Class 10-Real Number MCQs Multiple Choice Questions with Answers

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Class 10-Real Number MCQ Questions with Answers

Question 1.
In a seminar, the number of participants in English, German and Sanskrit are 45,75 and 135. Find the number of rooms required to house them, if in each room, the same number of participants are to be accommodated and they should be of the same language.
(a) 45
(b) 17
(c) 75
(d) 135

Answer

Answer: (b) 17
Since, in each room, the same number of participants, of the same language, are to be accommodated, their number in each room
HCF of 45, 75 and 135.
HCF (45, 75,135) = 15
Each room accommodates 15 participants
=> Total no. of rooms required for English = 45/15 = 3
Total no. of rooms required for German = 75/15 = 5
Total no. of rooms required for Sanskrit= 135/15 = 9
Total no. of rooms = 3 + 5 + 9 = 17

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Question 2.
If p = HCF (100,190) and q = LCM (100, 190); then p²q² is :
(a) 3.61 x 105
(b) 361 x 103
(c) 3.61 x 106
(d) 3.61 x 108

Answer

Answer:
pq = (HCF) (LCM) = Product of given numbers.
=> pq = 190×100 =19000
=> p²q² = 361 x 10⁶ = 3.61 x 10⁸

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Question 3.
A number \(\frac{p}{q}\), when expressed in decimal form, terminates after 7 digits, then factors of q are of the form x^m x yⁿ; the value of x + y should be :
(a) 7
(b) 14
(c) 9
(d) 26

Answer

Answer: (a) 7
\(\frac{p}{q}\) terminates after 8 digits
Decimal representation of \(\frac{p}{q}\) is terminating
=> Factors of q should be of the form x^m x yⁿ
x+y=2+5=7

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Question 4.
The condition to be satisfied by q, so that the rational number \(\frac{p}{q}\) has a non terminating decimal expansion is:
(a) The prime factorisation of q is of the form 2^m x 5ⁿ
(b) The prime factorisation of q is not of the form 2^m x 5ⁿ
(c) The prime factorisation of q is of the form m² x n⁵.
(d) The prime factorisation of q is not of the form m² x 5ⁿ.

Answer

Answer:
The prime factorisation is not of the form 2^m x 5ⁿ“, where m and n are non-negative integers

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Question 5.
The largest positive integer which divides 434 and 539 leaving remainders 9 and 12 respectively is:
(a) 9
(b) 108
(c) 17
(d) 539

Answer

Answer: (c) 17
Required number is the HCF of (434 – 9] and (539 -12)
= HCF of 425 and 527.
= 17

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Question 6.
There is a circular path around a field. Reema takes 22 minutes to complete one round while her friend Saina takes 20 minutes to complete the same. If they both start at the same time and move in the same direction, after how many minutes will they meet again at the starting
(a) 220
(b) 3.4
(c) 440
(d) 4.4

Answer

Answer: (a) 220
LCM of 20 and 22 = 220 (question state: after how many minutes will they meet)

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Question 7.
If HCF (306, 657) = 9, then LCM of 306 and 657 is:
(a) 1
(b) 22338
(c) 9
(d) 12

Answer

Answer: (b) 22338
LCM (306, 657) = \(\frac{306 × 657}{9}\)

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Question 8.
Euclid’s Lemma states that, for given positive integers a and b, there exist unique integers q and r, such that a = bq + r, where :
(a) 0 < r < b
(b) 0 ≤ r < b
(c) 0 < r ≤ b
(d) 0 ≤ r ≤ b

Answer

Answer: (b) 0 ≤ r < b

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Question 9.
A number which can be expressed in the form \(\frac{p}{q}\) where q ≠ 0 is a rational number is:
(a) p and q are co-prime numbers
(b) p and q are real numbers
(c) p and q are numbers
(d) p and q are integers

Answer

Answer: (d) p and q are integers

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Question 10.
In decimal expansion and representation, which of the following is not a rational numbers:
(a) terminating decimal
(b) terminates after 17 digits
(c) non-terminating repeating
(d) non-terminating non-repeating

Answer

Answer: (d) non-terminating non-repeating
In decimal representation, a rational number is either terminating or a non- terminating but repeating.

