Chapter 1: Real Numbers – Chapter-Wise Important Questions and Answers | Class 10 Mathematics CBSE Solutions

CBSE Class 10 Mathematics Chapter 1: Real Numbers Important Questions with Solutions and Answers

Chapter 1. Real Numbers

SECTION = A

Question: 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: (A) always irrational .

[Example : is a irrational; number.]

Question: The least number that is divisible by all the numbers from 1 to 10 (both inclusive) is :

(a) 10 (b) 100 (c) 504 (d) 2520

Answer: (d) 2520

[ We have, 2 = 2 × 1 , 3 = 3 × 1 , 4 = 2 × 2 , 5 = 1 × 5 , 6 = 2 × 3 , 7 = 7 × 1 , 8 = 2 × 2 × 2 = 2³ , 9 = 3 × 3 = 3² , 10 = 2 × 5

LCM (1 to 10) = 1 × 2³ × 3² × 5 × 7 = 8 × 9 × 35 = 2520 ]

Question: If the LCM of and 18 is 36 and the HCF of and 18 is 2 , then

(a) 2 (b) 3 (c) 4 (d) 1

Answer: (c) 4

[ Here, ,

We have,

]

Question: If the HCF of 65 and 117 is expressible in the form , then the value of is :

(a) 4 (b) 2 (c) 1 (d) 3

Answer: (b) 2

[ Since, 65 = 5 × 13 and 117 = 3 × 3 × 13

HCF (70 , 125) = 13

A/Q , ]

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

(a) 13 (b) 65 (c) 875 (d) 1750

Answer: (a) 13 .

[Since, 70 – 5 = 65 = 5 × 13 and 125 – 8 = 117 = 3 × 3 × 13

HCF (70 , 125) = 13 ]

Question: If two positive integers and are written as and are prime numbers, then HCF is :

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

Answer: (b) .

[ HCF = Product of the smallest power of each common prime factor in the numbers .

HCF ]

Question: If two positive integers and can be expressed as and are prime numbers, then HCF is :

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

Answer: (a)

[ HCF = Product of the smallest power of each common prime factor in the numbers .

Here, and

HCF]

Question: The LCM of 36 , 72 and 120 is :

(a) 306 (b) 360 (c) 380 (d) 300

Answer: (b) 360

[ We have, , and

]

Question: If LCM (91 , 26) = 182 , then HCF(91 , 26) =

(a) 13 (b) 26 (c) 7 (d) 9

Answer: (a) 13

[We have, ]

Question: 120 can be expressed as a product of its prime factors as : [CBSE 2020 basic]

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

Answer: (d)

[ We have , ]

Question: The H.C.F of 8 , 9 , 25 is :

(a) 8 (b) 9 (c) 25 (d) 1

Answer: (d) 1

[ We have , ; ;

HCF (8 , 9 , 25) = 1 ]

Question: Which of the following is an irrational number ? [SEBA 2020]

(a) 0.142857142857142857…………… (b) (c) π (d)

Answer: (c) π

Question: Which of the following is not irrational ?

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

Answer: (c)

[ We have, (c) is not irrational ]

Question: Given three statement such as :

(i) The sum or difference of a rational and an irrational number is irrational .

(ii) The product and quotient of a non-zero rational and irrational number is irrational .

(iii) The product of the two numbers is not equal to the product of their HCF and LCM .

(iv) The LCM is equal to the product of the greatest power of each common prime factor in the numbers.

(a) (i) , (ii) and (iii) are correct .

(b) (i) , (ii) and (iv) are correct .

(d) (ii) , (iii) and (iv) are correct .

Answer: (d) (ii) , (iii) and (iv) are correct .

Question: Which number is not divisible by 11 ?

(a) 253 (b) 1771 (c) 286 (d) 91

Answer: (d) 196

[ We have, 253 = 11×23 ; 1771 = 7 × 11 × 23 , 286 = 2 × 11 × 13 ; 91 = 7 × 13 ]

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: (a) 260

[ We have, 60 – 8= 52 = 2 × 2 × 13 ; 75 - 10 = 65 = 5 × 13

LCM (52 , 65) = 2 × 2 × 5 × 13 = 260 ]

Question: If LCM (91 , 26) = 182 , then HCF (91 , 26) is :[SEBA 2016]

(a) 13 (b) 26 (c) 7 (d) 9

Solution: (a) 13

[ We have, ]

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: (a) les than 7 .

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

[ A/Q , HCF (16 ,y) × LCM (16 , y) = 16 × y

⇒ 16 × y = 8 × 48

⇒ y = 8 × 3 = 24 ]

Question: Find the least number of 3 digits , that will gives us remainder of 9 when divided by 2 and 5 respectively .

