Math worksheet focusing on rational numbers, covering concepts such as reciprocals, multiplicative inverses, and identifying rational numbers.

Problem Analysis:

The task involves filling in blanks and choosing the correct answers related to rational numbers. Let's solve each part step by step.

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Part I: Fill in the blanks

#### 1. A rational number which has no reciprocal is ………………
- Explanation: The reciprocal of a number \( x \) is defined as \( \frac{1}{x} \). For the reciprocal to exist, \( x \neq 0 \). If \( x = 0 \), then \( \frac{1}{0} \) is undefined.
- Answer: 0

#### 2. The multiplicative inverse of \(-\frac{2}{3}\) is ………………
- Explanation: The multiplicative inverse (reciprocal) of a number \( x \) is \( \frac{1}{x} \). For \( -\frac{2}{3} \), the reciprocal is \( \frac{1}{-\frac{2}{3}} = -\frac{3}{2} \).
- Answer: \(-\frac{3}{2}\)

#### 3. The product of a non-zero rational number and its reciprocal is ………………
- Explanation: By definition, the product of a number \( x \) and its reciprocal \( \frac{1}{x} \) is \( x \cdot \frac{1}{x} = 1 \).
- Answer: 1

#### 4. There are ……………… rational numbers between two rational numbers.
- Explanation: Between any two rational numbers, there are infinitely many other rational numbers. This is because rational numbers are dense in the real number line.
- Answer: infinitely many

#### 5. The numbers ……………… and ……………… are their own reciprocals.
- Explanation: A number \( x \) is its own reciprocal if \( x = \frac{1}{x} \). Solving this equation:
\[
x = \frac{1}{x} \implies x^2 = 1 \implies x = 1 \text{ or } x = -1
\]
- Answer: 1 and -1

#### 6. The reciprocal of a negative rational number is ………………
- Explanation: The reciprocal of a negative rational number \( -x \) is \( \frac{1}{-x} = -\frac{1}{x} \), which is also a negative rational number.
- Answer: negative

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Part II: Choose the correct answer

#### 7. A number which can be expressed as \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \), is:
- Options:
- (a) Natural number
- (b) Whole number
- (c) Integer
- (d) Rational number
- Explanation: By definition, a rational number is any number that can be expressed as \( \frac{p}{q} \), where \( p \) and \( q \) are integers and \( q \neq 0 \).
- Answer: (d) Rational number

#### 8. Zero (0) is:
- Options:
- (a) The identity for addition of rational numbers.
- (b) The identity for subtraction of rational numbers.
- (c) The identity for multiplication of rational numbers.
- (d) The identity for division of rational numbers.
- Explanation:
- Addition identity: Adding 0 to any number leaves it unchanged. So, 0 is the additive identity.
- Subtraction identity: Subtraction does not have an identity element in the same sense.
- Multiplication identity: Multiplying any number by 0 results in 0, so 0 is not the multiplicative identity (the multiplicative identity is 1).
- Division identity: Division by 0 is undefined.
- Answer: (a) The identity for addition of rational numbers

#### 9. Multiplicative inverse of a negative rational number is:
- Options:
- (a) A positive rational number.
- (b) A negative rational number.
- (c) 0
- (d) 1
- Explanation: The multiplicative inverse of a negative rational number \( -x \) is \( \frac{1}{-x} = -\frac{1}{x} \), which is also negative.
- Answer: (b) A negative rational number

#### 10. The reciprocal of \(-1\) is:
- Options:
- (a) 1
- (b) \(-1\)
- (c) 0
- (d) Not defined
- Explanation: The reciprocal of \(-1\) is \( \frac{1}{-1} = -1 \).
- Answer: (b) \(-1\)

#### 11. Between two given rational numbers, we can find:
- Options:
- (a) One and only one rational number.
- (b) Only two rational numbers.
- (c) Only ten rational numbers.
- (d) Infinitely many rational numbers.
- Explanation: Rational numbers are dense in the real number line, meaning there are infinitely many rational numbers between any two rational numbers.
- Answer: (d) Infinitely many rational numbers

#### 12. \( \frac{x + y}{2} \) is a rational number:
- Options:
- (a) Between \( x \) and \( y \)
- (b) Less than \( x \) and \( y \) both.
- (c) Greater than \( x \) and \( y \) both.
- (d) Less than \( x \) but greater than \( y \).
- Explanation: The expression \( \frac{x + y}{2} \) represents the arithmetic mean of \( x \) and \( y \). The arithmetic mean of two numbers always lies between them.
- Answer: (a) Between \( x \) and \( y \)

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Final Answers:

1. 0
2. \(-\frac{3}{2}\)
3. 1
4. infinitely many
5. 1 and -1
6. negative

7. (d) Rational number
8. (a) The identity for addition of rational numbers
9. (b) A negative rational number
10. (b) \(-1\)
11. (d) Infinitely many rational numbers
12. (a) Between \( x \) and \( y \)

Boxed Final Answer:

\[
\boxed{
\begin{array}{l}
1. \, 0 \\
2. \, -\frac{3}{2} \\
3. \, 1 \\
4. \, \text{infinitely many} \\
5. \, 1 \, \text{and} \, -1 \\
6. \, \text{negative} \\
7. \, (d) \, \text{Rational number} \\
8. \, (a) \, \text{The identity for addition of rational numbers} \\
9. \, (b) \, \text{A negative rational number} \\
10. \, (b) \, -1 \\
11. \, (d) \, \text{Infinitely many rational numbers} \\
12. \, (a) \, \text{Between } x \text{ and } y
\end{array}
}
\]