator and the denominator:
[tex]\[ \frac{7}{9} \][/tex]
Now, multiply [tex]\(-2 / 3\)[/tex] by [tex]\(7 / 9\)[/tex]:
[tex]\[ \left( - \frac{2}{3} \right) \times \left( \frac{7}{9} \right) = - \frac{2 \times 7}{3 \times 9} = - \frac{14}{27} \][/tex]
Answer: [tex]\(- \frac{14}{27}\)[/tex]
---
Question 7:
Write [tex]\(\frac{2}{3}\)[/tex], [tex]\(-\frac{4}{9}\)[/tex], [tex]\(-\frac{8}{11}\)[/tex] in ascending order.
First, we convert these fractions into decimal to compare them more easily:
[tex]\[ \frac{2}{3} \approx 0.6667 \][/tex]
[tex]\[ -\frac{4}{9} \approx -0.4444 \][/tex]
[tex]\[ -\frac{8}{11} \approx -0.7273 \][/tex]
Now, arrange them in ascending order (from the smallest to the largest):
[tex]\[ -\frac{8}{11} \approx -0.7273 \][/tex]
[tex]\[ -\frac{4}{9} \approx -0.4444 \][/tex]
[tex]\[ \frac{2}{3} \approx 0.6667 \][/tex]
Thus, in ascending order:
[tex]\[ -\frac{8}{11}, -\frac{4}{9}, \frac{2}{3} \][/tex]
Answer: [tex]\(-\frac{8}{11}, -\frac{4}{9}, \frac{2}{3}\)[/tex]
---
These are the detailed solutions for each of the questions in the given problem set.
### Section A: Multiple Choice Questions
Question 1:
Which of the following is the reciprocal of [tex]\(a\)[/tex]?
- (a) [tex]\(1 \rightarrow a\)[/tex]
- (b) [tex]\(a\)[/tex]
- (c) [tex]\(1 / 2\)[/tex]
- (d) [tex]\(1 / a\)[/tex]
The reciprocal of a number [tex]\(a\)[/tex] is a number which, when multiplied by [tex]\(a\)[/tex], gives the product as [tex]\(1\)[/tex]. Therefore, the reciprocal of [tex]\(a\)[/tex] is [tex]\(1/a\)[/tex].
Answer: (d) [tex]\(1 / a\)[/tex]
---
Question 2:
Which of the following lies between 0 and -1?
- (a) 0
- (b) [tex]\(1 / 3\)[/tex]
- (c) [tex]\(-2 / 3\)[/tex]
- (d) [tex]\(4 / 3\)[/tex]
We need to find a fraction that lies between 0 and -1. Among the options, only [tex]\(-2 / 3\)[/tex] fits this criterion.
Answer: (c) [tex]\(-2 / 3\)[/tex]
---
Question 3:
Which of the following is neither a positive nor a negative rational number?
- (a) 1
- (b) 0
- (c) Such a rational number does not exist
- (d) None of these
A number that is neither positive nor negative is [tex]\(0\)[/tex].
Answer: (b) 0
---
Question 4:
_____ is the identity for the addition of rational numbers.
- (a) 1
- (b) 0
- (c) [tex]\(1/2\)[/tex]
- (d) None of these
The identity element for addition is the number which, when added to any other number, does not change the value of that number. For rational numbers, this is [tex]\(0\)[/tex].
Answer: (b) 0
---
Question 5:
Which of the following can be expressed as terminating or non-terminating?
- (a) [tex]\(1 / 3\)[/tex]
- (b) [tex]\(-14 / 15\)[/tex]
- (c) [tex]\(-38 / 81\)[/tex]
A fraction can be expressed as a terminating or non-terminating decimal. Any fraction can either terminate or continue indefinitely as a repeating decimal. Thus, the fraction [tex]\(-14 / 15\)[/tex], like the others, can be expressed as either terminating or non-terminating.
Answer: (b) [tex]\(-14 / 15\)[/tex]
---
### Section B: Short Answer Questions
Question 6:
Multiply the negative of [tex]\(2 / 3\)[/tex] by the inverse of [tex]\(9 / 7\)[/tex].
First, find the negative of [tex]\(2 / 3\)[/tex]:
[tex]\[ - \frac{2}{3} \][/tex]
Next, find the inverse of [tex]\(9 / 7\)[/tex], which means flipping the numerator and the denominator:
[tex]\[ \frac{7}{9} \][/tex]
Now, multiply [tex]\(-2 / 3\)[/tex] by [tex]\(7 / 9\)[/tex]:
[tex]\[ \left( - \frac{2}{3} \right) \times \left( \frac{7}{9} \right) = - \frac{2 \times 7}{3 \times 9} = - \frac{14}{27} \][/tex]
Answer: [tex]\(- \frac{14}{27}\)[/tex]
---
Question 7:
Write [tex]\(\frac{2}{3}\)[/tex], [tex]\(-\frac{4}{9}\)[/tex], [tex]\(-\frac{8}{11}\)[/tex] in ascending order.
First, we convert these fractions into decimal to compare them more easily:
[tex]\[ \frac{2}{3} \approx 0.6667 \][/tex]
[tex]\[ -\frac{4}{9} \approx -0.4444 \][/tex]
[tex]\[ -\frac{8}{11} \approx -0.7273 \][/tex]
Now, arrange them in ascending order (from the smallest to the largest):
[tex]\[ -\frac{8}{11} \approx -0.7273 \][/tex]
[tex]\[ -\frac{4}{9} \approx -0.4444 \][/tex]
[tex]\[ \frac{2}{3} \approx 0.6667 \][/tex]
Thus, in ascending order:
[tex]\[ -\frac{8}{11}, -\frac{4}{9}, \frac{2}{3} \][/tex]
Answer: [tex]\(-\frac{8}{11}, -\frac{4}{9}, \frac{2}{3}\)[/tex]
---
These are the detailed solutions for each of the questions in the given problem set.
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