CBSE Class 10 Mathematics Real Numbers MCQs Set 24

Mathematics Objective Questions and Answers: Chapter 01 Real Numbers

Review structured MCQ sets for Class 10 Mathematics Chapter 01 Real Numbers. Built according to official CBSE guidelines, these downloadable questions support daily revision and core concept reinforcement.

Download Chapter 01 Real Numbers MCQs with Answers

Navigate directly to the 50 objective questions for Chapter 01 Real Numbers using the digital viewer below. Each practice set includes verified answer keys, allowing students to instantly cross-check their work and identify areas requiring further revision.

Multiple Choice Questions

Question 1: Using Euclid's division algorithm, the HCF of 56 and 814 is:
(a) 2
(b) 4
(c) 14
(d) 26
Show Answer & Explanation

Answer: (a) 2

Explanation:
1. 814 = 56 x 14 + 30; 56 = 30 x 1 + 26; 30 = 26 x 1 + 4; 26 = 4 x 6 + 2; 4 = 2 x 2 + 0. The last non-zero remainder is 2.

Teacher's Note:
1. Euclid's division algorithm: divide the larger number by the smaller, then divide the divisor by the remainder, and so on. The last non-zero remainder is the HCF.

Question 2: Using Euclid's division algorithm, the HCF of 6265 and 76254 is:
(a) 895
(b) 1074
(c) 358
(d) 179
Show Answer & Explanation

Answer: (d) 179

Explanation:
1. 76254 = 6265 x 12 + 1074; 6265 = 1074 x 5 + 895; 1074 = 895 x 1 + 179; 895 = 179 x 5 + 0. The HCF is 179.

Teacher's Note:
1. Euclid's division algorithm: divide the larger number by the smaller, then divide the divisor by the remainder, and so on. The last non-zero remainder is the HCF.

Question 3: The HCF and LCM of 426 and 576 are respectively:
(a) 6 and 20448
(b) 12 and 40896
(c) 6 and 40896
(d) 3 and 81792
Show Answer & Explanation

Answer: (c) 6 and 40896

Explanation:
1. 426 = 2 x 3 x 71 and 576 = 2⁶ x 3². HCF = 2 x 3 = 6. LCM = 2⁶ x 3² x 71 = 40896. (Check: 6 x 40896 = 426 x 576.)

Teacher's Note:
1. Every composite number is a product of primes in a unique way. HCF = product of the smallest powers of the common primes and LCM = product of the greatest powers of all primes present.

Question 4: The HCF and LCM of 625, 1125 and 2125 are respectively:
(a) 25 and 95625
(b) 125 and 95625
(c) 125 and 19125
(d) 625 and 95625
Show Answer & Explanation

Answer: (b) 125 and 95625

Explanation:
1. 625 = 5⁴, 1125 = 3² x 5³, 2125 = 5³ x 17. HCF = 5³ = 125. LCM = 5⁴ x 3² x 17 = 95625.

Teacher's Note:
1. Every composite number is a product of primes in a unique way. HCF = product of the smallest powers of the common primes and LCM = product of the greatest powers of all primes present.

Question 5: Which of the following numbers is irrational?
(a) \( \frac{22}{7} \)
(b) \( \sqrt3 \)
(c) \( 0.25 \)
(d) \( \sqrt{16} \)
Show Answer & Explanation

Answer: (b) \( \sqrt3 \)

Explanation:
1. The square root of a prime number such as 3 is irrational. 22/7 is a fraction, 0.25 = 1/4 is rational and √16 = 4 is an integer.

Teacher's Note:
1. A rational number p/q (in lowest terms) has a terminating decimal expansion exactly when q = 2^m x 5^n.
2. The square root of a prime number is irrational.

Question 6: To prove that √5 is irrational by contradiction, from 5b² = a² (a, b co-prime) the fact used to conclude that 5 divides a is:
(a) if a prime divides a² then it divides a
(b) every prime is odd
(c) a² is always a multiple of 5
(d) 5 divides b only
Show Answer & Explanation

Answer: (a) if a prime divides a² then it divides a

Explanation:
1. 5 divides a², and since 5 is prime it divides a. Writing a = 5c gives b² = 5c², so 5 divides b as well, contradicting that a and b are co-prime.

Teacher's Note:
1. A rational number p/q (in lowest terms) has a terminating decimal expansion exactly when q = 2^m x 5^n.
2. The square root of a prime number is irrational.

Question 7: Can 7^n end with the digit zero for any natural number n?
(a) Yes, for n = 5
(b) Yes, for every even n
(c) Yes, for every odd n
(d) No, because 5 is not a prime factor of 7^n
Show Answer & Explanation

Answer: (d) No, because 5 is not a prime factor of 7^n

Explanation:
1. A number ending in 0 has both 2 and 5 as prime factors. The only prime factor of 7^n is 7, so it never ends in 0.

