Real Numbers - Bhange Coaching Classes, Nagpur

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Bhange Coaching ClassesX CBSE Maths - Objective Type Questions

Real Numbers1Chapter

Objective Type QuestionsA. Multiple Choice Questions

1. Euclids Division Lamma states that for any two positive integers a andb their exists unique integer q and r such that a = bq + r where r mustsatisfy.(A) 1 < r < b (B) 0 < r < b (C) 0 < r < b (D) 0 < r < b

2. The total number of factors of a prime even number is(A) 2 (B) 1 (C) 0 (D) 3

3. Given positive integers a, b there exists unique integers q and r satisfyinga = bq + r where(A) 0 > r < b (B) 0 < r < b (C) 0 < r < b (D) 0 > r > b

4. If n1, n2, n3 are three positive integers and l = LCM (n1, n2, n3),h = (HCF) (n1, n2, n3). Then which of the following condition holds good?(A) n1 x n2 x n3 = l x h (B) n1 x n3 = l x h(C) n1 x n2 = l x h (D) None of these

5. For some integer m, every even integer is of the form :(A) m (B) m + 1 (C) 2m (D) 2m + 1

6. For some integer q, every odd integer is of the form :(A) q (B) q + 1 (C) 2q (D) 2q + 1

7. What is the LCM of 10 and 3?(A) 3 (B) 10 (C) 30 (D) 1

8. HCF of 13 and 53 is ..........(A) 13 (B) 53 (C) 13 x 53 (D) 1

9. HCF of 26 and 91 is(A) 2 (B) 7 (C) 13 (D) 26

10. The HCF of the smallest composite number and the smallest prime numberis(A) 1 (B) 2 (C) 3 (D) 4

11. HCF of 52 × 32 and 35 × 53 is ............

(A) 53 × 35 (B) 5 × 33 (C) 53 × 32 (D) 52 × 32



12. If two positive integers ‘a’ and ‘b’ are written as a = x³y² and b = xy³,where x and y are prime numbers then HCF (a, b) is .........(A) xy (B) xy² (C) x³y³ (D) x2y2

13. The HCF and LCM of two numbers are 9 and 360 respectively. If onenumber is 45. Then other number is ......(A) 90 (B) 63 (C) 72 (D) 54

14. If HCF (26, 169) = 13 then the LCM (26, 169) will be.......(A) 26 (B) 52 (C) 338 (D) 13

15. LCM (x, 18) = 36 and HCF (x, 18) = 2 then x is ..........(A) 2 (B) 3 (C) 4 (D) 1

16. If two positive integers p & q can be expressed as p = ab² and q = a³b, a,b being prime no. then LCM (p, q) is,(A) ab (B) a²b² (C) a³b² (D) a³b³

17. If two positive integers a and b are written as a = x²y² and b = xy³, x, yare prime numbers, then HFC (a, b) is :(A) xy (B) xy² (C) x³y³ (D) x²y²

18. If a = 2³ x 3, b = 2 x 3 x 5, c = 3n x 5 and LCM (a, b, c) = 2³ x 3² x 5, then n =(A) 1 (B) 2 (C) 3 (D) 4

19. if HCF (n, 32) = 4 and LCM (n, 32) = 96 then n = ?(A) 3 (B) 4 (C) 8 (D) 12

20. If HCF of 65 & 117 is expresible in the form 65m – 117, then the value of m is(A) 4 (B) 2 (C) 1 (D) 3

21. The HCF of 45 and 105 is 15 then their LCM is :(A) 315 (B) 325 (C) 215 (D) 125

22. The decimal expansion of 147 will terminate after how many palces of 120

decimal?(A) 1 (B) 2 (C) 3 (D) will not terminate

23. 17 is a ...................... decimal.3125(A) terminating (B) non terminating repeating(C) non terminating and non repeating (D) none of these

24. After how many decimal places will the decimal expansion of 23457 2³x54

terminate.(A) 4 decimal places (B) 3 decimal places(C) 2 decimal places (D) 1 decimal place



25. The decimal expansion of 178 will terminate after how many places

of decimal(A) 1 (B) 2(C) 3 (D) will not terminate

26. 15 will terminate after ................. digits.1600(A) 2 (B) 4 (C) 6 (D) 8

27. The decimal expansion of the rational number 43 will ....... 24 x 53

(A) terminate after 4 decimal places (B) terminate after 3 decimal places(C) terminate after 2 decimal places (D) not terminate