(a) 121 (b) 141 (c) 110 (d) 109

Solution: (d) 109

[ We have , 109 = 10 × 10 + 9 ]

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

(a) 9 : 1 (b) 4 : 3 (c) 11 : 1 (d) 12 : 1

Answer: (d) 12 : 1

[ We have , 5 = 1 × 5 ; 15 = 3 × 5 ; 20 =

HCF (5 , 15 , 20) = 5

LCM (5 , 15 , 20) =

So, ]

Question: HCF of 52× 42 and 35 × 65 is :

(a) 52 × 35 (b) 5 × 33 (c) 65 × 32 (d) 7 × 13

Answer: (d) 7 × 13

[ We have, 52 × 42 = 2 × 2 × 13 × 2 × 3 × 7 =

and 35 × 65 = 5 × 7 × 5 × 13 = 52 × 7 × 13

HCF ( 52 × 42 , 35 × 65) = 7 × 13 ]

Question: Which one of the following is a rational number ? [SEBA 2018]

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

Answer: (d)

[ We have, ]

Question: The smallest number by which should be multiplied so as to get rational number is : [ SEBA 2017]

(a) (b) (c) (d) 3

Answer: (c)

[ We have, ]

Filled in the blanks :

Question: The HCF of two number is 27 and their LCM is 162 , if one of the number is 54 , then the other number is ......................................... .

Answer: 81

[ The other number ]

Question: If product of two numbers is 2366 and their LCM is 26 , then their HCF is ...............................

Answer: 91

[ We have, ]

Question: The HCF and LCM of two numbers are 33 and 264 respectively , When the first number is completely divided by 2 and the quotient is 33 , then other number is...................................... .

Answer: 132

[ We have ,

Other number ]

Question: If the prime factorisation of a natural number is , then number is.........................

Answer: 8232

[ We have , ]

Question: is ( irrational / a rational number ) .

Answer: a rational number .

[ We have , is a rational ]

Answers Following The Question :

Question: Find the LCM and HCF of 6 and 20 by prime factorization method .

Solution: We have, and

and .

Question: Express the number 0.104 in the form of rational number .

Solution: We have ,

is the form of rational number .

Question: Given that HCF(306 , 657) = 9 , find LCM (306 , 657) .

Solution: We have ,

Question: The LCM of two number is 182 and their HCF is 13 . If one of the numbers is 26 , find the other . [CBSE 2020 standard]

Solution: We have , LCM × HCF one number × other number

The other number

Question: Find HCF of 1001 and 385 .

Solution: We have,

and

Question: Find the LCM of the two digit smallest prime and smallest odd composite natural number .

Solution: The two digit smallest prime number is 11 and the smallest odd composite number is 15 .

So,

Question: Decompose 32760 into prime factors .

Solution: We have ,

Question: What is the HCF of smallest prime number and the smallest composite number ? [CBSE 2018]

Solution: The smallest prime number is 2 and the smallest composite number is 4 .

So,

Question: Find a rational number between and

Solution: We have , and

Thus , the rational number between and is .

Question: Find one irrational number between and .

Solution: We have , and

Thus , the one irrational number between and is 1.21021002100021………….. .

The decimal expansion of the rational number will terminate after 4 places of decimals .

Question: If HCF(336 , 54) = 6 , find LCM (336 , 54) . [CBSE 2019]

Solution: We have ,

Section = II

Case study based questions are compulsory . Attempt any four sub parts of each question .

Each subpart carries 1 marks.

Question: 36 ducks , 72 hens and 120 goats have to be taken across a river. There is only one boat which will have to make many trips in order to do so . The lazy boatman has his own conditions for transporting them . He insists that he will take the same number of animals in every trip and they have to be of the same kind. He will naturally like to take the largest possible number each time.

(a) Can you tell how many animals went in each trip ?

(i) 12 (ii) 13 (iii) 14 (iv) 15

(b) Given 36 , 72 and 120 are three number, then the HCF and LCM is ............................. .

(i) less than 63 × 72 × 12 (ii) greater than 63 × 72 × 12

(iii) equal to 63 × 72 × 12 (iv) not equal to 63 × 72 × 12

(i) 29 × 53 (ii) 53 (iii) 29 (iv) 1

(d) What is the LCM of 36 , 72 and 120 ?

(i) 620 (ii) 720 (iii) 360 (iv) 260

(e) The ratio of the two number is 3 : 7 . If the HCF is 13 , then the number are :

(i) 29 , 91 (ii) 39 , 72 (iii) 39 , 92 (iv) 29 , 82

Answer: (a) (i) 12 .

(b) (iv) not equal to 63 × 72 × 12 .

(d) (iii) 360 .

(e) (iii) 39 , 92

[ let the two number are and .

and

HCF

and ]

SECTION = B

Question: Find the LCM and HCF of 120 and 144 by using Fundamental theorem of Arithmetic . [CBSE2012]

Solution: We have ,

and

Question: Check whether can end with the digit 0 for any natural number .

Solution: We have,

Prime factors of are only 2 and 3 .

So, the prime factors of does not contain , where and are positive integers .