Teacher's Note:
1. Every composite number is a product of primes in a unique way. HCF = product of the smallest powers of the common primes and LCM = product of the greatest powers of all primes present.

Question 8: Without long division, the rational number 77/210 has a decimal expansion that is:
(a) terminating
(b) non-terminating and non-repeating
(c) non-terminating and repeating
(d) terminating after exactly 3 places
Show Answer & Explanation

Answer: (c) non-terminating and repeating

Explanation:
1. 77/210 = 11/30 in lowest terms. 30 = 2 x 3 x 5 has the prime factor 3, which is not 2 or 5. So the expansion is non-terminating and repeating.

Teacher's Note:
1. A rational number p/q (in lowest terms) has a terminating decimal expansion exactly when q = 2^m x 5^n.
2. The square root of a prime number is irrational.

Question 9: Without long division, the rational number 15/1600 has a decimal expansion that is:
(a) non-terminating and repeating
(b) non-terminating and non-repeating
(c) terminating
(d) recurring with period 2
Show Answer & Explanation

Answer: (c) terminating

Explanation:
1. 15/1600 = 3/320 and 320 = 2⁶ x 5. The denominator has only the prime factors 2 and 5, so the expansion terminates (0.009375).

Teacher's Note:
1. A rational number p/q (in lowest terms) has a terminating decimal expansion exactly when q = 2^m x 5^n.
2. The square root of a prime number is irrational.

Question 10: An army contingent of 616 members is to march behind an army band of 32 members, with both groups marching in the same number of columns. The maximum number of columns is:
(a) 4
(b) 8
(c) 16
(d) 24
Show Answer & Explanation

Answer: (b) 8

Explanation:
1. The number of columns must divide both 616 and 32. HCF(616, 32) = 8 (616 = 32 x 19 + 8, 32 = 8 x 4). So the maximum number of columns is 8.

Teacher's Note:
1. Euclid's division algorithm: divide the larger number by the smaller, then divide the divisor by the remainder, and so on. The last non-zero remainder is the HCF.

Question 11: Sonia takes 18 minutes and Ravi takes 12 minutes to drive one round of a circular path. Starting together at the same point in the same direction, they meet again at the starting point after:
(a) 36 minutes
(b) 6 minutes
(c) 30 minutes
(d) 72 minutes
Show Answer & Explanation

Answer: (a) 36 minutes

Explanation:
1. They meet when both have completed whole rounds, at the LCM of 18 and 12, which is 36 minutes.

Teacher's Note:
1. Every composite number is a product of primes in a unique way. HCF = product of the smallest powers of the common primes and LCM = product of the greatest powers of all primes present.

Question 12: A rational number lying between √2 and √3 is:
(a) \( \frac54 \)
(b) \( \frac74 \)
(c) \( 2 \)
(d) \( \frac32 \)
Show Answer & Explanation

Answer: (d) \( \frac32 \)

Explanation:
1. √2 = 1.414... and √3 = 1.732.... 3/2 = 1.5 lies between them; 5/4 = 1.25 is below √2, while 7/4 = 1.75 and 2 are above √3.

Teacher's Note:
1. Compare decimal values to see where a number lies.

Question 13: The square of any positive integer is of the form (m is an integer):
(a) \( 3m\ \text{or}\ 3m+2 \)
(b) \( 3m+1\ \text{or}\ 3m+2 \)
(c) \( 3m,\ 3m+1\ \text{or}\ 3m+2 \)
(d) \( 3m\ \text{or}\ 3m+1 \)
Show Answer & Explanation

Answer: (d) \( 3m\ \text{or}\ 3m+1 \)

Explanation:
1. By Euclid's division lemma any integer is 3q, 3q + 1 or 3q + 2. (3q)² = 3(3q²); (3q + 1)² = 3(3q² + 2q) + 1; (3q + 2)² = 3(3q² + 4q + 1) + 1. So the square is 3m or 3m + 1.

Teacher's Note:
1. Euclid's division lemma: a = bq + r with 0 ≤ r < b.

Free study material for Mathematics

Practice MCQs for Class 10 Mathematics Chapter 01 Real Numbers

Chapter MCQs with Answers for Class 10 Mathematics

Test your conceptual understanding of Chapter 01 Real Numbers with these targeted multiple-choice questions. Designed in alignment with the latest CBSE curriculum for Class 10 Mathematics, these problem sets build accuracy and prepare students for objective exams.

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Each question includes structured solution keys mapped directly to standard CBSE textbooks, helping students evaluate their reasoning and correct mistakes early in their revision.

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