MATHS EDUCATION : March 2024

Class-10. 1 .REAL NUMBERS. MCQS

1. The sum of exponents of prime factors in the prime-factorisation of 196 is :

(a) 3.

(b) 4.

(d.) 2

2. The total number of factors of prime number is.

(a) 1.

(b) 0.

(d) 3

3. The HCF and the LCM of 12, 21, 15 are respectively..

(a) 3, 140.

(b) 12, 420

(d) 420, 3

4. The decimal representation of 11/2³×5 will be..

(a) terminate after 1 decimal place.

(b) terminate after 2 decimal place

(d) not terminate

5. The LCM of smallest two digit composite number and smallest composite number is.

(a) 12.

(b) 4.

(d) 44

6. HCF of two numbers is 27 and their LCM is 162. If one of the numbers is 54, then the other number is.

(a) 36.

(b) 35.

(d) 81

7. HCF of 144 and 198 is.

(a) 9.

(b) 18.

(d) 12

8. 225 can be expressed as:

(a) 5 ×3²

(b) 5²×3.

(d) 5³×3

9. The decimal expansion of 23/2³×5² will terminate after how many places of decimal?

(a) 2.

(b) 4.

(d) 3.

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

(a) one decimal place.

(b) two decimal places.

(d) four decimal places

11. 2.35335… is ..

(a) an integer.

(b) a rational number.

(d) a natural number

12. 2√3 is ..

(a) an integer.

(b) a rational number.

(d) a whole number

13. For some integer q, every odd integer is of the form ....

(a) q.

(b) q+ 1.

(d) 2q+ 1

14. If two positive integers a and b are written as a= x³y² and b= xy³ , where x, y are prime numbers, then HCF(a,b), is.

(a) xy.

(b) xy²

(d)x²y²

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

(a) ab.

(b) a²b²

(d) a³b³

16. Find the HCF of first even number and first prime number…

(a) 4

(b) 6

( c) 8.

( d) 2

17. If p and q are two odd prime numbers such that p>q, then p²-q² is

(a) an even number.

(b) an odd number.

(d) a prime number

18. The rational number of the form p/q are positive integers, which represents as (0.1343434 ..........) is.

(a) 134 /999.

(b) 134 /990

(d) 133/990

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19. Which of the following will have a terminating decimal expansion..

(a). 77/210.

(b). 23/30.

(C). 125/441.

(d). 23 / 8

20. If x=0.777…, then 2x is…

(a) 1.444…

(b) 1.555…

(d) 1.454545…

21. Which of the following rational number have nonterminating repeating decimal

expansion?

(a) 31/3125.

(b) 71/512.

(d) None of these

22. The number 3¹³ - 3¹⁰ is divisible by ..

(a) 2 and 3.

(b) 3 and 10.

(d) 2, 3 and 13

23. The least number which is a perfect square and is divisible by each of 16, 20 and.

24 is…

(a) 240.

(b) 1600.

(d) 3600

24. n²-1 is divisible by 8, if n is ...

(a) an intege. .

(b) a natural number.

(c) an odd integer.

(d) an even integer

25. When 2²⁵⁶ is divided by 17 the remainder would be

(a) 1.

(b) 16.

(d) None of these

26 for some integer m ,every even numbers is the form of:

(a). 2m.

(b ). 2m + 1.

( c). m + 1.

(d) . 2m + 2

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27. Given that HCF (306, 657 ) = 9, then LCM (306, 657) will be......

(a). 33251

(b). 28954

(c). 22338

(d). 33822

28. Find the largest number that will divide 398 ,436, and 542 leaving remainders 7, 11, and 15....

(398-7=391,. 436-11= 425 ,. 542-15=527, HCF. of 391,425,527 )

(a). 11

(b). 21

(c). 15

(d). 17

29. a and b are two positive integers such that the least prime factor of a is 3 and the least prime factor of b is 5. Then the least prime factor of (a+b ) will be

(a) 1.

(b) 2.

