CBSE Class 10 Real Numbers Sure Shot Questions Set A

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1. Write whether every positive integer can be of the form 4q + 2, where q is an integer. Justify your answer.
2. “The product of two consecutive positive integers is divisible by 2”. Is this statement true or false? Give reasons.
3. “The product of three consecutive positive integers is divisible by 6”. Is this statement true or false”? Justify your answer.
4. Write whether the square of any positive integer can be of the form 3m + 2, where m is a natural number. Justify your answer.
5. A positive integer is of the form 3q + 1, q being a natural number. Can you write its square in any form other than 3m + 1, i.e., 3m or 3m + 2 for some integer m? Justify your answer.
6. Show that the square of an odd positive integer is of the form 8m + 1, for some whole number m.
7. Show that the square of any positive integer is either of the form 4q or 4q + 1 for some integer q.
8. Show that cube of any positive integer is of the form 4m, 4m + 1 or 4m + 3, for some integer m.
9. Show that the square of any positive integer cannot be of the form 5q + 2 or 5q + 3 for any integer q.
10. Show that the square of any positive integer cannot be of the form 6m + 2 or 6m + 5 for any integer m.
11. Show that the square of any odd integer is of the form 4q + 1, for some integer q.
12. If n is an odd integer, then show that n² – 1 is divisible by 8.
13. Prove that if x and y are both odd positive integers, then x² + y² is even but not divisible by 4.
14. Show that the square of an odd positive integer can be of the form 6q + 1 or 6q + 3 for some integer q.
15. Show that the cube of a positive integer of the form 6q + r, q is an integer and r = 0, 1, 2, 3, 4, 5 is also of the form 6m + r.
16. Prove that one and only one out of n, n + 2 and n + 4 is divisible by 3, where n is any positive integer.
17. Prove that one of any three consecutive positive integers must be divisible by 3.
18. 18. For any positive integer n, prove that n³ – n is divisible by 6.
19. 19. Show that one and only one out of n, n + 4, n + 8, n + 12 and n + 16 is divisible by 5, where n is any positive integer.
20. Show that the product of three consecutive natural numbers is divisble by 6.
21. Show that any positive odd integer is of the form 6q + 1 or 6q + 3 or 6q + 5 where q ε Z .
22. Show that any positive even integer is of the form 6q or 6q + 2 or 6q + 4 where q ε Z .
23. If a and b are two odd positive integers such that a > b, then prove that one of the two numbers a + b /2 and a - b/2 is odd and the other is even.
24. Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.
25. Using Euclid’s division algorithm to show that any positive odd integer is of the form 4q+1 or 4q+3, where q is some integer.
26. Use Euclid’s division algorithm to find the HCF of 441, 567, 693.
27. Using Euclid’s division algorithm, find the largest number that divides 1251, 9377 and 15628 leaving remainders 1, 2 and 3, respectively.
28. Using Euclid’s division algorithm, find which of the following pairs of numbers are co-prime: (i) 231, 396 (ii) 847, 2160
29. Show that 12n cannot end with the digit 0 or 5 for any natural number n.
30. In 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?
31. If LCM (480, 672) = 3360, find HCF (480,672).
32. Express 0.69 as a rational number in p / q form.
33. Show that the number of the form 7n, nÎN cannot have unit digit zero.
34. Using Euclid’s Division Algorithm find the HCF of 9828 and 14742.
35. The numbers 525 and 3000 are both divisible only by 3, 5, 15, 25 and 75. What is HCF (525, 3000)? Justify your answer.
36. Explain why 3 × 5 × 7 + 7 is a composite number.
37. Can two numbers have 18 as their HCF and 380 as their LCM? Give reasons.
38. Without actual division find whether the rational number 1323 /(63 x 352 ) has a terminating decimal or a non- terminating decimal.
39. Without actually performing the long division, find if 987 /10500 will have terminating or non- terminating (repeating) decimal expansion. Give reasons for your answer.
40. A rational number in its decimal expansion is 327.7081. What can you say about the prime factors of q, when this number is expressed in the form p /q? Give reasons.
41. Find the HCF of 81 and 237 and express it as a linear combination of 81 and 237.
42. Find the HCF of 65 and 117 and express it in the form 65m + 117n.
43. If the HCF of 210 and 55 is expressible in the form of 210x5 + 55y, find y.
44. If d is the HCF of 56 and 72, find x, y satisfying d = 56x + 72y. Also show that x and y are not unique.
45. Express the HCF of 468 and 222 as 468x + 222y where x, y are integers in two different ways.
46. Express the HCF of 210 and 55 as 210x + 55y where x, y are integers in two different ways.
47. If the HCF of 408 and 1032 is expressible in the form of 1032m – 408x5, find m.
48. If the HCF of 657 and 963 is expressible in the form of 657n + 963x(-15), find n.
49. A sweet seller has 420 kaju burfis and 130 badam burfis she wants to stack them in such a way that each stack has the same number, and they take up the least area of the tray. What is the number of burfis that can be placed in each stack for this purpose?
50. Find the largest number which divides 245 and 1029 leaving remainder 5 in each case.
51. Find the largest number which divides 2053 and 967 and leaves a remainder of 5 and 7 respectively.
52. Two tankers contain 850 litres and 680 litres of kerosene oil respectively. Find the maximum capacity of a container which can measure the kerosene oil of both the tankers when used an exact number of times.
53. In a morning walk, three persons step off together. Their steps measure 80 cm, 85 cm and 90 cm respectively. What is the minimum distance each should walk so that all can cover the same distance in complete steps?
54. Find the least number which when divided by 12, 16, 24 and 36 leaves a remainder 7 in each case.
