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MATHEMATICS CLASS- 10

MCQ- CH-1(Real Numbers)

CBSEChapter 1MCQ's

CLASS 10 MATHEMATICS

Chapter 1: Real Numbers

General Instructions: All questions are based on the NCERT textbook concepts. Select the most appropriate option.


  1. According to the Fundamental Theorem of Arithmetic, every composite number can be expressed as a product of:
    a) Integers
    b) Primes
    c) Even numbers
    d) Rational numbers

  2. The prime factorisation of a natural number is unique, except for the:
    a) Number of factors
    b) Value of the factors
    c) Order of its factors
    d) Sum of the factors

  3. Which mathematician is often referred to as the ‘Prince of Mathematicians’?
    a) Euclid
    b) Archimedes
    c) Carl Friedrich Gauss
    d) Isaac Newton

  4. The first correct proof of the Fundamental Theorem of Arithmetic was given by:
    a) Euclid
    b) Carl Friedrich Gauss
    c) Pythagoras
    d) Ramanujan

  5. If \( p \) is a prime number and \( p \) divides \( a^2 \), then \( p \) also divides:
    a) \( a \)
    b) \( 2a \)
    c) \( a^3 \)
    d) \( \sqrt{a} \)

  6. For any two positive integers \( a \) and \( b \), \( HCF(a, b) \times LCM(a, b) \) is equal to:
    a) \( a + b \)
    b) \( a - b \)
    c) \( a \times b \)
    d) \( a / b \)

  7. If the prime factorisation of a number \( 4^n \) does not contain the prime number 5, then \( 4^n \) can never end with the digit:
    a) 4
    b) 6
    c) 0
    d) 2

  8. The HCF of 6 and 20 is:
    a) 60
    b) 2
    c) 120
    d) 10

  9. The LCM of 6 and 20 is:
    a) 2
    b) 120
    c) 60
    d) 12

  10. HCF is the product of the ________ power of each common prime factor in the numbers.
    a) Greatest
    b) Smallest
    c) Average
    d) Highest

  11. LCM is the product of the ________ power of each prime factor involved in the numbers.
    a) Smallest
    b) Common
    c) Greatest
    d) Lowest

  12. A number \( s \) is called irrational if it cannot be written in the form \( p/q \), where \( p \) and \( q \) are integers and:
    a) \( q = 0 \)
    b) \( q \neq 0 \)
    c) \( p = 0 \)
    d) \( p \neq 0 \)

  13. Which of the following is an irrational number?
    a) \( \sqrt{16} \)
    b) \( 3.14 \)
    c) \( \sqrt{2} \)
    d) \( 22/7 \)

  14. The technique used to prove that \( \sqrt{2} \) is irrational is called:
    a) Proof by induction
    b) Proof by contradiction
    c) Direct proof
    d) Proof by exhaustion

  15. If we assume \( \sqrt{2} = a/b \) where \( a \) and \( b \) are coprime, then \( a \) and \( b \) have:
    a) No common factor
    b) No common factor other than 1
    c) At least 2 as a common factor
    d) Only 5 as a common factor

  16. The sum of a rational and an irrational number is always:
    a) Rational
    b) Irrational
    c) An integer
    d) Zero

  17. The product of a non-zero rational and an irrational number is always:
    a) Rational
    b) Irrational
    c) One
    d) Zero

  18. If \( 3 \) divides \( a^2 \), then \( 3 \) also divides \( a \). This statement is based on:
    a) Theorem 1.1
    b) Theorem 1.2
    c) Theorem 1.3
    d) Summary point 4

  19. The HCF of 96 and 404 is:
    a) 2
    b) 4
    c) 96
    d) 101

  20. The LCM of 96 and 404 is:
    a) 4
    b) 9696
    c) 38784
    d) 404

  21. The prime factorisation of 156 is:
    a) \( 2^2 \times 3 \times 13 \)
    b) \( 2 \times 3^2 \times 13 \)
    c) \( 2^2 \times 3^2 \times 13 \)
    d) \( 2 \times 3 \times 13 \)

  22. The number \( 0.10110111011110... \) is:
    a) Rational
    b) Irrational
    c) Terminating decimal
    d) Non-terminating repeating decimal

  23. For three positive integers \( p, q, r \), the product \( p \times q \times r \) is:
    a) Always equal to \( HCF(p, q, r) \times LCM(p, q, r) \)
    b) Not necessarily equal to \( HCF(p, q, r) \times LCM(p, q, r) \)
    c) Always less than \( HCF(p, q, r) \times LCM(p, q, r) \)
    d) Always zero

  24. The HCF of 6, 72 and 120 is:
    a) 6
    b) 12
    c) 2
    d) 3

  25. The LCM of 6, 72 and 120 is:
    a) 720
    b) 360
    c) 120
    d) 60

  26. A composite number can be producing by:
    a) Adding primes
    b) Subtracting primes
    c) Multiplying some or all primes
    d) Dividing primes