(d) 4

30. Select the least number that is divisible by all numbers between 1 and 10 (both inclusive).

(a) 2520.

(b) 5040.

(d) 2020

31. The number 7/75 will have -

(a) non-terminating repeating decimal expansion.

(b) terminating decimal expansion.

(d) terminating non repeating decimal expansion

32. The decimal representation of 21/16×25 will

(a) terminate after 2 decimal place

(b) terminate after 3 decimal place

(d) terminate after 5 decimal places

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33. The sum of exponents of prime factors in the prime-factorisation of 1764 is

(a) 3

(b) 4

(d) 6

34. What is the first composite number….

(a) 1.

(b) 2.

(c.) 3.

(d) 4

35. The HCF and LCM of 378, 180 and 420 of will be ...

(a) 6 and 3980

(b) 12 and 3780

(d) 12 and 3980

36. What is the HCF of 11 and 111 ?

(a) 1.

(b). 11.

(c.) 111.

(d) 0

37. What is the HCF of smallest primer number and the smallest composite number?

(a) 2.

(b) 4.

(d) 8

38. If HCF(336, 54) = 6 , LCM ( 336, 54) will be.

(a) 2024

(b) 3024

(d) 1512

39. (144)½ is a….

(a) Rational.

(b) Irrational

(c.) Even.

(d) None of these

40. How many prime number 80 to 100 ?

(a). 2

(b). 4

(c). 3. ( Ans. 83, 89 and 97. )

(d). 5

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41. Three bells toll at intervals of 9, 12, 15 minutes respectively. If they start tolling together, after what time will they next toll together?

(a). 2 Hours

(b). 3 Hours

(c). 5 Hours

(d). 2.5 Hours

Solution:
9 = 32, 12 = 22 × 3, 15 = 3 × 5
LCM = 22 × 32 × 5 = 4 × 9 × 5 = 180 minutes or 3 hours
They will next toll together after 3 hours.

42. Two tankers contain 850 liters and 680 liters of petrol. Find the maximum capacity of a container which can measure the petrol of each tanker in the exact number of times.

(a). 270 lit

(b). 170 lit

(c). 115 lit

(d). 25 lit

Solution:
To find the maximum capacity of a container which can measure the petrol of each tanker in the exact number of times, we find the HCF of 850 and 680.
850 = 2 × 52 × 17
680 = 23 × 5 × 17
HCF = 2 × 5 × 17 = 170
Maximum capacity of the container = 170 liters.

43. n² – 1 is divisible by 8, if n is:

(a) an integer

(b) a natural number

(c) an odd integer

(d) an even integer

44. If the HCF of 65 and 117 is expressible in the form 65m – 117, then the value of m is:

(a) 4

(b) 2

(c) 1

(d) 3

45. The largest number which divides 70 and 125, leaving remainders 5 and 8 respectively is:

(a) 13

(b) 65

(c) 875

(d) 1750

46. Euclid’s Division Lemma states that for two positive integers a and b, there exists unique integers q and r such that a = bq + r, where r must satisfy:

(a) 1 < r < b

(b) 0 < r ≤ b

(c) 0 ≤ r < b

(d) 0 < r < b

47. If p1 and p2 are two odd prime numbers such that p1 > p2, then

(p1)² – (p2)² is:

(a) an even number

(b) an odd number

(c) an odd prime number

(d) a prime number

(Let the two odd prime numbers and be 5 and 3.

Then,

And

Thus,

16 is even number.

