The concept of real numbers is a fundamental topic in mathematics, particularly in the field of algebra. Real numbers include all rational and irrational numbers and are represented on the number line. In this response, we will discuss multiple-choice questions (MCQs) related to real numbers, along with their solutions, for Class 10 Mathematics.

MCQs with Solutions:

1. Which of the following numbers is not a real number?
a) 4/5
b) √2
c) 2/0
d) π

Solution:
The correct answer is option c) 2/0. Division by zero is undefined in mathematics, making it an exception to the set of real numbers. Therefore, 2/0 is not a real number.

2. Which of the following is an irrational number?
a) 5
b) 3/4
c) √9
d) 2.5

Solution:
The correct answer is option c) √9. An irrational number is a number that cannot be expressed as a fraction or a ratio of two integers. Since √9 = 3, which is an integer, it is not an irrational number.

3. What is the decimal representation of the rational number 5/8?
a) 0.625
b) 0.75
c) 0.8
d) 0.5

Solution:
The correct answer is option a) 0.625. To convert a fraction to a decimal, divide the numerator by the denominator. In this case, 5 divided by 8 is equal to 0.625.

4. Which of the following is a terminating decimal?
a) 1/3
b) 2/5
c) 3/7
d) 1/9

Solution:
The correct answer is option b) 2/5. A terminating decimal is a decimal number that ends after a finite number of digits. When 2 is divided by 5, the decimal representation is 0.4, which terminates.

5. Which of the following is a non-terminating and non-repeating decimal?
a) 0.333...
b) 0.666...
c) 0.123456789101112...
d) 0.121212...

Solution:
The correct answer is option c) 0.123456789101112.... A non-terminating and non-repeating decimal is an irrational number. The given decimal continues indefinitely without any repeating pattern, making it non-repeating.

Summary:

- Real numbers include both rational and irrational numbers.
- Division by zero is undefined, making it an exception to the set of real numbers.
- Irrational numbers cannot be expressed as fractions or ratios of two integers.
- To convert a fraction to a decimal, divide the numerator by the denominator.
- Terminating decimals have a finite number of digits, while non-terminating decimals continue indefinitely.
- Non-repeating decimals have no repeating pattern in their decimal representation.

It is important to have a clear understanding of real numbers and their properties, as they form the foundation for various mathematical concepts and calculations. Practice solving MCQs to enhance your understanding and problem-solving skills.

Gud