28. The decimal expansion of the rational number 14587

will terminate 1250

after :(A) one decimal place (B) two decimal places(C) three decimal places (D) four decimal places

29. The number of decimal places after which the decimal expansion of the

rational number 23 will terminate, is 22 x 5(A) 2 (B) 3 (C) 9 (D) 7

30. The decimal expression of the rational no. is

(A) 0.115 (B) 1.15 (C) 11.5 (D) 0.0115

31. Termiating Decimal of 35 is : 4(A) 6.25 (B) 8.75 (C) 2.15 (D) 1.25

32. Which of the following is not an irrational number?(A) 2 (B) 3 (C) 9 (D) 7

33. The decimal representation of an irrational number is always...(A) terminating (B) terminating, repeating(C) non-terminating, repeating (D) Non-terminating, non repeating

34. The product of a non-zero rational and an irrational number is :(A) always irrational (B) always rational(C) rational or irrational (D) one

35. n² – 1 is divisible by 8 if it is -(A) an even integer (B) a natural number(C) an integer (D) an odd integer

232³.5²



36. The smallest number by which 27 should be multiplied so as to get arational number is(A) 27 (B) 3 3 (C) 3 (D) 3

37. Find a if a – 3 = 10 a

(A) 5, –2 (B) – 7, 7 (C) 7 , 7 (D) –5, 238. The positive root of 3x² + 6 = 9 is

(A) 3 (B) 4 (C) 5 (D) 739. For any positive integer n, [(n + 8) (n + 9), (n + 10)] is always divisible by

(A) only 3 (B) only 6 (C) both 3 & 6 (D) None of 3 & 640. A +ve integer when divided by 9, gives 7 as remainder; what will be the

remainder when (3n – 1) divided by 9?(A) 1 (B) 2 (C) 3 (D) 4

41. Which of the following can end with digit zero?(A) 6n (B) 4n (C) 6n x 4n (D) none of these

42. If n is a natural number, then 92n – 42n is always divisible by(A) 5 (B) 13 (C) both 5 and 13 (D) None of these[Hint : 92n – 42n is of the form a2n – b2n which is divisible by both a – b and a + b.So, 92n – 42n is divisible by both 9 – 4 = 5 and 9 + 4 = 13]

43. A product of greatest power of each prime factor involved in thenumber is :(A) LCM (B) HCF (C) both (A) & (B) (D) None of these

44. 6 x 5 x 4 x 3 x 2 x 1 + 5 is a ...........(A) Prime number (B) Co prime number(C) Composite number (D) Irrational number

45. Two +ve integers n1 and n2 have same prime factorisation then(A) n1 > n2 (B) n1 < n2 (C) n1 = n2 (D) n1 > n2

46. The sum of rational number and irrational number is always(A) rational (B) irrational (C) prime no. (D) composite no.

47. If x & y are both odd positive integers then x² + y² is(A) odd (B) even & divisible by 4(C) even but not divisible by 4 (D) either even or odd

48. Product of the smallest power of each common factor in the number is :(A) LCM (B) HCF(C) Both (A) & (B) (D) None of these

49. The number 313 – 310 is divisible by(A) 3, 13, 5 (B) 3, 10 (C) 2, 3, 13 (D) 2, 3, 10

50. The largest number which divides 70 and 125, leaving remainders 5and 8, respectively, is :(A) 13 (B) 65 (C) 875 (D) 1,750



Answer1. (C) 2. (A) 3. (B) 4. (D) 5. (C) 6. (D)

7. (C) 8. (D) 9. (C) 10. (B) 11. (D) 12. (B)

13. (C) 14.(C) 15. (C) 16. (C) 17.(B) 18. (B)

19. (D) 20. (B) 21. (A) 22. (C) 23. (A) 24. (A)

25. (C) 26. (C) 27. (A) 28. (D) 29. (A) 30. (A)

31. (B) 32. (C) 33. (D) 34. (A) 35. (D) 36. (C)

37. (A) 38. (C) 39. (C) 40. (B) 41. (D) 42. (C)

43. (A) 44. (C) 45. (C) 46. (B) 47. (C) 48. (B)

49. (C) 50. (A)

(1) CASE STUDY 1.To enhance the reading skills of grade X students, the school nominates youand two of your friends to set up a class library. There are two sections- sectionA and section B of grade X. There are 32 students in section A and 36 studentsin section B.