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Question 11.
If a positive integer ‘a’ is divided by 2, what can be the remainder ?
(a) 0 or 1
(b) 0,1 or 2
(c) 1 or 2
(d) Any positive number

Answer

Answer: (a) 0 or 1
5. When a is divided by 2, the remainder is either 0 or 1 using Euclid’s Lemma

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Question 12.
For some positive integer q, every even integer is of the form:
(a) q
(b) q + 1
(c)2q
(d)2q + l

Answer

Answer: (c)2q
(2q + r, 0≤r≤l but 2q +1 is odd)

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Question 13.
If a non-zero rational number is multiplied to an irrational number, we always get:
(a) an irrational number
(b) a rational number
(c) zero
(d) one

Answer

Answer: (a) an irrational number
The product of a rational (non-zero) and m irrational number is always an irrational number.

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Question 14.
HCF and LCM of two positive integers a and b satisfy a relationship, that:
(a) (HCF)(LCM) = \(\frac{a}{b}\)
(b) (HCF) (LCM) = 1
(c) (HCF)(LCM) = ab
(d) No defined relation

Answer

Answer: (c) (HCF)(LCM) = ab
7. (HCF) (LCM) = product of nos. a and b.

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Question 15.
If we write 0.9 as a rational number, we get:
(a) \(\frac{9}{10}\)
(b) 1
(c) \(\frac{1}{2}\)
(d) \(\frac{1}{10}\)

Answer

Answer: (b) 1
8. Let x = 0.9
10x =9.9
9x = 9
x = 1

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Question 16.
Write an irrational number between 2 and 3.
(a) 2.5
(b) 2.001
(c) 2.1333333456…
(d) 2.13

Answer

Answer:(c) 2.1333333456…
non terminating non repeating

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Question 17.
Which of the following are irrational whose sum and product are both rationals :
(a) √2 + 3,√2 – 3
(b) √2 + √3,√2 – √3
(c) 3 + √2, 3 – √2
(d) √2 + 1,√2 – 1

Answer

Answer: (c) 3 + √2, 3 – √2
3 + √2 + 3 – √2 = 6 and (3+√2) (3-√2) =7

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Question 18.
Find the value of x from the following such that x² is irrational but x⁴ is rational:
(a) √2
(b) 3√2
(c) 4√2
(d) 2

Answer

Answer: (c) 4√2
√2 is a irrational number.

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Question 19.
A rational number between 72 and 73 is:
(a) \(\frac{√2 + √3}{2}\)
(b) \(\frac{√2 – √3}{2}\)
(c) 1.5
(d) 1.8

Answer

Answer: (c) 1.5

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Question 20.
The rational number that corresponds
to 0.6+0.\(\bar{7}\)+0.4\(\bar{7}\) is :
(a) \(\frac{83}{90}\)
(b) \(\frac{7}{9}\)
(c) \(\frac{43}{90}\)
(d) \(\frac{167}{90}\)

Answer

Answer: (d) \(\frac{167}{90}\)
0.7 = 7/9
0.6 + 0.7 + 0.47
= 6/10 + 7/9 + 43/90
= \(\frac{167}{90}\)

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Question 21.
The greatest number which divides 87 and 97, leaving 7 as remainder is :
(a) 10
(b) 1
(c) 87 x 97
(d) 6300

Answer

Answer: (a) 10
Greatest number which divides 87 and 97, leaving 7 as remainder = HCF of 80 and 90

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Question 22.
The least number, which when divided by 10,14 and 18, leaves remainder 4, is :
a) 630
(b) 634
(c) 252
(d) 496