  27. The factor tree of 32760 shows that one of its prime factors is:
    a) 11
    b) 13
    c) 17
    d) 19

  28. If \( p \) is a prime, then \( \sqrt{p} \) is always:
    a) Rational
    b) Irrational
    c) A whole number
    d) A negative integer

  29. In the proof that \( \sqrt{3} \) is irrational, we assume \( 3b^2 = a^2 \). This implies \( a^2 \) is divisible by:
    a) 2
    b) 3
    c) 9
    d) 6

  30. If \( a \) and \( b \) are coprime, then their HCF is:
    a) \( a \times b \)
    b) 0
    c) 1
    d) 2

  31. The number \( 5 - \sqrt{3} \) is:
    a) Rational
    b) Irrational
    c) An integer
    d) A natural number

  32. If \( 3 \sqrt{2} = a/b \), then \( \sqrt{2} \) equals:
    a) \( a/3b \)
    b) \( 3a/b \)
    c) \( a/b - 3 \)
    d) \( a + 3b \)

  33. The number of prime factors of 5005 is:
    a) 2
    b) 3
    c) 4
    d) 5

  34. The largest power of 2 in the prime factorisation of 32760 is:
    a) \( 2^2 \)
    b) \( 2^3 \)
    c) \( 2^4 \)
    d) \( 2^5 \)

  35. In Theorem 1.1, the term "apart from the order" means:
    a) The factors can be changed
    b) The sequence of primes doesn't affect uniqueness
    c) The factors are not unique
    d) The number is not composite

  36. If \( n \) is a natural number, then \( 6^n \) ends with 0 only if its prime factorisation contains:
    a) Only 2
    b) Only 3
    c) Both 2 and 5
    d) Both 2 and 3

  37. Does \( 6^n \) ever end with the digit 0 for any natural number \( n \)?
    a) Yes, for \( n = 5 \)
    b) Yes, for \( n = 10 \)
    c) No
    d) Yes, for all \( n \)

  38. The HCF of 17, 23 and 29 is:
    a) 1
    b) 17
    c) 23
    d) 29

  39. The value of \( 1.1(v) \) in Exercise (factorisation of 7429) is:
    a) \( 17 \times 19 \times 23 \)
    b) \( 13 \times 17 \times 19 \)
    c) \( 17 \times 23 \times 29 \)
    d) \( 11 \times 13 \times 17 \)

  40. If \( p = 2 \) and \( a = 4 \), then \( p \) divides \( a^2 \) means:
    a) 2 divides 8
    b) 2 divides 16
    c) 4 divides 16
    d) 2 divides 4

  41. The decimal expansion of an irrational number is:
    a) Terminating
    b) Non-terminating repeating
    c) Non-terminating non-repeating
    a) Finite

  42. The HCF of two consecutive even numbers is:
    a) 1
    b) 2
    c) 4
    d) The numbers themselves

  43. In the formula for \( LCM(p, q, r) \), the numerator is:
    a) \( p \times q \times r \)
    b) \( p \times q \times r \times HCF(p, q, r) \)
    c) \( HCF(p, q) \times HCF(q, r) \times HCF(p, r) \)
    d) \( p + q + r \)

  44. If \( HCF(306, 657) = 9 \), then \( LCM(306, 657) \) is:
    a) 22338
    b) 22383
    c) 23238
    d) 306

  45. Which of the following is a composite number?
    a) \( 7 \times 11 \times 13 + 13 \)
    b) 17
    c) 23
    d) 2

  46. If Sonia takes 18 mins and Ravi takes 12 mins to complete a round, they meet at the start after:
    a) 6 mins
    b) 30 mins
    c) 36 mins
    d) 48 mins

  47. The product of HCF and LCM of 26 and 91 is:
    a) 2366
    b) 182
    c) 13
    d) 91

  48. Is \( \pi \) a rational number?
    a) Yes, because it is \( 22/7 \)
    b) No, it is irrational
    c) Yes, it is a terminating decimal
    d) It is an integer

  49. The HCF of 8, 9 and 25 is:
    a) 1
    b) 2
    c) 5
    d) 1800

  50. Every positive integer \( a \) can be divided by \( b \) to leave a remainder \( r \) such that:
    a) \( r > b \)
    b) \( r < b \)
    c) \( r = b \)
    d) \( r = a \)


Answer Key

1(b), 2(c), 3(c), 4(b), 5(a), 6(c), 7(c), 8(b), 9(c), 10(b), 11(c), 12(b), 13(c), 14(b), 15(c), 16(b), 17(c), 18(b), 19(b), 20(b), 21(a), 22(b), 23(b), 24(a), 25(b), 26(c), 27(b), 28(b), 29(b), 30(c), 31(b), 32(a), 33(c), 34(b), 35(b), 36(c), 37(c), 38(a), 39(a), 40(b), 41(c), 42(b), 43(b), 44(a), 45(a), 46(c), 47(a), 48(b), 49(a), 50(b)