Class 10 Maths Chapter 1 Real Numbers MCQs Quiz

Are you feeling anxious about Chapter 1, ‘Real Numbers,’ in your Class 10 Maths curriculum? Do concepts like Euclid’s Division Lemma and the Fundamental Theorem of Arithmetic seem complex and overwhelming? Worry no more!

We bring you the perfect solution: our Class 10 Maths Chapter 1 Real Numbers MCQs Quiz. This free online test is designed to help you build a strong foundation in this crucial chapter and understand the exam pattern. So, let’s start your board exam preparation with this powerful Real Numbers quiz and build your confidence from day one!

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Real Numbers MCQs Quiz

Click the Start button to begin the quiz and attempt all questions within the given time. Read each MCQ carefully, select the correct option, and move forward to complete the quiz. Your score will appear at the end to help you check your preparation level..

Q1: Find the HCF of 135 and 225 by the prime factorization method.

A: 15
B: 25
C: 35
D: 45

Explanation: By prime factorization method:
225 = 3 × 3 × 5 × 5 = 3² × 5²
135 = 3 × 3 × 3 × 5 = 3³ × 5
Therefore, HCF = 3² × 5 = 9 × 5 = 45.

Q2: According to the Fundamental Theorem of Arithmetic, every composite number can be represented as:

A: a sum of prime numbers
B: a product of even numbers
C: a product of prime numbers
D: a sum of odd numbers

Explanation: The Fundamental Theorem of Arithmetic states that every composite number greater than 1 can be expressed uniquely as a product of prime numbers, apart from the order of the factors.
Example: 30 = 2 × 3 × 5

Q3: The number \( \sqrt{5} \) is:

A: a rational number
B: an irrational number
C: an integer
D: a natural number

Explanation: \( \sqrt{5} \) is an irrational number because it cannot be expressed in the form \( \frac{p}{q} \), where p and q are integers and q ≠ 0.
It can be proved that 5 is a prime number and its square root is irrational.

Q4: Find the HCF of 96 and 404 by the prime factorization method.

A: 4
B: 2
C: 8
D: 12

Explanation: Prime factorization of 96 and 404:
96 = 2 × 2 × 2 × 2 × 2 × 3 = 2⁵ × 3
404 = 2 × 2 × 101 = 2² × 101
The common prime factor is 2 with the smallest power 2².
Therefore, HCF = 2² = 4.

Q5: Which of the following numbers is irrational?

A: \( \frac{22}{7} \)
B: 3.1416
C: \( \pi \)
D: \( \frac{3.\overline{142857}}{1} \)

Explanation: \( \pi \) (pi) is an irrational number because it cannot be expressed as a ratio of two integers.
Option A is a rational number (numerator and denominator are integers).
Option B is a terminating decimal, hence rational.
Option D is a recurring decimal, hence rational.

Q6: Find the HCF of 6, 72, and 120 by the prime factorization method.

A: 8
B: 6
C: 12
D: 24

Explanation: Prime factorization:
6 = 2 × 3
72 = 2³ × 3²
120 = 2³ × 3 × 5
Common prime factors are 2 and 3.
HCF = Product of the common factors with the smallest powers = 2¹ × 3¹ = 6.

Q7: The rational form of the number \( 0.\overline{35} \) is:

A: \( \frac{35}{99} \)
B: \( \frac{35}{90} \)
C: \( \frac{35}{100} \)
D: \( \frac{7}{20} \)

Explanation: Let \( x = 0.\overline{35} = 0.353535… \)
Then, \( 100x = 35.353535… \)
Subtracting: \( 100x – x = 35.3535… – 0.3535… \)
\( 99x = 35 \)
\( x = \frac{35}{99} \)
Thus, \( 0.\overline{35} = \frac{35}{99} \).

Q8: Which of the following statements is true?

A: The product of two irrational numbers is always irrational
B: The sum of two irrational numbers is always irrational
C: Every integer is an irrational number
D: Every integer is a rational number

Explanation: Options A and B are false: Example: \( \sqrt{2} \times \sqrt{2} = 2\) (rational), \( (2+\sqrt{3}) + (2-\sqrt{3}) = 4\) (rational).
Option C is false.
Option D is true: Any integer \( m \) can be written as \( \frac{m}{1} \), so it is a rational number.

Q9: If \( p \) is a prime number and \( p \) divides \( a^2 \), then \( p \) will divide:

A: 3a
B: 2a
C: a
D: \( a^3 \)

Explanation: This is a theorem: If \( p \) is a prime number and \( p \) divides the product \( a^2 \), then \( p \) must divide \( a \). This follows from the Fundamental Theorem of Arithmetic.

Q10: The HCF of 96 and 404 is 4. What is their LCM?

A: 9600
B: 9696
C: 10000
D: 10100

Explanation: Formula: HCF × LCM = Product of the two numbers
Given: HCF = 4, numbers are 96 and 404
Product = 96 × 404 = 38784
Therefore, LCM = \( \frac{96 \times 404}{4} = \frac{38784}{4} = 9696 \).

Q11: Find the HCF of 306 and 657 by the prime factorization method.

A: 3
B: 9
C: 18
D: 27

Explanation: Prime factorization of the two numbers:
306 = 2 × 3 × 3 × 17 = 2 × 3² × 17
657 = 3 × 3 × 73 = 3² × 73
The common prime factor is 3 with the smallest power 3².
Therefore, HCF = 3² = 9.