55. The length, breadth and height of a room are 825 cm, 675 cm and 450 cm respectively. Find the longest tape which can measure the three dimensions of the room exactly.
56. Determine the smallest 3-digit number which is exactly divisible by 6, 8 and 12.
57. Determine the greatest 3-digit number exactly divisible by 8, 10 and 12.
58. The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108 seconds respectively. If they change simultaneously at 7 a.m., at what time will they change simultaneously again?
59. Three tankers contain 403 litres, 434 litres and 465 litres of diesel respectively. Find the maximum capacity of a container that can measure the diesel of the three containers exact number of times.
60. Find the least number which when divided by 6, 15 and 18 leave remainder 5 in each case.
61. Find the smallest 4-digit number which is divisible by 18, 24 and 32.
62. Renu purchases two bags of fertiliser of weights 75 kg and 69 kg. Find the maximum value of weight which can measure the weight of the fertiliser exact number of times.
63. In a seminar, the number, the number of participants in Hindi, English and Mathematics are 60, 84 and 108, respectively. Find the minimum number of rooms required if in each room the same number of participants are to be seated and all of them being in the same subject.
64. 144 cartons of Coke cans and 90 cartons of Pepsi cans are to be stacked in a canteen. If each stack is of the same height and is to contain cartons of the same drink, what would be the greatest number of cartons each stack would have?
65. A merchant has 120 litres of oil of one kind, 180 litres of another kind and 240 litres of third kind. He wants to sell the oil by filling the three kinds of oil in tins of equal capacity. What would be the greatest capacity of such a tin?
66. Express each of the following positive integers as the product of its prime factors: (i) 3825 (ii) 5005 (iii) 7429
67. Express each of the following positive integers as the product of its prime factors: (i) 140 (ii) 156 (iii) 234
68. There is circular path around a sports field. Priya takes 18 minutes to drive one round of the field, while Ravish takes 12 minutes for the same. 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 again at the starting point?
69. In a morning walk, three persons step off together and their steps measure 80 cm, 85 cm and 90 cm, respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps?
70. A circular field has a circumference of 360 km. Three cyclists start together and can cycle 48, 60 and 72 km a day, round the field. When will they meet again?
71. Find the smallest number which leaves remainders 8 and 12 when divided by 28 and 32 respectively.
72. Find the smallest number which when increased by 17 is exactly divisible by 520 and 468.
73. Find the greatest numbers that will divide 445, 572 and 699 leaving remainders 4, 5 and 6 respectively.
74. Find the greatest number which divides 2011 and 2423 leaving remainders 9 and 5 respectively
75. Find the greatest number which divides 615 and 963 leaving remainder 6 in each case.
76. Find the greatest number which divides 285 and 1249 leaving remainders 9 and 7 respectively.
77. Find the largest possible positive integer that will divide 398, 436, and 542 leaving remainder 7,11, 15 respectively.
78. If d is the HCF of 30, 72, find the value of x & y satisfying d = 30x + 72y.
79. State Euclid’s Division Lemma.
80. State the Fundamental theorem of Arithmetic.
81. Given that HCF (306, 657) = 9, find the LCM(306, 657).
82. Why the number 4n, where n is a natural number, cannot end with 0?
83. Why is 5 x 7 x 11 + 7 is a composite number?
84. Explain why 7 x 11 + 13 + 13 and 7 x 6 x 5 x 4 x 3 x 2 x 1 + 5 are composite numbers.
85. In a school there are two sections – section A and section B of class X. There are 32 students in section A and 36 students in section B. Determine the minimum number of books required for their class library so that they can be distributed equally among students of section A or section B.
86. Determine the number nearest 110000 but greater than 100000 which is exactly divisible by each of 8, 15 and 21.
87. Three sets of English, Hindi and Mathematics books have to be stacked in such a way that all the books are stored topic wise and the height of each stack is the same. The number of English books is 96, the number of Hindi books is 240 and the number of Mathematics books is 336. Assuming that the books are of the same thickness, determine the number of stacks of English, Hindi and Mathematics books.
88. Using Euclid’s division algorithm, find the HCF of 2160 and 3520.
89. Find the HCF and LCM of 144, 180 and 192 by using prime factorization method.
90. Find the missing numbers in the following factorization:
91. Find the HCF and LCM of 17, 23 and 37 by using prime factorization method.
92. If HCF(6, a) = 2 and LCM(6, a) = 60 then find the value of a.
93. If remainder of (5m +1)(5m +3)(5m + 4) /5 is a natural number then find it.
94. A rational number p / q has a non-terminating repeating decimal expansion. What can you say about q?
95. If 278 /2^m m has a terminating decimal expansion and m is a positive integer such that 2 < m < 9, then find the value of m.
96. Write the condition to be satisfied by q so that a rational number p / q has a terminating expression.
97. If a and b are positive integers. Show that √2 always lies between a / b and a² - 2b² / b(a + b)
98. Find two rational number and two irrational number between √2 and √3 .
99. Prove that 5 - 2 √3 is an irrational number.
100. Prove that 15 +17√3 is an irrational number.
101. Prove that 2√3 /5 is an irrational number.
102. Prove that 7 +3 √2 is an irrational number.
103. Prove that 2 +3 √5 is an irrational number.
104. Prove that √2 +√3 is an irrational number.
105. Prove that√3 +√5 is an irrational number.
106. Prove that 7 -2 √3 is an irrational number.
107. Prove that 3 -√5 is an irrational number.
108. Prove that √2 is an irrational number.
109. Prove that 7 -√5 is an irrational number
110. Show that there is no positive integer ‘n’ for which

