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MCQs for Class 10 Maths Chapter 1 Real Numbers

1. The product of a rational and irrational number is

(a) rational

(b) irrational

(c) both of above

(d) none of above

Answer- b) 

2. For some integer m, every odd integer is of the form

(a) m                                      

(b) m + 1

(c) 2m                                    

(d) 2m + 1

Answer:  (d)

3.Decide whether 52.123456789 is a rational number or not. If rational (in the form p/q), what can you say about the prime factors of q?

(a) Rational Number, Prime factor of q will be only 2.

(b) Rational Number, Prime factor of q will have a factor other than 2 or 5.

(c) Not rational number

(d) Rational Number, Prime factors of q will have either 2 or 5 or both

Answer: (d)Rational Number, Prime factors of q will have either 2 or 5 or both

4. If HCF (a, b) = 12 and a × b = 1800 then LCM (a, b) is

(a) 3600

(b) 900

(c) 150

(d) 90

Answer: (c) 150

5. If HCF (a, b) = 12 and a × b = 1800 then LCM (a, b) is

(a) 3600

(b) 900

(c) 150

(d) 90

Answer: (c) 150

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6. There is a circular path around a sports field. Priya takes 18 minutes to drive one round of the field. Harish takes 12 minutes. Suppose they both start at the same point and at the same time and go in the same direction. After how many minutes will they meet ?

(a) 36 minutes

(b) 18 minutes

(c) 6 minutes

(d) They will not meet

Answer: a

7. For some integer p, every even integer is of the form

(a) 2p + 1

(b) 2p

(c) p + 1

(d) p

Answer: b

8.  m² – 1 is divisible by 8, if m is

(a) an even integer

(b) an odd integer

(c) a natural number

(d) a whole number

Answer: b

9. If two positive integers a and b are written as a = p³q² and b = pq³; p, q are prime numbers, then HCF (a, b) is:

(a) pq                                                 

(b) pq²

(c) p³q³                                               

(d) p²q²

Answer:  b

10. The least number that is divisible by all the numbers from 1 to 10 (both inclusive) is 

(a) 10

(b) 100

(c)  504

(d)  2520

Anwer: d

11.  For any two positive integers a and b, there exist (unique) whole numbers q and r such that

(a) q = ar + b , 0 ⩽ r < b.

(b) a = bq + r , 0 ⩽ r < b.

(c) b = aq + r , 0 ⩽ r < b.

(d) none of these

Answer:  (b) 

12.  If two positive integers A and B can be expressed as A = xy³ and B = x⁴y²z; x, y being prime numbers then HCF (A, B) is

(a) xy²

(b) x⁴y²z

(c) x⁴y³

(d) x⁴y₃z

Answer: a

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13. Find the HCF of 1848, 3058 and 1331.

(a) 9

(b) 14

(c) 13

(d) 11

Answer: (d) 11

14. If mn = 32, where m and n are positive integers, then the value of (n)mn is

(a) 9765625

(b) 9775625

(c) 9785625

(d) 9865625

Answer: (a) 976562

15.  If two positive integers p and q can be expressed as p = ab² and q = a³b; a, b being prime numbers, then LCM (p, q) is

 (a) ab

 (b) a²b²

 (c) a³b²

 (d)   a³b³

Answer: (c)

16. The decimal expansion of n is

(a) terminating

(b) non-terminating and non-recurring

(c) non-terminating and recurring

(d) does not exist.

Answer: (b) non-terminating and non-recurring

17. The product of a non-zero number and an irrational number is:

(a) always irrational                          

(b) always rational

(c) rational or irrational                     

(d) one

Answer:  a

18. Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating decimal expansion: 23/8

(a) non-terminating non – repeating decimal

(b) non-terminating repeating decimal

(c) non-terminating decimal

(d) terminating decimal

Answer: (d) terminating decimal

19. If A = 2n + 13, B = n + 7 where n is a natural number then HCF of A and B

(a) 2

(b) 1

(c) 3

(d) 4

Answer: (b) 1

20. The least number that is divisible by all the numbers from 1 to 8 (both inclusive) is

(a) 840

(b) 2520

(c) 8

(d) 420

Answer: a

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21. The decimal expansion of the rational number 97/2×5⁴ will terminate after:

(a) one decimal place

(b) two decimal places

(c) three decimal places

(d) four decimal places

Answer: d

22. A rational number in its decimal expansion is 327.7081. What would be the prime factors of q when the number is expressed in the p/q form?

