CBSE Class 10 Mathematics Real Numbers MCQs Set 31

Mathematics Objective Questions and Answers: Chapter 01 Real Numbers

Access targeted multiple-choice questions for Chapter 01 Real Numbers designed to align with the latest CBSE academic syllabus for Class 10 Mathematics. These objective practice sets help students evaluate their conceptual understanding and improve exam readiness.

Download Chapter 01 Real Numbers MCQs with Answers

View or download the dedicated Chapter 01 Real Numbers MCQ resource below. Practicing these 50 objective questions regularly builds familiarity with standard exam patterns and helps secure higher marks in final Mathematics evaluations.

Multiple Choice Questions

Question 1: If a, b ≠ 1, ab > 0, a ≠ b and log_b a = log_a b, then ab =
(a) 1/2
(b) 1
(c) 2
(d) 10
Show Answer & Explanation

Answer: (b) 1

Explanation:
1. Let x = log_b a. Then x = 1/x, so x = ±1. Since a ≠ b, x = −1, so a = 1/b and ab = 1.

Teacher's Note:
1. log_a b is the reciprocal of log_b a.

Question 2: If 2009 = pᵃ·qᵇ, where p and q are prime numbers, then p + q =
(a) 3
(b) 48
(c) 51
(d) 2009
Show Answer & Explanation

Answer: (b) 48

Explanation:
1. 2009 = 7² × 41, so p + q = 7 + 41 = 48.

Teacher's Note:
1. Factorise 2009.

Question 3: If HCF(p, q) = 12 and p × q = 1800, then LCM(p, q) is
(a) 3600
(b) 900
(c) 150
(d) 90
Show Answer & Explanation

Answer: (c) 150

Explanation:
1. LCM = 1800/12 = 150.

Teacher's Note:
1. HCF × LCM = product.

Question 4: Given √600 = 24.49, the value of (√2 + √3)/(√3 − √2) is
(a) 9.978
(b) 8.989
(c) 9.989
(d) 9.898
Show Answer & Explanation

Answer: (d) 9.898

Explanation:
1. Multiply by (√3 + √2)/(√3 + √2): the value = (√2 + √3)² = 5 + 2√6 = 5 + √24.
2. √6 = 24.49/10 = 2.449, so the value = 5 + 4.898 = 9.898.

Teacher's Note:
1. Rationalise the denominator.

Question 5: If x < 0 and log₇(x² − 5x − 65) = 0, then x is
(a) −13
(b) −11
(c) −6
(d) −5
Show Answer & Explanation

Answer: (c) −6

Explanation:
1. x² − 5x − 65 = 1, so x² − 5x − 66 = 0, giving x = 11 or −6. Since x < 0, x = −6.

Teacher's Note:
1. Remember the condition on x.

Question 6: If the eight-digit number 2575d568 is divisible by 54 and 87, the value of the digit d is
(a) 4
(b) 7
(c) 0
(d) 8
Show Answer & Explanation

Answer: (b) 7

Explanation:
1. The number must be divisible by lcm(54, 87) = 1566. Checking d = 0 to 9, only d = 7 gives 25757568 = 1566 × 16448.

Teacher's Note:
1. Test each digit against the LCM.

Question 7: The product of three consecutive natural numbers is 124850054994. Their average is
(a) 4993
(b) 4994
(c) 4997
(d) 4998
Show Answer & Explanation

Answer: (d) 4998

Explanation:
1. (n − 1)n(n + 1) = n³ − n ≈ 1.2485 × 10¹¹, so n ≈ 4998.
2. Check: 4997 × 4998 × 4999 = 124850054994, so the average is 4998.

Teacher's Note:
1. The middle number is the average.

Question 8: In a certain number system 363 + 1056 = 1452. The value of (654 − 456) in the same system is
(a) 156
(b) 165
(c) 178
(d) 198
Show Answer & Explanation

Answer: (b) 165

Explanation:
1. Trying the base from the digits, base 7 works: 363₇ = 192, 1056₇ = 384 and 1452₇ = 576 = 192 + 384.
2. 654₇ − 456₇ = 333 − 237 = 96 = 165₇.
3. The answer is 165.

Teacher's Note:
1. Find the base from the units digits.

Question 9: Let T be the number of 4-digit integers, each ending in 3 and each divisible by 11. Then
(a) 20 ≤ T ≤ 79
(b) 80 ≤ T ≤ 89
(c) 90 ≤ T ≤ 99
(d) T ≥ 100
Show Answer & Explanation

Answer: (b) 80 ≤ T ≤ 89

Explanation:
1. Numbers of the form 10m + 3 that are multiples of 11 repeat every 110. From 1000 to 9999 the count is 82.

Teacher's Note:
1. Such numbers form an arithmetic progression with difference 110.

Question 10: The number plate of a vehicle has 4 digits. The first digit is the square of the second, the third digit is thrice the second and the fourth digit is twice the second. The sum of all 4 digits is thrice the first. The number is
(a) 1132
(b) 4264
(c) 1642
(d) 9396
Show Answer & Explanation

Answer: (d) 9396

Explanation:
1. With second digit b: b² + b + 3b + 2b = 3b², so b = 3.
2. The number is 9, 3, 9, 6.

Teacher's Note:
1. Express all digits through the second.

Question 11: A number x is rational if there exist integers p and q such that x = p/q. This definition of rational numbers requires that
(a) both p and q can be zero
(b) both p and q should not be zero
(c) q can be zero but not p
(d) p can be zero but not q
Show Answer & Explanation

Answer: (d) p can be zero but not q

Explanation:
1. The denominator q cannot be zero, while the numerator p may be zero (0 is rational).