48. If the HCF of 210 and 55 is expressible as 210x – 19 × 55, then x is:

(a) 7

(b) 4

(c) 5

(d) 1

( 5 is the HCFof 210 and 55)

∴5=210×x- 19× 55

X= 5

49. If the HCF of 81 and 237 can be written as 237m + 81n, then the values of m and n are:

(a) m = 13, n = –38

(b) m = 13, n = 38

(c) m = –13, n = 38

(d) m = –13, n = –38

50. The largest number that divides 398, 436 and 542, leaving remainders 7, 11 and 15

respectively is:

(a) 15

(b) 17

(c) 19

(d) 13

51. If two positive integers p and q are written as p = x²y² and q = xy³, x and y are prime

numbers, then HCF (p, q) is:

(a) xy

(b) xy²

(c) x³y

(d) x²y²

52. If HCF and LCM of two numbers are respectively (n – 1) and (n² – 1) (n² – 4), then the

product of the two numbers will be:

(a) (n² – 1) (n² – 4)

(b) (n² + 1) (n² – 4) (n² – 1)

(c) (n² – 4) (n + 1) (n – 1)².

(d) (x² – 1) (x² + 1) (x – 4)

53. For any natural number n, 6n – 5n is always ends with :

(a) 2

(b) 1

(c) 3

(d) 4

(The numbers 6n and 5n always end with 6 and 5 respectively for any natural number. For n = 5; 65 - 55 = 7776 - 3125 = 1 and so on. ∴ 6n - 5n always ends with 1 for any natural numbers )

54. In m is a rational number and n is an irrational number, then m + n, m – n and mn are

all :

(a) rational numbers

(b) irrational numbers

(c) either rational or irrational

(d) neither rational nor irrational

55. If the LCM of P and 18 is 36 and the HCF of P and 18 is 2, then P equals :

(a) 2

(b) 3

(c) 1

(d) 4

56. If HCF (26, 169) = 13, then LCM (26, 169) equals

(a) 26

(b) 52

(c) 20

(d) 338

57. The (HCF × LCM) for the numbers 50 and 20 is:

(a) 10

(b) 100

(c) 1000

(d) 50

58. If HCF (105, 120) = 15, then LCM (105, 120) is:

(a) 210

(b) 420

(c) 840

(d) 1680

59. The HCF of two numbers is 16 and their product is 3072. Their LCM is:

(a) 182

(b) 192

(c) 200

(d) 210

60. HCF × LCM for the numbers 45 and 117 is:

(a) 5165

(b) 5265

(c) 3365

(d) 3065

61. HCF of 960 and 432 is :

(a) 45

(b) 46

(c) 48

(d) 50

62. If HCF (45, 105) = 15, then LCM (45, 105) is equal to :

(a) 315

(b) 320

(c) 400

(d) 410

63. After how many decimal places will the decimal expanssion of 47 /2⁵ ×5⁴

terminates?

(a) 3

(b) 4

(c) 5

(d) 6

64. Product of HCF and LCM of numbers 50 and 95 :

(a) 2350

(b) 4750

(c) 5530

(d) 4900

65. Prime factors of 98 are :

(a) 2 × 7³

(b) 3 × 7³

(c) 2 × 7²

(d) 2 × 7⁵

66. HCF of 726 and 275 is :

(a) 17

(b) 16

(c) 15

(d) 11

67. 15/ 2⁷ ×5³ will terminate after how many places of decimal

(a) 2

(b) 3

(c) 4

(d) 7

68. The product of HCF and LCM of 72 and 132 is :

(a) 9540

(b) 9045

(c) 9504

(d) 9054

69. The HCF × LCM for the numbers 100 and 190 is

(a) 17000

(b) 18000

(d) 20000

70. The decimal representation of

11/ 25³ will

(a) terminate after 1 decimal place

(b) terminate after 2 decimal places

(c) terminate after 6 decimal places

(d) not terminate

71. LCM of smallest two digit composite number and smallest composite number is ..

(a) 12

(b) 4

(c) 20

(d) 44

72. HCF of 168 and 126 is

(a) 21

(b) 42

(c) 14

(d) 18

73. 325 can be expressed as a product at its primes as :

(a) 5² × 7

(b) 5² × 13

(c) 5 × 13²

(d) 2 × 3² × 5²

74. HCF of x²y⁷ and x⁷y² :

(a) x²y³

(b) x⁶y⁶

(c) x⁷y⁷

(d) x²y²

75. HCF of two prime numbers is

(a) 0

(b) 1

(c) 2

(d) 3

76. 7 +√ 7 is :