1. What is the minimum number of books you will acquire for the class library, sothat they can be distributed equally among students of Section A or Section B?a) 144 b) 128 c) 288 d) 272

B. Case Study



2. If the product of two positive integers is equal to the product of their HCF andLCM is true then, the HCF (32 , 36) isa) 2 b) 4 c) 6 d) 8

3. 36 can be expressed as a product of its primes asa) 22 x 32 b) 21 x 33 c) 23 x 31 d) 20 x 30

4. 7 x 11 x 13 x 15 + 15 is aa) Prime number b) Composite numberc) Neither prime nor composite d) None of the above

5. If p and q are positive integers such that p = ab² and q = a²b, where a, b areprime numbers, then the LCM (p, q) isa) ab b) a2b2 c) a3b2 d) a3b3

(2) CASE STUDY 2.A seminar is being conducted by an Educational Organisation, where theparticipants will be educators of different subjects. The number of participantsin Hindi, English and Mathematics are 60, 84 and 108 respectively

1. In each room the same number of participants are to be seated and all of thembeing in the same subject, hence maximum number participants that canaccommodated in each room area) 14 b) 12 c) 16 d) 18

2. What is the minimum number of rooms required during the event?a) 11 b) 31 c) 41 d) 21

3. The LCM of 60, 84 and 108 isa) 3780 b) 3680 c) 4780 d) 4680

4. The product of HCF and LCM of 60,84 and 108 isa) 55360 b) 35360 c) 45500 d) 45360

5. 108 can be expressed as a product of its primes asa) 23 x 32 b) 23 x 33 c) 22 x 32 d) 22 x 33



(3) CASE STUDY 3.A Mathematics Exhibition is being conducted in your School and one of yourfriends is making a model of a factor tree. He has some difficulty and asks foryour help in completing a quiz for the audience.Observe the following factor tree and answer the following:

1. What will be the value of x?a) 15005 b) 13915 c) 56920 d) 17429

2. What will be the value of y?a) 23 b) 22 c) 11 d) 19

3. What will be the value of z?a) 22 b) 23 c) 17 d) 19

4. According to Fundamental Theorem of Arithmetic 13915 is aa) Composite number b) Prime numberc) Neither prime nor composite d) Even number

5. The prime factorisation of 13915 isa) 5 x 113 x 132 b) 5 x 113 x 232 c) 5 x 112 x 23 d) 5 x 112 x 132

(4) CASE STUDY 4.Old age homes are mean for senior citizens who are unable to stay with theirfamilies or destitute. These old age homes have special medical facilities forsenior citizens such as mobile health care systems, ambulances nurses andprovision of well balanced meals.



Amita, Suneha and Raghav start preparing cards for greeting each person ofan old age home on new year. In order to complete one card, they take 10, 16and 20 min respectively.

(1) If all of them started together, after what time will they start preparing a newcard together.(a) 85 min (b) 80 min (c) 60 min (d) 90 min

(2) What is the common time to make card?(a) 1 min (b) 2 min (c) 4 min (d) None of these

(3) The smallest prime even number is(a) 1 (b) 3 (c) 2 (d) 4

(4) A largest positive integer that divic given two positive integers is called the(a) LCM (b) HCF(c) Do not say anything (d) None of the above

(5) Coprime number are those number which do not have any common factor otherthen(a) 0 (b) 1 (c) 2 (d) None of these

(5) CASE STUDY 5.Find the missing numbers a, b, c and din the given factor tree :

(1) The number a is(a) 3 (b) 7(c) 11 (d) 13

(2) The number b is(a) 5 (b) 7(c) 8 (d) 3

(3) The number c is(a) 7 (b) 13 (c) 11 (d) both (b) & (c)

(4) The number d is(a) 7 (b) 11 (c) both (b) & (d) (d) 13

(6) CASE STUDY 6.Traffic Light SignalIn our daily life we all see traffic lights. A traffic controller set the timmingsof traffic lights in such a way that all light are not green at the same time orspecially not in the rush hour. It may create problem in an hour because lights



are for few minutes only. So, he take the timmings of nearby places in samearea and calculate I cm of all traffic stops and he easily manage the traffic byincrease the duration or set at different times. There are two traffic lights on aparticular highway which shows green light on time of 90 seconds and 144seconds respectively.

