MATHEMATICS CLASS- 10
MCQ- CH-1(Real Numbers)
Class 10 Mathematics
Chapter 1: Real Numbers
Multiple Choice Questions with Solutions
Q1. The HCF of 26 and 91 is
(a) 7
(b) 13
(c) 26
(d) 91
Answer: (b) 13
Solution: 26 = 2 × 13 and 91 = 7 × 13. Therefore, HCF = 13.
Q2. The LCM of 12 and 18 is
(a) 6
(b) 18
(c) 36
(d) 72
Answer: (c) 36
Solution: 12 = 2² × 3 and 18 = 2 × 3². Hence, LCM = 2² × 3² = 36.
Q3. If HCF(72, 120) = 24, then LCM(72, 120) is
(a) 240
(b) 360
(c) 480
(d) 720
Answer: (b) 360
Solution: HCF × LCM = Product of numbers
24 × LCM = 72 × 120
LCM = 360.
Q4. The decimal expansion of 13/8 is
(a) Terminating
(b) Non-terminating recurring
(c) Non-recurring
(d) Irrational
Answer: (a) Terminating
Solution: 8 = 2³. Denominator contains only factor 2. Hence decimal expansion terminates.
Q5. The decimal expansion of 7/15 is
(a) Terminating
(b) Non-terminating recurring
(c) Irrational
(d) Whole number
Answer: (b) Non-terminating recurring
Solution: 15 = 3 × 5. Since denominator contains factor 3, decimal expansion is non-terminating recurring.
Q6. Which of the following is irrational?
(a) 0.25
(b) 5/7
(c) √3
(d) 0.125
Answer: (c) √3
Solution: √3 cannot be expressed as p/q, so it is irrational.
Q7. The HCF of two consecutive even numbers is
(a) 1
(b) 2
(c) 4
(d) 8
Answer: (b) 2
Solution: Consecutive even numbers always have HCF = 2.
Q8. Which of the following has a terminating decimal expansion?
(a) 7/12
(b) 9/40
(c) 11/18
(d) 13/21
Answer: (b) 9/40
Solution: 40 = 2³ × 5. Hence decimal expansion terminates.
Q9. The Fundamental Theorem of Arithmetic states that every composite number can be expressed as
(a) Product of even numbers
(b) Product of odd numbers
(c) Product of primes uniquely
(d) Sum of primes
Answer: (c) Product of primes uniquely
Solution: Prime factorisation of a composite number is unique except for the order of factors.
Q10. The prime factorisation of 540 is
(a) 2² × 3³ × 5
(b) 2² × 3² × 5
(c) 2³ × 3² × 5
(d) 2² × 3³ × 5²
Answer: (a) 2² × 3³ × 5
Solution: 540 = 54 × 10 = (2 × 3³) × (2 × 5) = 2² × 3³ × 5.
Q11. The decimal expansion of a rational number is always
(a) Terminating only
(b) Non-terminating only
(c) Either terminating or recurring
(d) Irrational
Answer: (c) Either terminating or recurring
Solution: Rational numbers have either terminating or non-terminating recurring decimals.
Q12. Which of the following is a rational number?
(a) π
(b) √5
(c) √49
(d) √7
Answer: (c) √49
Solution: √49 = 7, which is rational.
Q13. The HCF of 96 and 404 is
(a) 2
(b) 4
(c) 8
(d) 12
Answer: (b) 4
Solution: Using Euclid's Division Algorithm:
404 = 96 × 4 + 20
96 = 20 × 4 + 16
20 = 16 × 1 + 4
16 = 4 × 4 + 0
HCF = 4.
Q14. Which of the following is not irrational?
(a) √2
(b) √11
(c) √81
(d) √17
Answer: (c) √81
Solution: √81 = 9, which is rational.
Q15. If the denominator of a rational number in lowest form is 2³ × 5², then its decimal expansion is
(a) Non-terminating recurring
(b) Irrational
(c) Terminating
(d) Non-recurring
Answer: (c) Terminating
Solution: Denominator contains only factors 2 and 5.