Q12: The product of \( \sqrt{2} \) and \( \sqrt{3} \) is:

A: \( \sqrt{5} \)
B: \( 3\sqrt{2} \)
C: \( 2\sqrt{3} \)
D: \( \sqrt{6} \)

Explanation: \( \sqrt{2} \times \sqrt{3} = \sqrt{2 \times 3} = \sqrt{6} \).
Since 6 is not a perfect square, \( \sqrt{6} \) is an irrational number.

Q13: The smallest number divisible by 7, 8, and 9 is:

A: 126
B: 252
C: 504
D: 756

Explanation: The smallest number divisible by 7, 8, and 9 is the LCM of these numbers.
7 = 7 (prime)
8 = 2³
9 = 3²
LCM = 2³ × 3² × 7 = 8 × 9 × 7 = 504.
Thus, the smallest number is 504.

Q14: The number \( 1.\overline{27} \) is:

A: an integer
B: a rational number
C: an irrational number
D: a natural number

Explanation: Any recurring decimal number can be expressed in the form \( \frac{p}{q} \), where p and q are integers and q ≠ 0.
Therefore, \( 1.\overline{27} \) is a rational number.
It can be simplified to \( \frac{126}{99} \) or \( \frac{14}{11} \).

Q15: According to the Fundamental Theorem of Arithmetic, the prime factorization of 32760 is:

A: \( 2^3 \times 3^2 \times 5 \times 7 \times 13 \)
B: \( 2^3 \times 3^2 \times 5 \times 7 \times 11 \)
C: \( 2^4 \times 3^2 \times 5 \times 7 \times 13 \)
D: \( 2^3 \times 3^3 \times 5 \times 7 \times 13 \)

Explanation: Prime factorization of 32760:
32760 ÷ 2 = 16380
16380 ÷ 2 = 8190
8190 ÷ 2 = 4095
4095 ÷ 3 = 1365
1365 ÷ 3 = 455
455 ÷ 5 = 91
91 ÷ 7 = 13
13 ÷ 13 = 1
Thus, 32760 = 2³ × 3² × 5 × 7 × 13.

Quiz Results

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Class 10 Maths Chapter 1 Real Numbers MCQs Quiz

Chapter 1, ‘Real Numbers,’ delves deep into the world of numbers, building on your previous knowledge. You will learn how to use Fundamental Theorem of Arithmetic to find the Highest Common Factor (HCF) of any two positive integers. The chapter introduces the Fundamental Theorem of Arithmetic, which states that every composite number can be uniquely expressed as a product of prime numbers. You’ll also learn to apply this theorem to find the HCF and LCM of numbers. Furthermore, the chapter explores the difference between rational and irrational numbers and their decimal expansions, which are vital concepts for the CBSE Class 10 Maths Quiz and your board exams.

Conclusion

The chapter on ‘Real Numbers’ is foundational not only for scoring well in your board exams but also for understanding higher-level mathematics. Our Class 10 Maths Chapter 1 Real Numbers MCQs Quiz is crafted to help you strengthen this foundation. This quiz provides ample practice on key theorems like Euclid’s Division Lemma and the Fundamental Theorem of Arithmetic, helping you master concepts like HCF and LCM by Prime Factorization. By solving these objective questions, you can assess your preparation, identify weak areas, and gain the confidence to score full marks in this chapter. Try this Free Online Maths Quiz for Class 10 now!

FAQs on Real Numbers MCQs

1. Question: How many marks from the chapter Real Numbers are asked in the board exam?
Answer: Typically, questions totaling 6-7 marks from this chapter appear in the CBSE board exam, including MCQs, short answer, and long answer questions.

2. Question: What are the most important theorems in this chapter?
Answer: The most important theorems are Euclid’s Division Lemma and the Fundamental Theorem of Arithmetic.

3. Question: Is this Real Numbers quiz based on the NCERT syllabus?
Answer: Yes, our Real Numbers Class 10 MCQ quiz is completely based on the NCERT curriculum and the latest CBSE syllabus.

4. Question: What is Euclid’s Division Lemma?
Answer: Euclid’s Division Lemma states that for any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b.

5. Question: What is the Fundamental Theorem of Arithmetic?
Answer: It states that every composite number greater than 1 can be expressed as a unique product of prime numbers, irrespective of the order of the prime factors.

6. Question: Is this online quiz free?
Answer: Yes, this Free Online Maths Quiz for Class 10 is completely free for you to practice anytime, anywhere.

7. Question: How should I prepare this chapter for the board exam?
Answer: First, understand all the theorems and their proofs. Then, solve the examples and exercise questions from the NCERT book. Finally, test your knowledge with our Real Numbers Online Test.

8. Question: What is the difference between a rational and an irrational number?
Answer: A rational number can be expressed in the form p/q, where q is not zero. An irrational number cannot be expressed in such a form.

9. Question: How do you find HCF and LCM using the prime factorization method?
Answer: First, find the prime factorization of the numbers. The HCF is the product of the smallest powers of common prime factors, and the LCM is the product of the greatest powers of all prime factors.

10. Question: What is the benefit of taking this online quiz?
Answer: This quiz helps you improve time management, test the accuracy of your concepts, and understand the exam pattern, which boosts your confidence for better performance.

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