Answer

Answer: (b) 634
Required number is the LCM of 10, 14 and 18 (is 630) + 4
= 630 + 4
= 634

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Question 23.
The greatest number which divides 17, 28 and 34 leaving remainders 2, 3 and 4 respectively is:
(a) 5
(b) 24
(c) 1
(d) 17

Answer

Answer: (a) 5
Here 17 – 2 = 15, 28 – 3 = 25, 34 – 4 = 30 Required number is the HCF of 15,25 and 30 = 5

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Question 24.
72 litres of liquid A and 108 litres of liquid B are to be packed in containers of the same size. The minimum number of containers required are:
a) 36
(b) 18
(c) 5
(d) 10

Answer

Answer: (c) 5
For minimum number of containers required, each container should contain the maximum liquid.
Volume of each container should be the HCF of 72 and 108, which is 36. Number of containers required for liquid
A = \(\frac{72}{36}\) = 2
Number of containers required for liquid
B = \(\frac{108}{36}\) = 3
total = 5

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Question 25.
he HCF of two consecutive rational numbers x and x +1 is :
(a) x
(b)x + 1
(c)1
(d) 0

Answer

Answer: (c)1
HCF of two consecutive numbers is always 1.

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Question 26.
The HCF of a number which is neither prime nor composite and any other number x is:
(a) x
(b) x + 1
(c) 1
(d) Any number

Answer

Answer: (c) 1
As, 1 is neither prime nor a composite number, thus, required number is the HCF of 1 and x, which is 1

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Question 27.
Find q and r, if 12560 = 215 q + r.
(a) q = 58, r = 0
(b) q = 58, r = 10
(c) q = 58, r = 4
(d) q = 58, r = 90

Answer

Answer: (d) q = 58, r = 90
Using Euclid’s Lemma, 12560 when divided by 215, gives quotient as 58 and remainder as 90.
=> 12560 = 215 (58) + 90
=>q = 58 and r = 90

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Question 28.
Using Euclid’s Lemma, if d is the HCF of 1155 and 506, find ‘d’
(a) 11
(b) 143
(c) 77
(d) 66

Answer

Answer: (a) 11
1155=506×2 + 143
506 = 143 x 3 + 77
143 =77×1 + 66
77 =66 x 1 + 11
66 =11×6 + 0
Last divisor = 11 =>
11= HCF (66,11) =
HCF (1155,506)
=>d = 11

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Question 29.
An army contingent of 616 members is to march behind an army band of 32 members in a parade. The two groups have to march in the same number of columns. Find the maximum number of columns in which they can march ?
(a) 32
(b) 60
(c) 40
(d) 8

Answer

Answer: (d) 8
Required number is the HCF of 616 and 32 616 = 32 x 19 + 8 32 =8 x 4 + 0
HCF (616,32) = 8

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Question 30.
Select the incorrect answer,
(2 + √5)(2-√5) is:
(a) a natural number
(b) a rational number
(c) a whole number
(d) an irrational number

Answer

Answer: (d) an irrational number
which is a natural number, a whole number and also a rational number.

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Question 31.
Which of the following rational numbers, in decimal form, terminates ?
(a) \(\frac{64}{455}\)
(b) \(\frac{77}{210}\)
(c) \(\frac{31}{200}\)
(d) \(\frac{29}{343}\)

Answer

Answer: (c) \(\frac{31}{200}\)
(a) \(\frac{64}{455}\)
= \(\frac{64}{5 × 7 × 13}\)
= non terminating
(b) \(\frac{77}{210}\)
= \(\frac{77}{7 × 2 × 3 x 5}\)
= non terminating
(c) \(\frac{31}{200}\)
= 31/2³ × 5²
= terminating
(d) \(\frac{29}{343}\)
= 29/7³
= non terminating

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Question 32.
\(\frac{91}{625}\) ,when written in decimal form 625 terminates; as factors of denominator are in the form m² x n⁵. This number will terminate after how many digits ?
(a) 3
(b) 2
(c) 1
(d) 4

Answer

Answer: (d) 4
= 91 x 16/10⁴
= 1456/10⁴ = 0.1456

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Question 33.
Select the incorrect answer,
(2 + √5)(2-√5) is:
(a) a natural number
(b) a rational number
(c) a whole number
(d) an irrational number

Answer

Answer: (d) an irrational number
which is a natural number, a whole number and also a rational number.