Q16. Which of the following numbers is irrational?
(a) 0.333...
(b) 0.125
(c) 1.41421356...
(d) 2.75
Answer: (c) 1.41421356...
Solution: This is a non-terminating non-recurring decimal, hence irrational.
Q17. The LCM of two coprime numbers 8 and 15 is
(a) 8
(b) 15
(c) 120
(d) 23
Answer: (c) 120
Solution: For coprime numbers, LCM = Product = 8 × 15 = 120.
Q18. Which of the following fractions has a non-terminating recurring decimal expansion?
(a) 11/20
(b) 3/16
(c) 17/125
(d) 5/18
Answer: (d) 5/18
Solution: 18 = 2 × 3². Since factor 3 is present, decimal expansion is recurring.
Q19. The product of HCF and LCM of two positive integers is equal to
(a) Sum of integers
(b) Difference of integers
(c) Product of integers
(d) Square of integers
Answer: (c) Product of integers
Solution: HCF × LCM = Product of the two numbers.
Q20. Which of the following statements is true?
(a) Every irrational number is rational
(b) Every rational number is irrational
(c) Every natural number is rational
(d) No integer is rational
Answer: (c) Every natural number is rational
Solution: Any natural number n can be written as n/1, therefore it is rational.
Q21. The HCF of 18, 24 and 30 is
(a) 2
(b) 3
(c) 6
(d) 12
Answer: (c) 6
Solution: 18 = 2 × 3², 24 = 2³ × 3, 30 = 2 × 3 × 5. Common factors = 2 × 3 = 6.
Q22. The LCM of 8, 12 and 15 is
(a) 60
(b) 120
(c) 240
(d) 360
Answer: (b) 120
Solution: LCM = 2³ × 3 × 5 = 120.
Q23. The HCF of 144 and 180 is
(a) 12
(b) 18
(c) 24
(d) 36
Answer: (d) 36
Solution: 144 = 2⁴ × 3² and 180 = 2² × 3² × 5. HCF = 2² × 3² = 36.
Q24. The least number divisible by both 16 and 24 is
(a) 24
(b) 32
(c) 48
(d) 96
Answer: (c) 48
Solution: LCM(16, 24) = 2⁴ × 3 = 48.
Q25. If the HCF of two numbers is 8 and their LCM is 96, then the product of the numbers is
(a) 768
(b) 864
(c) 7680
(d) 104
Answer: (a) 768
Solution: Product = HCF × LCM = 8 × 96 = 768.
Q26. Which of the following fractions has a terminating decimal expansion?
(a) 11/24
(b) 13/40
(c) 7/18
(d) 9/28
Answer: (b) 13/40
Solution: 40 = 2³ × 5. Only factors 2 and 5 are present.
Q27. The decimal expansion of 29/125 is
(a) 0.232
(b) 0.223
(c) 2.32
(d) 0.322
Answer: (a) 0.232
Solution: 29 ÷ 125 = 0.232.
Q28. Which of the following numbers is irrational?
(a) 22/7
(b) 1.272727...
(c) √18
(d) 0.125
Answer: (c) √18
Solution: √18 = 3√2, which is irrational.
Q29. The HCF of two consecutive odd numbers is
(a) 1
(b) 2
(c) 3
(d) 5
Answer: (a) 1
Solution: Consecutive odd numbers are always co-prime.
Q30. If HCF = 12 and one number is 60, then the other number can be
(a) 25
(b) 36
(c) 42
(d) 55
Answer: (b) 36
Solution: HCF(60, 36) = 12.
Q31. Three bells ring at intervals of 12 min, 15 min and 20 min. They ring together at 8:00 AM. They will ring together again after
(a) 30 min
(b) 45 min
(c) 60 min
(d) 120 min
Answer: (c) 60 min
Solution: LCM(12, 15, 20) = 60.
Q32. The smallest number that leaves remainder 0 when divided by 12, 15 and 18 is
(a) 60
(b) 120
(c) 180
(d) 240
Answer: (c) 180
Solution: LCM(12, 15, 18) = 180.
