Real Numbers (Class 10 Mathematics)
Real numbers include all rational and irrational numbers. Understanding properties like divisibility, prime factorization, and the Euclidean algorithm is f
TL;DR: Real numbers include all rational and irrational numbers. Understanding properties like divisibility, prime factorization, and the Euclidean algorithm…
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Real numbers include all rational and irrational numbers. Understanding properties like divisibility, prime factorization, and the Euclidean algorithm is f
Real Numbers MCQs with Answers & Explanations
Q1. Which of the following is an irrational number?
- √2 ✓ (correct)
- 0.5
- 0.333...
- 22/7
Explanation: √2 is irrational because it cannot be expressed as a ratio of two integers. Its decimal expansion is non-terminating and non-repeating.
Q2. The Euclidean algorithm is used to find:
- Prime factors
- Greatest Common Divisor (GCD) ✓ (correct)
- Least Common Multiple (LCM)
- Square roots
Explanation: The Euclidean algorithm repeatedly applies division to find the GCD of two numbers efficiently.
Q3. If HCF(a, b) = 12 and LCM(a, b) = 72, then a × b equals:
- 60
- 720
- 864 ✓ (correct)
- 144
Explanation: For any two numbers a and b: a × b = HCF(a, b) × LCM(a, b) = 12 × 72 = 864.
Q4. Which of the following is a prime number?
- 87
- 93
- 91
- 97 ✓ (correct)
Explanation: 97 is prime. 91 = 7×13, 87 = 3×29, and 93 = 3×31 are composite numbers.
Q5. The decimal expansion of a rational number is always:
- Terminating or non-terminating repeating ✓ (correct)
- Non-terminating
- Non-terminating and non-repeating
- Terminating
Explanation: Rational numbers have either terminating decimals (like 1/4 = 0.25) or non-terminating repeating decimals (like 1/3 = 0.333...).
Q6. By the Fundamental Theorem of Arithmetic, every composite number can be uniquely expressed as:
- Difference of squares
- Product of primes ✓ (correct)
- Sum of two squares
- Sum of primes
Explanation: Every composite number has a unique prime factorization (up to order), which is the basis of the Fundamental Theorem of Arithmetic.
Q7. The HCF of 15 and 25 is:
- 25
- 15
- 5 ✓ (correct)
- 75
Explanation: 15 = 3 × 5 and 25 = 5 × 5. The common factor is 5, so HCF = 5.
Q8. If p/q is a rational number in lowest terms, then q must be:
- A multiple of p
- A perfect square
- Not equal to 1
- Coprime to p ✓ (correct)
Explanation: For a fraction p/q to be in lowest terms, HCF(p, q) = 1, meaning p and q are coprime (share no common factors).
Q9. The LCM of 12 and 18 is:
- 36 ✓ (correct)
- 6
- 72
- 216
Explanation: 12 = 2² × 3 and 18 = 2 × 3². LCM = 2² × 3² = 4 × 9 = 36.
Q10. Which statement is true about irrational numbers?
- They cannot be negative
- Their decimal expansion never terminates or repeats ✓ (correct)
- They can be expressed as p/q
- They are less common than rational numbers
Explanation: Irrational numbers have non-terminating, non-repeating decimal expansions and cannot be expressed as simple fractions.
Q11. 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.
Q12. The decimal expansion of a rational number is:
- Always non-terminating
- Non-terminating and non-repeating
- Always terminating
- Terminating or non-terminating repeating ✓ (correct)
Explanation: Rational numbers have either terminating decimals or non-terminating repeating decimals.
Q13. If a = 2^3 x 3^2 x 5 and b = 2^2 x 3 x 5^2, then LCM(a,b) is:
- 2^3 x 3^2 x 5^2 ✓ (correct)
- 2^3 x 3^2 x 5
- 2^2 x 3 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.
Q14. Euclid's division lemma states that for integers a and b, a = bq + r where:
- r < 0
- 0 <= r < b ✓ (correct)
- r >= b
- r >= 0
Explanation: In Euclid's division lemma, the remainder r must satisfy 0 <= r < b for any integer division.
Q15. The sum of two rational numbers is always:
- Sometimes rational
- Irrational
- Rational ✓ (correct)
- Neither rational nor irrational
Explanation: The sum of two rational numbers is always rational because the set of rational numbers is closed under addition.
Q16. sqrt(2) + sqrt(3) is:
- Rational
- Can be rational or irrational
- Neither
- Irrational ✓ (correct)
Explanation: The sum of two irrational numbers can be irrational. In this case, sqrt(2) + sqrt(3) is irrational.
Q17. If p and q are primes and p divides q, then:
- p = q ✓ (correct)
- p < q
- p divides q
- p > q
Explanation: If a prime p divides another prime q, then p must equal q because primes have no other divisors.
Q18. The HCF of 36 and 48 is:
- 6
- 12 ✓ (correct)
- 24
- 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.
Q19. The product of a non-zero rational and an irrational number is:
- Zero
- Rational
- Irrational ✓ (correct)
- Neither
Explanation: The product of a non-zero rational and an irrational number is always irrational.
Frequently Asked Questions
What is the relationship between HCF and LCM?
For any two positive integers a and b, their product equals the product of their HCF and LCM: a × b = HCF(a, b) × LCM(a, b). This relationship helps solve many problems involving divisibility.
How do you prove that √2 is irrational?
Assume √2 = p/q in lowest terms. Then 2q² = p², meaning p is even (say p = 2k). Substituting: 2q² = 4k², so q² = 2k², meaning q is also even. This contradicts our assumption that p/q is in lowest terms, so √2 is irrational.
More Class 10 Mathematics MCQs
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