(1) Find the HCF between two green lights.(a) 18 (b) 20 (c) 16 (d) 22

(2) Find the LCF between two green light.(a) 720 (b) 730 (c) 710 (d) 740

(3) Factor tree is used for determining the(a) HCF (b) LCM (c) prime factor (d) None of these

(4) Identify the correct option.

(a) HCF (a, b) x LCM (a, b) = a (b) HCF (a, b) = a b LCM (a, b) b(c) HCF (a, b) x LCM (a, b) = a – b (d) HCF (a, b) x LCM (a, b) = ab

(5) A number which do not have any factor other than 1, is(a) coprime number (b) prime number(c) coprime or prime number (d) None of the above

(7) CASE STUDY 7.Morning WalkIn a morning walk, Nirmaljeet, Puneet and Raju step off together, their stepsmeasuring 240 cm, 90 cm, 120 cm respectively. They want to go lor a juice shopfor a health issue, which is situated near by them.



(1) Find the minimum distance of shop from where they start to walk together, sothat one can cover the distance in complete steps?(a) 720 cm (b) 700 cm (c) 620 cm (d) 740 m

(2) Find the number of common steps cover by all of them to reach the juice shop.(a) 20 (b) 30 (c) 35 (d) 40

(3) If a and b are two number, then find the relation between LCM and HCF.(a) a

b = LCM (a, b) HCF (a, b) (b) a x b = LCM (a, b) x HCF (a, b)

(c) a x LCM (a, b) = b x HCF (a, b) (d) None of the above(4) A largest positive integer that divides given two positive integers is called

(a) HCF (b) LCM (c) Coprime (d) None of these(5) Factor tree is a chain of factors, which is represented in the form of a

(a) tree (b) division (c) flower (d) None of these

(8) CASE STUDY 8.Book SellerA book seller has 420 science stream books and 130 Arts stream books. Hewants to stack them in such a way that each stack has the same number andthey take up the least area ol the surface.

(1) If a number has no factors other than 1 and number itself is(a) composite (b) prime (c) do not say any thing (d) None of the above

(2) What is the maximum number of books that can be placed in each stackfor this purpose?(a) 10 (b) 14 (c) 12 (d) 15

(3) Which mathematical concept is used to solve the problem?(a) Prime factorisation method (b) Area of triangle(c) Arithmetic progression (d) None of the above

(4) If the book seller double the quantity, then the maximum number of booksthat can be placed in each stack is(a) remain same (b) double (c) triple (d) None of these

(5) Find the LCM of the given book streams.(a) 5450 (b) 5460 (c) 2730 (d) None of these



(9) CASE STUDY 9.

The basis of the Euclidean division algorithm is Euclid’s division lemma. Tocalculate the Highest Common Factor (HCF) of two positive integers a and bwe use Euclid’s division algorithm. HCF is the largest number which exactlydivides two or more positive integers. That means, on dividing both the integersa and b the remainder is zero.

(1) What is the Highest Common Factor (HCF) of 4052 and 12576?(a) 4 (b) 12 (c) 8 (d) 16

(2) State true or false : HCF (420, 272) = HCF (4052, 420)(a) True (b) False

(3) State true or false : Euclid’s Division algorithm can be extended for all integersexcept zero.(a) True (b) False

(4) State true or false : HCF (c, d) = HCF (d, r) where the symbol HCF (c, d) denotesthe HCF of c and d is equal to HCF of d and r.(a) True (b) False

(5) A proven statement used for proving another statement is called as(a) Algorithm (b) Lemma (c) Postulate (d) None of the above

(10) CASE STUDY 10.

Euclid’s division lemma algorithm has several applications related to findingproperties of numbers like every positive even integer is of the form 6q, andevery positive odd integer is of the form 6q + 1, where q is some integer.



(1) Any positive odd integer can be in which of the following form(s)?(a) 2q + 1 (b) 4q + 1 (c) 4q + 3 (d) All of the above

(2) Cube of any positive integer can be in which of the following form(s)?(a) 9q (b) 9q + 1 (c) 9q + 8 (d) All of the above

(3) Square of any positive integer can be in which of the following form(s)?(a) 3q (b) 3q + 1 (c) Both A & B (d) 3q + 2

(4) Any positive even integer can be in which of the following form(s)?

(a) 4q + 2 (b) 4q + 1 (c) 4q + 3 (d) None of the above

(5) Using Euclid’s division algorithm what is H.C.F of 420 and 130?(a) 5 (b) 10 (c) 15 (d) 20

(11) CASE STUDY 11.

