Real Numbers (Class 10 Mathematics)
Real numbers include all rational and irrational numbers, and understanding their properties is essential for higher mathematics. This chapter covers the E
TL;DR: Real numbers include all rational and irrational numbers, and understanding their properties is essential for higher mathematics. This chapter covers…
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Real numbers include all rational and irrational numbers, and understanding their properties is essential for higher mathematics. This chapter covers the E
Real Numbers MCQs with Answers & Explanations
Q1. The HCF of two numbers is 12 and their LCM is 180. If one number is 36, what is the other?
- 72
- 96
- 60 ✓ (correct)
- 84
Explanation: Using HCF x LCM = Product of two numbers: 12 x 180 = 36 x other, so other = 2160/36 = 60.
Q2. Which of the following is an irrational number?
- 2.4
- 0.375
- 0.5
- sqrt(2) ✓ (correct)
Explanation: sqrt(2) is irrational because it cannot be expressed as a ratio of two integers, unlike the others which are rational.
Q3. The decimal expansion of a rational number is:
- Terminating or non-terminating repeating ✓ (correct)
- Always terminating
- Non-terminating and non-repeating
- Always non-terminating
Explanation: Rational numbers have either terminating decimals or non-terminating repeating decimals.
Q4. If a = 2^3 x 3^2 x 5 and b = 2^2 x 3 x 5^2, then LCM(a,b) is:
- 2^2 x 3 x 5
- 2^3 x 3^2 x 5^2 ✓ (correct)
- 2^3 x 3^2 x 5
- 2^2 x 3^2 x 5^2
Explanation: LCM is found by taking the highest power of each prime factor: 2^3 x 3^2 x 5^2.
Q5. Euclid's division lemma states that for integers a and b, a = bq + r where:
- r >= b
- r < 0
- 0 <= r < b ✓ (correct)
- r >= 0
Explanation: In Euclid's division lemma, the remainder r must satisfy 0 <= r < b for any integer division.
Q6. The sum of two rational numbers is always:
- Irrational
- Sometimes rational
- Neither rational nor irrational
- Rational ✓ (correct)
Explanation: The sum of two rational numbers is always rational because the set of rational numbers is closed under addition.
Q7. sqrt(2) + sqrt(3) is:
- Irrational ✓ (correct)
- Neither
- Rational
- Can be rational or irrational
Explanation: The sum of two irrational numbers can be irrational. In this case, sqrt(2) + sqrt(3) is irrational.
Q8. If p and q are primes and p divides q, then:
- p > q
- p = q ✓ (correct)
- p < q
- p divides q
Explanation: If a prime p divides another prime q, then p must equal q because primes have no other divisors.
Q9. The HCF of 36 and 48 is:
- 6
- 24
- 12 ✓ (correct)
- 36
Explanation: 36 = 2² x 3² and 48 = 2⁴ x 3. The HCF takes the lowest power of each common prime: 2² x 3 = 12.
Q10. The product of a non-zero rational and an irrational number is:
- Rational
- Neither
- Zero
- Irrational ✓ (correct)
Explanation: The product of a non-zero rational and an irrational number is always irrational.
Frequently Asked Questions
What is the difference between rational and irrational numbers?
Rational numbers can be expressed as a ratio of two integers, while irrational numbers cannot be expressed in this form.
How is Euclid's algorithm used to find the HCF of two numbers?
Euclid's algorithm repeatedly applies division lemma until the remainder becomes zero, with the last non-zero remainder being the HCF.
More Class 10 Mathematics MCQs
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