Teacher's Note:
1. Zero is rational but division by zero is not defined.

Question 12: The least positive integer n such that 2 divides n, 3 divides n + 1, 4 divides n + 2, 5 divides n + 3 and 6 divides n + 4 is
(a) 62
(b) 120
(c) 720
(d) 52
Show Answer & Explanation

Answer: (a) 62

Explanation:
1. n − 2 is divisible by 3, 4, 5, 6 (LCM 60) and n is even. The least positive n is 62.

Teacher's Note:
1. Write each condition as n ≡ 2 modulo the divisor.

Question 13: What will be the remainder if the number 7²⁰¹⁵ is divided by 25?
(a) 1
(b) 7
(c) 18
(d) 24
Show Answer & Explanation

Answer: (c) 18

Explanation:
1. 7⁴ ≡ 1 (mod 25), and 2015 = 4 × 503 + 3, so the remainder is 7³ = 343 ≡ 18.

Teacher's Note:
1. Use the cycle of length 4.

Question 14: The sum of all three-digit even numbers divisible by seventeen is
(a) 18846
(b) 18684
(c) 14688
(d) 16848
Show Answer & Explanation

Answer: (c) 14688

Explanation:
1. Even multiples of 17 are multiples of 34: 102, 136, ..., 986, a total of 27 terms.
2. Sum = 27 × (102 + 986)/2 = 14688.

Teacher's Note:
1. They form an AP with difference 34.

Question 15: The fifteen-digit number 8888888*8888888, where * is the missing eighth digit, is divisible by 22. The eighth digit is
(a) 1
(b) 3
(c) 5
(d) 8
Show Answer & Explanation

Answer: (c) 5

Explanation:
1. Odd-place digits are all 8: sum 64. Even-place digits: six 8s = 48 plus *.
2. 64 − 48 − * = 16 − * must be a multiple of 11, so * = 5 (the number is even, so divisible by 2 as required).

Teacher's Note:
1. Use the alternating-sum test for 11.

Question 16: What will be the remainder if (1000000)³ is divided by 143?
(a) 9
(b) 6
(c) 1
(d) 0
Show Answer & Explanation

Answer: (c) 1

Explanation:
1. 143 = 11 × 13. 10⁶ ≡ 1 (mod 143) because 10⁶ − 1 = 999999 = 143 × 6993.
2. So (10⁶)³ ≡ 1.

Teacher's Note:
1. 10⁶ − 1 is divisible by 143.

Question 17: If √338 − √288 = m, then m =
(a) √50
(b) √32
(c) √18
(d) √2
Show Answer & Explanation

Answer: (d) √2

Explanation:
1. √338 = 13√2 and √288 = 12√2, so m = √2.

Teacher's Note:
1. Simplify each surd.

Question 18: How many four-digit numbers divisible by 29 have the sum of their digits equal to 29?
(a) 4
(b) 5
(c) 13
(d) none of these
Show Answer & Explanation

Answer: (b) 5

Explanation:
1. Checking all multiples of 29 from 1000 to 9999 gives exactly 5 with digit sum 29.

Teacher's Note:
1. List the multiples and test the digit sum.

Question 19: 5901AB04 is an eight-digit number divisible by 792. A + B =
(a) 6
(b) 7
(c) 8
(d) 9
Show Answer & Explanation

Answer: (c) 8

Explanation:
1. 792 = 8 × 9 × 11. Checking digits gives A = 3, B = 5.
2. A + B = 8.

Teacher's Note:
1. Use divisibility rules for 8, 9 and 11.

Question 20: What is the smallest natural number which, when multiplied by 15 and divided by 63, gives remainder 21?
(a) 13
(b) 14
(c) 17
(d) 20
Show Answer & Explanation

Answer: (b) 14

Explanation:
1. 15n ≡ 21 (mod 63) gives 5n ≡ 7 (mod 21), so n ≡ 14 (mod 21). The smallest is 14.

Teacher's Note:
1. Divide the congruence by the common factor 3.

Question 21: How many three-digit numbers are divisible by 13 and have middle digit 5?
(a) 5
(b) 7
(c) 10
(d) 13
Show Answer & Explanation

Answer: (b) 7

Explanation:
1. Listing the multiples of 13 from 104 to 988 with tens digit 5 gives 7 numbers.

Teacher's Note:
1. List the multiples of 13 and test the tens digit.

Question 22: The tenth term in the sequence 12, 18, 20, 28, ... (numbers with exactly six divisors) is
(a) 336
(b) 63
(c) 216
(d) 68
Show Answer & Explanation

Answer: (b) 63

Explanation:
1. The numbers with exactly six divisors are 12, 18, 20, 28, 32, 44, 45, 50, 52, 63, so the tenth is 63.

Teacher's Note:
1. Numbers with six divisors have the form p²q or p⁵.

Question 23: The number of integers n such that 1 + n is a divisor of 1 + n² is
(a) 0
(b) 1
(c) 4
(d) 2
Show Answer & Explanation

Answer: (c) 4

Explanation:
1. n² + 1 = (n + 1)(n − 1) + 2, so n + 1 divides 2, giving n + 1 = ±1, ±2 and n = 0, −2, 1, −3.

Teacher's Note:
1. Use polynomial division.

Question 24: The number of integer triples (a, b, c) for which a² + b² − 8c = 3 is
(a) 2
(b) infinite
(c) 0
(d) 4
Show Answer & Explanation

Answer: (c) 0

Explanation:
1. a² + b² = 8c + 3 ≡ 3 (mod 8), but sums of two squares are only ≡ 0, 1, 2, 4, 5 (mod 8), so there are no solutions.

Teacher's Note:
1. Look at squares modulo 8.

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FAQs

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