Real Numbers – Important MCQs with Solutions
Section 1.1: Fundamental Theorem of Arithmetic
1. Which of the following is the prime factorization of 140?
a) 2 × 5 × 7
b) 2² × 5 × 7
c) 2 × 5² × 7
d) 2² × 5² × 7
Answer: b) 2² × 5 × 7
Explanation: 140 = 2 × 2 × 5 × 7 = 2² × 5 × 7
2. The Fundamental Theorem of Arithmetic states that:
a) Every natural number can be expressed as a product of primes
b) The prime factorization of a number is unique
c) Both a) and b)
d) None of the above
Answer: c) Both a) and b)
Explanation: The theorem states both that every composite number can be expressed as a product of primes and that this factorization is unique.
3. Which of the following numbers ends with the digit 0?
a) 4ⁿ
b) 6ⁿ
c) 2ⁿ × 5ⁿ
d) 3ⁿ × 5ⁿ
Answer: c) 2ⁿ × 5ⁿ
Explanation: A number ends with 0 if it has both 2 and 5 as prime factors.
4. The HCF of 96 and 404 is:
a) 4
b) 6
c) 12
d) 2
Answer: a) 4
Explanation: 96 = 2⁵ × 3 and 404 = 2² × 101. HCF is product of smallest powers of common primes.
5. The LCM of 6, 72, and 120 is:
a) 360
b) 720
c) 180
d) 240
Answer: a) 360
Explanation: LCM is product of greatest powers of all primes present in the numbers.
Section 1.2: Irrational Numbers
6. Which of the following is irrational?
a) √4
b) √5
c) √9
d) √16
Answer: b) √5
Explanation: √5 cannot be expressed as a fraction of integers.
7. The sum of a rational and an irrational number is:
a) Rational
b) Irrational
c) Can be either rational or irrational
d) None of the above
Answer: b) Irrational
Explanation: The sum is always irrational as shown in proofs.
8. The product of a non-zero rational and an irrational number is:
a) Rational
b) Irrational
c) Can be either rational or irrational
d) None of the above
Answer: b) Irrational
Explanation: The product is always irrational as demonstrated in proofs.
9. Which of the following is a correct proof that √3 is irrational?
a) It cannot be written as a fraction
b) Its decimal expansion is non-terminating and non-repeating
c) Proof by contradiction assuming it is rational
d) All of the above
Answer: d) All of the above
Explanation: All these are valid approaches to prove irrationality.
10. The number 3 + 2√5 is:
a) Rational
b) Irrational
c) An integer
d) None of the above
Answer: b) Irrational
Explanation: Sum of rational (3) and irrational (2√5) is irrational.