(a) 2 and 3                                         

(b) 3 and 5

(c) 2, 3 and 5                                     

(d) 2 and 5

Answer: d

23. The number in the form of 4p + 3, where p is a whole number, will always be

(a) even

(b) odd

(c) even or odd

(d) multiple of 3

Answer: b

24. The decimal expansion of the rational number 14587/1250  will terminate after

 (a) one decimal place

 (b) two decimal places

 (c) three decimal places

 (d) four decimal places

Answer: d

25. The least number that is divisible by all the numbers from 1 to 8 (both inclusive) is

(a) 840

(b) 2520

(c) 8

(d) 420

Answer: a

26. Every positive even integer is of the form ____ for some integer ‘q’.

(a) 2q

(b) 2q – 1

(c) 2q + 1

(d) none of these

Answer:  (a) 2q

27. (-1)ⁿ + (-1)⁸ⁿ = 0 when n is

(a) any positive integer

(b) any odd natural number

(c) any even numeral number

(d) any negative integer

Answer: (b) 

28. (6 + 5 √3) – (4 – 3 √3) is

(a) a rational number

(b) an irrational number

(c) a natural number

(d) an integer

Answer: b

29. When a number is divided by 7, its remainder is always:

(a) greater than 7

(b) at least 7

(c) less than 7

(d) at most 7

Answer: c

30. A number 10x + y is multiplied by another number 10a + b and the result comes as 100p + 10q +r, where r = 2y, q = 2(x + y) and p = 2x; x, y < 5, q ≠ 0. The value of 10a + b may be:

(a) 11

(b) 13

(c) 31

(d) 22

Answer: d

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31.  If n is a rational number, then 52n − 22n is divisible by

(a) 3                                                                           

(b) 7

(c) Both 3 and 7                                                        

(d) None of these

Answer: c

32. On a morning walk, three persons step off together and their steps measure 40 cm, 42 cm and 45 cm, respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps?

(a) 2520cm                                                                

(b) 2525cm

(c) 2555cm                                                                

(d) 2528cm

Answer: a

33. If the LCM of 12 and 42 is 10 m + 4 then the value of m is

(a) 50

(b) 8

(c) 1/5

(d) l

Answer: (b) 8

34.  For any two positive integers a and b, there exist (unique) whole numbers q and r such that

(a) q = ar + b , 0 ⩽ r < b.

(b) a = bq + r , 0 ⩽ r < b.

(c) b = aq + r , 0 ⩽ r < b.

(d) none of these

Answer: b

35. s’ is called irrational if it cannot be written in the form of _____ where p and q are integers and ______

(a) p/q, p = 0

(b) p/q, p ≠ 0

(c) p/q, q ≠ 0

(d) p/q, q = 0

Answer: c

36. What is the greatest possible speed at which a man can walk 52 km and 91 km in an exact number of minutes?

(a) 17 m/min

(b) 7 m/min

(c) 13 m/min

(d) 26 m/min

Answer. C

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37. Express 98 as a product of its primes

(a) 2² × 7

(b) 2² × 7²

(c) 2 × 7²

(d) 23 × 7

Answer: c

38.  If the HCF of 408 and 1032 is expressible in the form 1032 x 2 + 408xp, then the value of p is

(a) 5

(b) -5

(c) 4

(d) -4

Answer: b

39.  The HCF and LCM of two numbers are 33 and 264 respectively. When the first number is completely divided by 2 the quotient is 33. The other number is:

(a)  66

(b) 130

(c) 132

(d) 196

Answer: c

40.  LCM of the given number ‘x’ and ‘y’ where y is a multiple of ‘x’ is given by

(a) x

(b) y

(c) xy

(d) x/y

Answer: b

41. The decimal expansion of number 441/2²×5³×7 is

(a) A terminating decimal

(b) Non-terminating but repeating

(c) Non-terminate non repeating

(d) terminating after two places of decimal

Answer: (a) A terminating decimal

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42. For any positive integer a and b, there exist unique integers q and r such that a = 3q + r, where r must satisfy.

