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Practice Class 10 Maths Chapter 1 Real Numbers MCQ Questions with Answers based on NCERT syllabus. Learn Fundamental Theorem of Arithmetic, prime factorization, irrational numbers, HCF, LCM, and decimal expansion with detailed solutions and formulas for CBSE board exams.
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Introduction:
Real Numbers is the foundation of many mathematical concepts taught in higher classes. In this chapter, students learn about Euclid’s Division Lemma, prime numbers, composite numbers, HCF, LCM, irrational numbers, and the decimal expansion of rational numbers. These concepts are frequently tested in school examinations and board exams. The following MCQ practice set is designed to strengthen conceptual understanding through detailed explanations and useful formulas.
What You Will Learn?
✔ Fundamental Theorem of Arithmetic
✔ Prime Factorization
✔ Rational and Irrational Numbers
✔ Decimal Expansion
✔ HCF and LCM Concepts
✔ Board Exam Oriented MCQs
Why is this Topic Important?
Real Numbers forms the basis of many chapters in algebra and number systems. Understanding prime factorization and decimal expansions helps students solve complex mathematical problems more efficiently. Questions from this chapter are regularly asked in board examinations and competitive tests.
Q1. According to the Fundamental Theorem of Arithmetic, every composite number can be expressed as:
A) Sum of prime numbers
B) Product of prime numbers
C) Difference of prime numbers
D) Square of a prime number
Answer:
Product of prime numbers
Useful Formula for this Question:
Every composite number can be expressed as a product of prime numbers.
This factorization is unique except for the order of the prime factors.
Solution:
The Fundamental Theorem of Arithmetic states that every composite number can be expressed as a product of prime numbers in a unique way.
Example:
$$60 = 2 \times 2 \times 3 \times 5$$
Therefore, the correct answer is:
Product of prime numbers
Q2. The prime factorization of $$90$$ is:
A) $$2 \times 3 \times 5$$
B) $$2 \times 3^2 \times 5$$
C) $$2^2 \times 3 \times 5$$
D) $$3^2 \times 5$$
Answer:
$$2 \times 3^2 \times 5$$
Useful Formula for this Question:
Prime factorization means expressing a number as a product of prime numbers only.
Solution:
$$90 = 2 \times 45$$
$$45 = 3 \times 15$$
$$15 = 3 \times 5$$
Therefore:
$$90 = 2 \times 3 \times 3 \times 5$$
$$= 2 \times 3^2 \times 5$$
Hence, the correct answer is:
$$2 \times 3^2 \times 5$$
Q3. Which of the following numbers is not a rational number?
A) $$\frac{7}{9}$$
B) $$0.75$$
C) $$\sqrt{7}$$
D) $$0.125$$
Answer:
$$\sqrt{7}$$
Useful Formula for this Question:
A rational number can be written in the form:
$$\frac{p}{q}$$
where $$q \ne 0$$.
Solution:
$$\sqrt{7}$$ cannot be expressed in the form:
$$\frac{p}{q}$$
Its decimal expansion is non-terminating and non-recurring.
Therefore, $$\sqrt{7}$$ is irrational.
Hence, the correct answer is:
$$\sqrt{7}$$
Q4. The LCM of $$15$$ and $$20$$ is:
A) $$30$$
B) $$45$$
C) $$60$$
D) $$120$$
Answer:
$$60$$
Useful Formula for this Question:
$$LCM = Product\ of\ highest\ powers\ of\ prime\ factors$$
Solution:
Prime factorization:
$$15 = 3 \times 5$$
$$20 = 2^2 \times 5$$
Taking highest powers:
$$LCM = 2^2 \times 3 \times 5$$
$$= 60$$
Therefore, the correct answer is:
$$60$$
Q5. Which of the following numbers has a terminating decimal expansion?
A) $$\frac{3}{14}$$
B) $$\frac{7}{15}$$
C) $$\frac{9}{40}$$
D) $$\frac{5}{21}$$
Answer:
$$\frac{9}{40}$$
Useful Formula for this Question:
A rational number has a terminating decimal expansion if the denominator contains only the prime factors:
$$2$$ and/or $$5$$
Solution:
For:
$$\frac{9}{40}$$
Denominator:
$$40 = 2^3 \times 5$$
Only factors $$2$$ and $$5$$ are present.
Therefore, the decimal expansion is terminating.
Hence, the correct answer is:
$$\frac{9}{40}$$
Important Formulas:
$$a = bq + r$$
where
$$0 \le r < b$$
$$HCF \times LCM = Product\ of\ two\ numbers$$
A rational number is of the form:
$$\frac{p}{q}$$
where $$q \ne 0$$
A rational number has a terminating decimal expansion if the denominator contains only:
$$2$$ and/or $$5$$
FAQs:
Q. What is the Fundamental Theorem of Arithmetic?
Answer:
Every composite number can be expressed as a unique product of prime numbers.
Q. What is prime factorization?
Answer:
Expressing a number as a product of prime numbers only.
Q. Which numbers have terminating decimal expansions?
Answer:
Numbers whose denominators contain only the prime factors:
$$2$$ and/or $$5$$
Common Mistakes Students Make
❌ Considering every non-terminating decimal as irrational.
❌ Forgetting repeated prime factors during factorization.
❌ Confusing HCF with LCM.
❌ Assuming 1 is a prime number.
Chapter 1 Quick Revision
✔ 2 is the smallest prime number.
✔ 1 is neither prime nor composite.
✔ Every composite number can be written as a product of prime factors.
✔ HCF × LCM = Product of two numbers.
✔ Irrational numbers cannot be written in the form:
$$\frac{p}{q}$$
Conclusion:
These Class 10 Maths Chapter 1 Real Numbers MCQs with detailed solutions help students master important concepts such as prime factorization, irrational numbers, LCM, and decimal expansion. Regular practice of these questions can improve conceptual understanding and board exam performance.
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