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Real Numbers Class 10 MCQs Questions with Answers
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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
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
Question 3.
A number \(\frac{p}{q}\), when expressed in decimal form, terminates after 7 digits, then factors of q are of the form xm 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 xm x yn
x+y=2+5=7
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 2m x 5n
(b) The prime factorisation of q is not of the form 2m 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 2m x 5n“, where m and n are non-negative integers
Question 5.
If – 2 ≤ x ≤ 2, which of the following is not true for x:
(a) x can take only 3 values if x is a natural number.
(b) if x is an irrational number, then only one value of x is possible.
(c) if x is a non-negative integer, then only 3 values of x are possible.
(d) if x is a rational number, infinite values of x are possible.
Answer
Answer: (d) if x is a rational number, infinite values of x are possible.
– 2 ≤ x ≤ 2
Question 6.
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
Question 7.
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)
Question 8.
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) = \(\frac{306 × 657}{9}\)
Question 9.
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
Question 10.
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
Question 11.
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.
Question 12.
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
Question 13.
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)
Question 14.
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.
Question 15.
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.
Question 16.
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
Question 17.
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
Question 18.
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
Question 19.
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.
Question 20.
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
Question 21.
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
= \(\frac{167}{90}\)
Question 22.
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
Question 23.
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
Question 24.
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
Question 25.
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
Question 26.
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.
Question 27.
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
Question 28.
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
Question 29.
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
Question 30.
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
Question 31.
The factor tree of a number has some unknowns, find the values of a, b, c, d, e,f and g.
Answer
Answer: a = 277720, b = 6930, c = 3465, d = 3, e = 385, f = 77, g = 11
Question 32.
Which of the following is a rational number ?
(a) 
(b) 
(c) 
(d) 
Answer
Answer:
Question 33.
\(\frac{91}{625}\) ,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
Question 34.
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.
Question 35.
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
Question 36.
\(\frac{91}{625}\) ,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
Question 37.
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.
Question 38.
The least positive integer divisible by 20 and 24 is
(a) 360
(b) 120
(c) 480
(d) 240
Answer
Answer: (b) 120
Question 39.
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
Question 40.
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
Question 41.
The product of three consecutive integers is divisible by
(a) 5
(b) 6
(c) 7
(d) none of these
Answer
Answer: (b) 6
Question 42.
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
Question 43.
The smallest composite number is:
(a) 1
(b) 2
(c) 3
(d) 4
Answer
Answer: (c) 3
Question 44.
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
Question 45.
The smallest composite number is:
(a) 1
(b) 2
(c) 3
(d) 4
Answer
Answer: (c) 3
Question 46.
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
Question 47.
HCF of 8, 9, 25 is
(a) 8
(b) 9
(c) 25
(d) 1
Answer
Answer: (d) 1
Question 48.
√7 is
(a) An integer
(b) An irrational number
(c) A rational number
(d) None of these
Answer
Answer: (b) An irrational number
Question 49.
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
Question 50.
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
Question 51.
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.
Question 52.
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
Question 53.
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
Question 54.
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
Question 55.
\(\frac { 1 }{ \sqrt { 3 } }\) is –
(a) A rational number
(b) An irrational number
(c) A whole number
(d) None of these
Answer
Answer: (b) An irrational number
Question 56.
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
Question 57.
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
Question 58.
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
Question 59.
Which number is divisible by 11?
(a) 1516
(b) 1452
(c) 1011
(d) 1121
Answer
Answer: (b) 1452
Question 60.
For positive integers a and 3, 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
Answer: (a) 0 < r < 3
Question 61.
The decimal expansion of the rational number \(\frac { 47 }{ { 2 }^{ 3 }{ 5 }^{ 2 } }\) will terminate after:
(a) one decimal place
(b) two decimal places
(c) three decimal places?
(d) more than three decimal places
Answer
Answer: (c) three decimal places
Question 62.
The sum of a rational and an irrational number is
(a) Rational
(b) Irrational
(c) Both (a) and (c)
(d) Either (a) or (b)
Answer
Answer: (b) Irrational
Fill in the blanks:
1. Given any two positive integers a and b, there exist unique integers q and r satisfying a = bq + r and ______
Answer
Answer: 0<r<b
2. A rational number \(\frac{p}{q}\) will have a terminating decimal representation, if prime factors of q are of the form _____
Answer
Answer: where m and n are positive integers.
3. Rational number is a number which can be expressed in the form \(\frac{p}{q}\), q and p, q are ______
Answer
Answer: integers
4. If p is a prime number then √p is ________
Answer
Answer: irrational number
5. Every _________ can be expressed as a product of prime numbers.
Answer
Answer: composite number
Say true or false:
Question 1.
HCF of two consecutive natural numbers n and n +1 is n.
Answer
Answer: false
HCF of two consecutive natural numbers is 1
Question 2.
Every composite number can be expressed as a product of primes.
Answer
Answer: true
Question 3.
HCF of three numbers a, b, c; when multiplied by their LCM gives the product abc
Answer
Answer: false
this property is true only for 2 numbers.
Question 4.
The number 6″ can never have a value which ends with the digit 0.
Answer
Answer: true
Question 5.
Non-terminating and Non-recurring decimals are called Irrational Numbers.
Answer
Answer: true
Match the following:
Question 1.
| 1. 4.37\(\bar{2}\) | a. \(\frac{1093}{250}\) |
| 2. 4.3\(\bar{72}\) | b. \(\frac{787}{180}\) |
| 3. 4. \(\bar{372}\) | c. \(\frac{1456}{333}\) |
| 4. 4.372 | d. \(\frac{4329}{990}\) |
Answer
Answer:
| 1. 4.37\(\bar{2}\) | b. \(\frac{787}{180}\) |
| 2. 4.3\(\bar{72}\) | d. \(\frac{4329}{990}\) |
| 3. 4. \(\bar{372}\) | c. \(\frac{1456}{333}\) |
| 4. 4.372 | a. \(\frac{1093}{250}\) |
Question 2.
| a. it is the greatest positive integer which divides both the intergers. | Rational number | 4.37 |
| b. it is the lowest of their common multiples. | HCF | Of two consecutive numbers is 1 |
| c. terminating decimal expansion. | Irrational number | Of two consecutive numbers is their product. |
| d. non terminating non repeating decimal expansion. | LCM | 2.101001000110000……. |
Answer
Answer:
| a. it is the greatest positive integer which divides both the intergers. | HCF | Of two consecutive numbers is 1 |
| b. it is the lowest of their common multiples. | LCM | Of two consecutive numbers is their product. |
| c. terminating decimal expansion. | Rational number | 4.37 |
| d. non terminating non repeating decimal expansion. | Irrational number | 2.101001000110000……. |
Classify the following as Rational and Irrational:
Question 1.
√9
Answer
Answer: Rational
Question 2.
2 + √2
Answer
Answer: Irrational
Question 3.
\(\frac { π }{ 2 }\)
Answer
Answer: Irrational
Question 4.
0.666………
Answer
Answer: Rational
Question 5.
1.7324
Answer
Answer: Rational
Question 6.
√4 + √7
Answer
Answer: Irrational
Question 7.
0.7347896……..
Answer
Answer: Irrational
Question 8.
4.363636……
Answer
Answer: Rational
Question 9.
– 7 – √3
Answer
Answer: Irrational
Question 10.
– 3 + √16
Answer
Answer: Rational
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