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Question 34.
The least positive integer divisible by 20 and 24 is
(a) 360
(b) 120
(c) 480
(d) 240

Answer

Answer: (b) 120

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Question 35.
The HCF and LCM of two numbers is 9 and 459 respectively. If one of the numbers is 27, then the other number is
(a) 459
(b) 153
(c) 135
(d) 150

Answer

Answer: (b) 153

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Question 36.
The largest number which divides 615 and 963 leaving remainder 6 in each case is
(a) 82
(b) 95
(c) 87
(d) 93

Answer

Answer: (c) 87

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Question 37.
The product of three consecutive integers is divisible by
(a) 5
(b) 6
(c) 7
(d) none of these

Answer

Answer: (b) 6

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Question 38.
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

Answer: (a) 840

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Question 39.
The smallest composite number is:
(a) 1
(b) 2
(c) 3
(d) 4

Answer

Answer: (c) 3

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Question 40.
For some integer p, every odd integer is of the form
(a) 2p + 1
(b) 2p
(c) p + 1
(d) p

Answer

Answer: (a) 2p + 1

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Question 41.
The smallest composite number is:
(a) 1
(b) 2
(c) 3
(d) 4

Answer

Answer: (c) 3

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Question 42.
A lemma is an axiom used for proving
(a) other statement
(b) no statement
(c) contradictory statement
(d) none of these

Answer

Answer: (a) other statement

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Question 43.
HCF of 8, 9, 25 is
(a) 8
(b) 9
(c) 25
(d) 1

Answer

Answer: (d) 1

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Question 44.
√7 is
(a) An integer
(b) An irrational number
(c) A rational number
(d) None of these

Answer

Answer: (b) An irrational number

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Question 45.
The product of a rational and irrational number is
(a) rational
(b) irrational
(c) both of above
(d) none of above

Answer

Answer: (b) irrational

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Question 46.
A number when divided by 61 gives 27 as quotient and 32 as remainder. find the number
(a) 1967
(b) 1796
(c) 1679
(d) 1569

Answer

Answer: (c) 1679

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Question 47.
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

Answer: (b) a = bq + r , 0 = r < b.

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Question 48.
Two natural numbers whose difference is 66 and the least common multiple is 360, are:
(a) 120 and 54
(b) 90 and 24
(c) 180 and 114
(d) 130 and 64

Answer

Answer: (b) 90 and 24

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Question 49.
The product of a non zero rational and an irrational number is
(a) Always irrational
(b) Always rational
(c) Rational or irrational
(d) One

Answer

Answer: (a) Always irrational

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Question 50.
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

Answer: (a) 2q

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Question 51.
\(\frac { 1 }{ \sqrt { 3 } }\) is –
(a) A rational number
(b) An irrational number
(c) A whole number
(d) None of these

Answer

Answer: (b) An irrational number

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Question 52.
The largest number which divides 60 and 75, leaving remainders 8 and 10 respectively, is
(a) 260
(b) 75
(c) 65
(d) 13

Answer

Answer: (d) 13

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Question 53.
p is
(a) a rational number
(b) an irrational number
(c) both (a) & (b)
(d) neither rational nor irrational

Answer

Answer: (b) an irrational number

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Question 54.
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

Answer: (a) 17

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Question 55.
Which number is divisible by 11?
(a) 1516
(b) 1452
(c) 1011
(d) 1121

Answer

Answer: (b) 1452

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