A sweetseller has 420 kaju barfis and 130 badam barfis. He wants to stackthem in such a way that each stack has the same number, and they take up theleast area of the tray. He takes help of his friend for this who tolds that he canmake stacks of 10 for both kinds of barfi.

(1) Sweetseller friend might have used which of the method(s) to solve the givenproblem?(a) Hit and Trail (b) Highest Common Factor(c) Both A and B (d) Least Common Multiple.

(2) Which of the following is the most optimal method to solve the given problem?(a) Hit and Trail (b) Highest Common Factor(c) Least Common Multiple (d) None of the above

(3) If sweetseller has 272 kaju barfis and 148 badam barfis, what is the stack sizegiven that each stack has the same number, and they take up the least area ofthe tray?(a) 10 (b) 8 (c) 5 (d) 4

(4) State true or false : Euclid’s division algorithm is a technique to compute theHighest Common Factor (HCF) of two given positive integers.(a) True (b) False

(5) State true or false : HCF of two positive integers a and b is the largest positiveinteger d that divides both a and b.(a) True (b) False



(12) CASE STUDY 12.

According to Fundamental Theorem of arithmetic, Every composite number canbe expressed (factorised) as a product of primes, and this factorisation is unique,apart from the order in which the prime factors occur.

(1) State true or false? The prime factorisation of a natural number is unique,except for the order of its factors.(a) True (b) False

(2) Is a composite number?(a) Yes (b) No

(3) How is 156 expressed as a product of its prime factors?

(4) Highest Common Factor (HCF) of two numbers can be found out by using whichof the following method(s)?(a) Prime Factorisation (b) Euclid's division algorithm(c) Both (a) and (b) (d) Hit and trial.

(5) What is Least Common Multiple (LCM) of 96 and 404?(a) 9696 (b) 9600 (c) 9500 (d) None of the above

(13) CASE STUDY 13.

There is a circular path around a sports field. Sonia takes 18 minutes to driveone round of the field, while Ravi takes 12 minutes for the same. If they bothstart at the same point and at the same time, and go in the same direction theymeet again at the starting point after 36 minutes.

(a)(b)(c)(d)



(1) Which of the following concept(s) is used to find the time after which they willmeet again?(a) Least Common Multiple (b) Prime factorisation(c) Both (a) and (b) (d) Highest Common Factor

(2) What is the LCM of 6, 72, 120?(a) 360 (b) 48 (c) 300 (d) 420

(3) State true or false : Product of three numbers is equal to the product of theirHCF and LCM.(a) True (b) False

(4) If Sonia takes 6 minutes to drive one round of the field, while Ravi takes 20minutes for the same. If they both start at the same point and at the same time,and go in the same direction they meet again at the starting point after howmany miinutes?(a) 25 (b) 30 (c) 36 (d) 60

(5) State true or false : Product of three numbers is not equal to the product oftheir HCF and LCM.(a) True (b) False

Answer1. 1. c) 288 2. b) 4 3. a) 22 x 32 4. b) composite number 5. b) a2b2

2. 1. b) 12 2. d) 21 3. a) 3780 4. d) 45360 5. d) 22 x 33

3. 1. b) 13915 2. c) 11 3. b) 23 4. a) composite number 5. c) 5 x 112 x 234. 1. b) 80 min 2. b) 2 min 3. c) 2 4. b) HCF 5. b) 15. 1. a) 3 2. b) 7 3. d) both b & c 4. c) both b & d6. 1. a) 18 2. a) 720

3. c) prime factor 4. d) HCF (a, b) x LCM (a, b) = ab5. a) coprime number

7. 1. a) 720 cm 2. b) 30 3. b) a x b = LCM (a, b) x HCF (a, b)4. a) HCF 5. a) tree

8. 1. b) prime 2. a) 10 3. a) Prime factorisation method 4. b) double5. b) 5460

9. 1. a) 4 2. a) True 3. a) True 4. a) True 5. b) lemma10.1. d) all of above 2. d) all of above 3. c) Both a and b

4. a) 4q + 2 5. b) 1011.1. b) HCF 2. b) HCF 3. d) 4 4. a) True 5. a) True12.1. a) True 2. a) Yes 3. c) 2 x 2 x 3 x 13

4. c) both a) and b) 5. a) 969613.1. a) LCM 2. a) 360 3. b) False 4. d) 60 5. a) True

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