Q33. The largest number that divides 72 and 120 exactly is
(a) 12
(b) 24
(c) 36
(d) 48
Answer: (b) 24
Solution: HCF(72, 120) = 24.
Q34. The denominator of a rational number in lowest form is 2² × 5³. Its decimal expansion is
(a) Non-terminating recurring
(b) Terminating
(c) Irrational
(d) Non-recurring
Answer: (b) Terminating
Solution: Denominator contains only 2 and 5.
Q35. Which of the following is not a rational number?
(a) 0.4444...
(b) √64
(c) π
(d) -7
Answer: (c) π
Solution: π is irrational.
Q36. A number is divisible by 6 and 8. It must also be divisible by
(a) 12
(b) 16
(c) 24
(d) 48
Answer: (c) 24
Solution: LCM(6, 8) = 24.
Q37. The HCF of 135 and 225 is
(a) 15
(b) 25
(c) 35
(d) 45
Answer: (d) 45
Solution: 135 = 3³ × 5 and 225 = 3² × 5². HCF = 3² × 5 = 45.
Q38. Two ropes of lengths 36 m and 48 m are to be cut into equal pieces of maximum length. The length of each piece is
(a) 6 m
(b) 12 m
(c) 18 m
(d) 24 m
Answer: (b) 12 m
Solution: Required length = HCF(36, 48) = 12 m.
Q39. The smallest number exactly divisible by 9, 12 and 15 is
(a) 90
(b) 120
(c) 180
(d) 360
Answer: (c) 180
Solution: LCM(9, 12, 15) = 180.
Q40. If p is prime, then the HCF of p and p² is
(a) 1
(b) p
(c) p²
(d) 2p
Answer: (b) p
Solution: Common highest factor of p and p² is p.
Q41. The HCF of 0 and 25 is
(a) 0
(b) 1
(c) 25
(d) Not defined
Answer: (c) 25
Solution: HCF(0, a) = a.
Q42. Which of the following is an irrational number?
(a) √121
(b) √169
(c) √196
(d) √200
Answer: (d) √200
Solution: √200 = 10√2, which is irrational.
Q43. The product of three consecutive integers is always divisible by
(a) 2 only
(b) 3 only
(c) 6
(d) 12
Answer: (c) 6
Solution: Among three consecutive integers, one is divisible by 3 and one is even.
Q44. The decimal expansion of 17/80 is
(a) 0.2125
(b) 0.1257
(c) 0.2175
(d) 0.2157
Answer: (a) 0.2125
Solution: 17 ÷ 80 = 0.2125.
Q45. If two numbers are co-prime, then their HCF is
(a) 0
(b) 1
(c) 2
(d) Product of numbers
Answer: (b) 1
Solution: Co-prime numbers have HCF = 1.
Q46. A tank can be emptied by three pipes in 24 min, 36 min and 54 min respectively. They will empty together after every
(a) 108 min
(b) 216 min
(c) 432 min
(d) 648 min
Answer: (b) 216 min
Solution: LCM(24, 36, 54) = 216.
Q47. The HCF of 999 and 111 is
(a) 9
(b) 27
(c) 37
(d) 111
Answer: (d) 111
Solution: 999 = 111 × 9. Hence HCF = 111.
Q48. The least number that should be added to 560 so that it becomes exactly divisible by 24 is
(a) 8
(b) 10
(c) 12
(d) 16
Answer: (d) 16
Solution: 560 ÷ 24 leaves remainder 8. Required number = 24 − 8 = 16.
Q49. If the decimal expansion of a rational number terminates after 4 decimal places, then its denominator in lowest form can be
(a) 3⁴
(b) 7²
(c) 2² × 5⁴
(d) 11 × 13
Answer: (c) 2² × 5⁴
Solution: Only factors 2 and 5 are allowed.
Q50. The greatest 4-digit number exactly divisible by 12, 15 and 18 is
(a) 9720
(b) 9900
(c) 9990
(d) 9980
Answer: (b) 9900
Solution: LCM(12,15,18)=180.
Greatest multiple of 180 less than 10000 = 180 × 55 = 9900.