(a) an interger

(b) a rational number

(c) an irrational number

(d) none of these

77. The ratio between the LCM and HCF of 5, 15, 20 is :

(a) 9 : 1

(b) 4 : 3

(c) 11 : 1

(d) 12 : 1

78. 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

The correct option is D 4

Rational number = \frac{14587}{1250} = \frac{14587}{2^{1} \times 5^{4}} = \frac{14587}{10 \times 5^{3}} \times \frac{(2^{3})}{(2^{3})} = \frac{14587 \times 8}{10 \times 1000} = \frac{116696}{10000} = 11.6696

Hence, given rational number will terminate after 4 decimal places.

79. If 112=6q +r, then the possible value of r.....

(a)1,2,3

(b) 1,2,3,4

(c) ,2,3,4,5

(d) 1,2,3,4,5,6

80. If the LCM of P and 12 is 24 and the HCF of P and 12 is 4, then the value of P :

(a) 3

(b) 4

(c) 6

(d) 8

81. Prime factorisation of 8232 is

(a) 2³ × 3× 7³

(b) 72 × 3 × 23

(c) 2 × 3 × 74

(d) 22 × 32 × 72

82. If the product of two numbers a and b is 1152 and HCF of a and b is 12, then LCM of

a and b :

(a) 88

(b) 90

(c) 92

(d) 96

83. If the HCF of 18 and 117 is expressible in the form 18 m – 117, then the value of m is:

(a) 5

(b) 6

(c) 7

(d) 8.

84. The LCM of two numbers is 740 and their HCF is 37. If one of the numbers is 185, then the other number is...

(a) 136

(b) 140

(c) 142

(d) 148

85. Greatest number that will divide 382, 509 and 636 leaving reminders 4, 5 and 6 respectively

is :

(a) 126

(b) 127

(c) 128

(d) 129

86. The decimal expanssion

63 / 72×175

is :

(a) terminating

(b) non-terminating

(c) non-terminating and repeating

(d) none of these

87. The decimal expansion of

111 / 2⁵ is :

(a) terminating

(b) non-terminating non-repeating

(c) non-terminating

(d) does not exist

88. The decimal expansion of the number 31 / 2⁵ × 5² will terminate after :

(a) one decimal place

(b) two decimal places

(c) three decimal places

(d) more than three decimal places

89. The decimal expansion of the number 441 / 2² × 5³ × 7 is :

(a) terminating

(b) non-terminating but repeating

(c) non-terminating non-repeating

(d) terminating after two places of decimal

90. If the HCF of 85 and 153 is expressible in the form 85 m – 153, then the value of m is :

(a) 4

(b) 3

(c) 1

(d) 2

91. Decimal expansion of 1717 /2² ×5³

is _____________.Ans 3.434

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92. If a positive integer n is divided by 3, then the possible reminder = ____________Ans . 0,1,2

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93. The decimal expansion of 7
125 will terminate after _____________ places.

Ans. 3

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94. If HCF (225, 60) = 225 × 5 – 10x, then the value of x = _____________.Ans. 111

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95. Product of smallest prime number and smallest composite number is _____________. Ans 8

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96. If p = 2³ × 3 × 5², q = 2³ × 3³, then HCF (p, q) = ____________. Ans 24

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97. How many irrational numbers will be between √2 and √3 . Ans Infinite

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98. Greatest number that will divide 445, 572 and 699 leaving reminders 4, 5 and 6 respectively
is ___________. Ans. 63

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99. The LCM of two numbers is 2079 and their HCF is 27. If one of the numbers is 297,
then other number is ______189______.

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100. LCM (72, 126, 168) = ____504_______.

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MATHS EDUCATION

Sunday, 31 March 2024

10.Real Numbers (SAQ+LAQ)

10. Real Numbers MCQ

Worksheet of A.P

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