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Practice Class 10 Maths Chapter 1 Real Numbers MCQ Questions with Answers based on NCERT syllabus. Learn Euclid’s Division Algorithm, HCF, LCM, prime factorization, rational numbers, irrational numbers, and decimal expansion with detailed solutions and formulas for CBSE and board exams.
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Introduction:
Real Numbers is one of the most important chapters in Class 10 Mathematics because it develops the foundation of number theory and arithmetic operations. Students learn how to find HCF and LCM using prime factorization, apply Euclid’s Division Lemma, identify rational and irrational numbers, and determine whether a decimal expansion is terminating or non-terminating recurring. These concepts are frequently tested in school examinations, board exams, and scholarship tests. The following MCQ set contains fresh questions with detailed solutions and useful formulas to strengthen conceptual understanding.
What You Will Learn?
✔ Euclid’s Division Algorithm
✔ Prime Factorization
✔ HCF and LCM Applications
✔ Rational Numbers
✔ Irrational Numbers
✔ Decimal Expansion of Rational Numbers
✔ Board Exam Oriented Questions
Why is this Topic Important?
Real Numbers provides the foundation for many higher mathematical concepts. Understanding HCF, LCM, and decimal expansions helps students solve algebraic and arithmetic problems efficiently. Questions from this chapter regularly appear in Class 10 board examinations and competitive tests.
Q1. The HCF of $$45$$ and $$75$$ is:
A) $$5$$
B) $$10$$
C) $$15$$
D) $$25$$
Answer:
$$15$$
Useful Formula for this Question:
HCF is the greatest common factor of two or more numbers.
Solution:
Prime factorization:
$$45 = 3^2 \times 5$$
$$75 = 3 \times 5^2$$
Common prime factors:
$$3 \times 5$$
Therefore:
$$HCF = 15$$
Hence, the correct answer is:
$$15$$
Q2. Using Euclid’s Division Lemma, when $$29$$ is divided by $$6$$, the remainder is:
A) $$3$$
B) $$4$$
C) $$5$$
D) $$6$$
Answer:
$$5$$
Useful Formula for this Question:
$$a = bq + r$$
where
$$0 \le r < b$$
Solution:
Divide:
$$29 \div 6$$
We get:
$$29 = 6 \times 4 + 5$$
Therefore:
$$r = 5$$
Hence, the correct answer is:
$$5$$
Q3. Which of the following numbers is irrational?
A) $$\frac{11}{13}$$
B) $$0.625$$
C) $$\sqrt{11}$$
D) $$0.4444\ldots$$
Answer:
$$\sqrt{11}$$
Useful Formula for this Question:
An irrational number cannot be written in the form:
$$\frac{p}{q}$$
where $$q \ne 0$$.
Solution:
$$\sqrt{11}$$ cannot be expressed as:
$$\frac{p}{q}$$
Its decimal expansion is non-terminating and non-recurring.
The remaining options are rational numbers.
Therefore, the correct answer is:
$$\sqrt{11}$$
Q4. The LCM of $$18$$ and $$24$$ is:
A) $$48$$
B) $$60$$
C) $$72$$
D) $$96$$
Answer:
$$72$$
Useful Formula for this Question:
LCM is obtained by taking the highest powers of all prime factors.
Solution:
Prime factorization:
$$18 = 2 \times 3^2$$
$$24 = 2^3 \times 3$$
Taking highest powers:
$$LCM = 2^3 \times 3^2$$
$$= 8 \times 9$$
$$= 72$$
Therefore, the correct answer is:
$$72$$
Q5. Which of the following fractions has a terminating decimal expansion?
A) $$\frac{7}{24}$$
B) $$\frac{11}{25}$$
C) $$\frac{13}{27}$$
D) $$\frac{5}{18}$$
Answer:
$$\frac{11}{25}$$
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:
$$\frac{11}{25}$$
Denominator:
$$25 = 5^2$$
Only factor $$5$$ is present.
Therefore, the decimal expansion is terminating.
$$\frac{11}{25}=0.44$$
Hence, the correct answer is:
$$\frac{11}{25}$$
Important Formulas:
Euclid’s Division Lemma:
$$a = bq + r$$
where
$$0 \le r < b$$
Relationship between HCF and LCM:
$$HCF \times LCM = Product\ of\ two\ numbers$$
A rational number can be written as:
$$\frac{p}{q}$$
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 remainder in Euclid’s Division Lemma?
Answer:
The remainder is represented by:
$$r$$
and satisfies:
$$0 \le r < b$$
Q. What is an irrational number?
Answer:
An irrational number cannot be expressed in the form:
$$\frac{p}{q}$$
Q. How do we find LCM using prime factorization?
Answer:
Take the highest powers of all prime factors present in the given numbers.
Common Mistakes Students Make
❌ Forgetting repeated prime factors while finding HCF and LCM.
❌ Assuming every square root is irrational.
❌ Taking the divisor itself as the remainder.
❌ Ignoring denominator factorization while checking decimal expansion.
Chapter 1 Quick Revision
✔ HCF is the greatest common factor.
✔ LCM is obtained using highest powers of prime factors.
✔ Euclid’s Division Lemma is:
$$a = bq + r$$
✔ Irrational numbers cannot be written as:
$$\frac{p}{q}$$
✔ A terminating decimal expansion requires denominator factors only:
$$2$$ and/or $$5$$
Conclusion:
These Class 10 Maths Chapter 1 Real Numbers MCQs with answers and detailed solutions help students strengthen concepts related to HCF, LCM, Euclid’s Division Lemma, irrational numbers, and decimal expansions. Regular practice of such questions can improve accuracy, confidence, and board exam performance.
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