(a) 0 ≤ r < 3

(b) 1 < r < 3

(c) 0 < r < 3

(d) 0 < r ≤ 3

Answer: (a) 0 ≤ r < 3

43.  π is

(a) a rational number

(b) an irrational number

(c) both (a) & (b)

(d) neither rational nor irrational

Answer: (b) an irrational number

44. Which among the following options is irrational?

(a) 3.1415926535… (non-repeating and non-terminating)

(b) 10.2

(c) (0.2)2

(d) 0.2

Answer: (a) 3.1415926535… (Non-repeating and non-terminating)

45.  m² – 1 is divisible by 8, if m is

(a) an even integer

(b) an odd integer

(c) a natural number

(d) a whole number

Answer: b

46. If x and y are the two digits of the number 653xy such that this number is divisible by 80, then x + y =?

(a) 4 or 8

(b)6

(c)4

(d)8

Answer: (b) 6

47. Pairs of natural numbers whose least common multiple is 78 and the greatest common divisor is 13 are:

(a) 58 and 13 or 16 and 29 

(b) 68 and 23 or 36 and 49     

 (c) 18 and 73 or 56 and 93

 (d) 78 and 13 or 26 and 39

Answer: d

48. The smallest composite number is:

(a) 1

(b) 2

(c) 3

(d) 4

Answer:  (d) 4

49. 1/√3 is –

(a) A rational number

(b) An irrational number

(c) A whole number

(d) None of these

Answer:  (b) An irrational number

50.  If b = 3, then any integer can be expressed as a =

(a) 3q, 3q+ 1, 3q + 2

(b) 3q

(c) none of the above

(d) 3q+ 1

Answer: a

51. The largest number that will divide 398,436 and 542 leaving remainders 7,11 and 15 respectively is

(a) 17

(b) 11

(c) 34

(d) 45

Answer: a

52. Which of the following is not irrational?

(a) (2 – √3)2

(b) (√2 + √3)2

(c) (√2 -√3)(√2 + √3)

(d)27/√7

Answer: c

53. If the number 91876y2 is completely divisible by 8, then the smallest whole number in place of y will be:

(a) 2

(b) 4

(c) 3

(d) 1

Answer: (c) 3

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54. a and b, when divided by 7 and 6 respectively, leave remainders p and q respectively. What is the maximum value of p + q?

(a) 5

(b) 6

(c) 12

(d) 11

Answer: (d) 11

55. The decimal expansion of the rational number 47/2352 will terminate after:

(a) one decimal place

(b) two decimal places

(c) three decimal places​

(d) more than three decimal places

Answer: (c) three decimal places

56. 987/10500 will have 

(a) Terminating decimal expansion   

(b) Non-Terminating Non repeating decimal expansion       

(c) Non-Terminating repeating decimal expansion   

(d) None of these

Answer: a

57. There is a circular path around a sports field. Priya takes 18 minutes to drive one round of the field. Harish takes 12 minutes. Suppose they both start at the same point and at the same time and go in the same direction. After how many minutes will they meet ?

(a) 36 minutes

(b) 18 minutes

(c) 6 minutes

(d) They will not meet

Answer: a

58. The difference of two numbers is 1365. On dividing the larger number by the smaller, we get 6 as quotient and the 15 as remainder. What is the smaller number?

(a) 360

(b) 295

(c) 270

(d) 240

Answer: (c) 270

59. The values of the remainder r, when a positive integer a is divided by 3 are:

(a) 0, 1, 2, 3                           

(b) 0, 1

(c) 0, 1, 2                               

(c) 2, 3, 4

Answer: c

60. The product of three consecutive positive integers is divisible by

(a) 4

(b) 6

(c) no common factor

(d) only 1

Answer: b

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