√(n -1) + √(n + 1) is rational.

Multiple Choice Questions

Question. The decimal form of \( \frac{129}{2^2 \times 5^7 \times 7^5} \) is
(a) terminating
(b) non-terminating repeating
(c) non-terminating non-repeating
(d) none of the above
Answer: (b) non-terminating repeating

Question. HCF of 8, 9, 25 is
(a) 8
(b) 9
(c) 25
(d) 1
Answer: (d) 1

Question. Which of the following is not irrational?
(a) \( (2 - \sqrt{3})^2 \)
(b) \( (\sqrt{2} + \sqrt{3})^2 \)
(c) \( (\sqrt{2} - \sqrt{3})(\sqrt{2} + \sqrt{3}) \)
(d) \( \frac{2\sqrt{7}}{7} \)
Answer: (c) \( (\sqrt{2} - \sqrt{3})(\sqrt{2} + \sqrt{3}) \)

Question. The product of a rational and irrational number is
(a) rational
(b) irrational
(c) both of above
(d) none of above
Answer: (b) irrational

Question. The sum of a rational and irrational number is
(a) rational
(b) irrational
(c) both of above
(d) none of above
Answer: (b) irrational

Question. The product of two different irrational numbers is always
(a) rational
(b) irrational
(c) rational or irrational
(d) none of above
Answer: (c) rational or irrational

Question. The sum of two irrational numbers is always
(a) irrational
(b) rational
(c) rational or irrational
(d) one
Answer: (a) irrational

Question. 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) \( 3q, 3q + 1, 3q + 2 \)

Question. The product of three consecutive positive integers is divisible by
(a) 4
(b) 6
(c) no common factor
(d) only 1
Answer: (b) 6

Question. The set \( A = \{0, 1, 2, 3, 4, ...\} \) represents the set of
(a) whole numbers
(b) integers
(c) natural numbers
(d) even numbers
Answer: (a) whole numbers

Question. Which number is divisible by 11?
(a) 1516
(b) 1452
(c) 1011
(d) 1121
Answer: (b) 1452

Question. 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) \( \frac{x}{y} \)
Answer: (b) \( y \)

Question. 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) 17
Explanation: Algorithm \( 398 – 7 = 391; 436 – 11 = 425; 542 – 15 = 527 \).
\( \therefore \) HCF of 391, 425, 527 = 17

Question. There are 312, 260 and 156 students in class X, XI and XII respectively. Buses are to be hired to take these students to a picnic. Find the maximum number of students who can sit in a bus if each bus takes equal number of students
(a) 52
(b) 56
(c) 48
(d) 63
Answer: (a) 52
Explanation: HCF of 312, 260, 156 = 52

Question. 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) 36 minutes
Explanation: LCM of 18 and 12 = 36

Question. Express 98 as a product of its primes
(a) \( 2^2 \times 7 \)
(b) \( 2^2 \times 7^2 \)
(c) \( 2 \times 7^2 \)
(d) \( 2^3 \times 7 \)
Answer: (c) \( 2 \times 7^2 \)

Question. Three farmers have 490 kg, 588 kg and 882 kg of wheat respectively. Find the maximum capacity of a bag so that the wheat can be packed in exact number of bags.
(a) 98 kg
(b) 290 kg
(c) 200 kg
(d) 350 kg
Answer: (a) 98 kg
Explanation: HCF of 490, 588, 882 = 98 kg

Question. For some integer p, every even integer is of the form
(a) \( 2p + 1 \)
(b) \( 2p \)
(c) \( p + 1 \)
(d) \( p \)
Answer: (b) \( 2p \)

Question. For some integer p, every odd integer is of the form
(a) \( 2p + 1 \)
(b) \( 2p \)
(c) \( p + 1 \)
(d) \( p \)
Answer: (a) \( 2p + 1 \)

Question. \( m^2 – 1 \) 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) an odd integer

Question. If two positive integers A and B can be expressed as \( A = xy^3 \) and \( B = x^4y^2z \); x, y being prime numbers, the LCM (A, B) is
(a) \( xy^2 \)
(b) \( x^4y^2z \)
(c) \( x^4y^3 \)
(d) \( x^4y^3z \)
Answer: (d) \( x^4y^3z \)

Question. The product of a non-zero rational and an irrational number is
(a) always rational
(b) rational or irrational
(c) always irrational
(d) zero
Answer: (c) always irrational

Question. If two positive integers A and B can be expressed as \( A = xy^3 \) and \( B = x^4y^2z \); x, y being prime numbers then HCF (A, B) is
(a) \( xy^2 \)
(b) \( x^4y^2z \)
(c) \( x^4y^3 \)
(d) \( x^4y^3z \)
Answer: (a) \( xy^2 \)

Question. The largest number which divides 60 and 75, leaving remainders 8 and 10 respectively, is
(a) 260
(b) 75
(c) 65
(d) 13
Answer: (d) 13

Question. The least number that is divisible by all the numbers from 1 to 5 (both inclusive) is
(a) 5
(b) 60
(c) 20
(d) 100
Answer: (b) 60
Explanation: LCM of 2, 3, 4, 5 = 60

Question. 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) 840

Question. The decimal expansion of the rational number \( \frac{14587}{250} \) will terminate after:
(a) one decimal place
(b) two decimal places
(c) three decimal places
(d) four decimal places
Answer: (c) three decimal places

Question. The decimal expansion of the rational number \( \frac{97}{2 \times 5^4} \) will terminate after:
(a) one decimal place
(b) two decimal places
(c) three decimal places
(d) four decimal places
Answer: (d) four decimal places

Question. The product of two consecutive natural numbers is always:
(a) prime number
(b) even number
(c) odd number
(d) even or odd
Answer: (b) even number

Question. If the HCF of 408 and 1032 is expressible in the form \( 1032 \times 2 + 408 \times p \), then the value of p is
(a) 5
(b) –5
(c) 4
(d) –4
Answer: (b) –5
Explanation: HCF of 408 and 1032 is 24, \( \therefore 1032 \times 2 + 408 \times (–5) \)

Question. 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) odd

Question. 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) less than 7

Question. \( (6 + 5\sqrt{3}) – (4 – 3\sqrt{3}) \) is
(a) a rational number
(b) an irrational number
(c) a natural number
(d) an integer
Answer: (b) an irrational number

Question. If HCF (16, y) = 8 and LCM (16, y) = 48, then the value of y is
(a) 24
(b) 16
(c) 8
(d) 48
Answer: (a) 24

Question. According to the fundamental theorem of arithmetic, if T (a prime number) divides \( b^2 \), b > 0, then
(a) T divides b
(b) b divides T
(c) \( T^2 \) divides \( b^2 \)
(d) \( b^2 \) divides \( T^2 \)
Answer: (a) T divides b

Question. The number ‘\( \pi \)’ is
(a) natural number
(b) rational number
(c) irrational number
(d) rational or irrational
Answer: (c) irrational number

Question. If LCM (77, 99) = 693, then HCF (77, 99) is
(a) 11
(b) 7
(c) 9
(d) 22
Answer: (a) 11

Question. Euclid’s division lemma states that for two positive integers a and b, there exist unique integer q and r such that \( a = bq + r \), where r must satisfy
(a) \( a < r < b \)
(b) \( 0 < r \le b \)
(c) \( 1 < r < b \)
(d) \( 0 \le r < b \)
Answer: (d) \( 0 \le r < b \)

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