Therefore, does not end with the digit 0 .

Question: Prove that is irrational . [SEBA 2016]

Solution: Let us assume , to the contrary that is rational .

We can find co-prime and ( ) such that

Since, 2 , and are integers , is rational and so, is rational . But this contradicts the fact that is irrational . So , we conclude that is irrational .

Question: Find the largest number that will divides 398 , 436 and 542 leaving remainders 7 , 11 and 15 respectively .

Solution: We have , 398 - 7 = 391 = 17 × 23 , 436 - 11 = 425 = 5 × 5 × 17 and 542 - 15 = 527 = 17 × 31

HCF(391 , 425 , 527) = 17

Therefore, 17 is the largest number that will divide given numbers.

Question: Show that cannot end with the digit 0 or 5 for any natural number n.

Solution: We have,

The prime factorization of contains only the primes 2 and 3. It has no factor of 5 at all.

Since, is never divisible by 5, it can never end in 0 or 5.

Thus, cannot end with the digit 0 or 5 for any natural number n.

Question: On 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?

Solution: We have, , and 45=3×3×5=3²×5

The minimum distance each should walk is 2520 cm, so that each covers the distance in complete steps.

Question: Express 5050 as product of its prime factors . Is it unique ?

Solution: We have,

It is not unique .

Question: Check whether can end with the digit 0 for any natural number .

Solution: Since,

Prime factors of are only 2 and 3 .

So, the prime factors of does not contain , where and are positive integers .

Therefore, does not end with the digit 0 .

Question: Find the HCF and LCM of 12 , 15 and 21 ,using the prime factorization method .

Solution: We have , , and

and

Question: Prove that is irrational .

Solution: Let us assume , to the contrary that is rational .

We can find co-prime and ( ) such that,

Since and are integers , is rational and so, is rational .

But this contradicts the fact that is irrational . So, is irrational .

Question: An army contingent of 616 members is to march behind an army band of 32 members in a parade . The two groups are to march in the same number of columns . What is the maximum number of columns in which they can march ?

Solution: The maximum number of column HCF (616 , 32)

Using Euclid’s division algorithm , we have

The HCF (616 , 32) is 8 .

Thus , the maximum number of column is 8 .

Question: Write the smallest number which is divisible by both 306 and 657 . [CBSE 2019]

Solution: We have , 306 = 2 × 3 × 3 × 17

and 657 = 3 × 3 × 73

HCF of 306 and 657 is 9 .

SECTION = C

Question: There is a circular path around a sports field . Sonia takes 18 minutes to drive one round of the field , while Ravi 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 ?

Solution: The required number of minutes is LCM (18 , 12) .

We find the LCM by prime factorization method ,

and

Therefore, the LCM (18 , 12)

Hence, Sonia and Ravi will meet again at the starting point after 36 minutes .

Question: The soldiers in a regiment can be stood in some raws consisting of 15 , 20 or 25 numbers of soldiers . Find the least number of soldiers in the regiment .

Solution: We need to determine the LCM of the numbers 15, 20, and 25 .

( Because the regiment can arrange soldiers in rows consisting of 15 , 20 and 25 .)

We have , , and

LCM

So, the least number of soldiers in the regiment is 300 .

Question: Find LCM and HCF of 6 , 72 and 120 by prime factorization method . Is HCF × LCM of three numbers equal to the product of the three numbers ?

Solution: We have,

and

LCM (6 , 72 , 120)

HCF (6 , 72 , 120)

LCM (6 , 72 , 120 ) × HCF (6 , 72 , 120 )

So, the product of three numbers is not equal to the product of their HCF and LCM .

Question: Find the LCM and HCF of 144 , 112 and 418 by prime factorization .

Solution: We have,

and

LCM (144 , 112 , 418)

HCF (144 , 112 , 418)

Question: Prove that is irrational .

Solution: Let us assume , to the contrary that is rational .

So, we can find integers and () such that , [ and are co-prime]

Therefore , is divisible by 3 and so is also divisible by 3 .

let , , for some integer .

From and , we get

Therefore , is divisible by 3 and so is also divisible by 3 .

So, and have at least 3 as a common factor .

But this contradicts the fact that and are co-prime . So, is irrational .

Question: Prove that is irrational .

Solution: Let, us assume to the contrary that is rational . There exists co-prime integers and () such that

[Squaring both side]

Therefore , is divisible by 5 . So, is also divisible by 5 .

Let , for some integer .

From and , we get

Therefore , is divisible by 5 . So, is also divisible by 5 .

Therefore, and have at least as a common factor . But this contradicts the fact that and are co-prime . This contradiction has arisen because of our incorrect assumption that is rational . So, we conclude that is irrational .

Question: Prove that is an irrational .

Solution: Let us assume , to the contrary , that is rational .

We can find co-prime and ( ) such that

–

Since , 2 , and are integers , is rational and so, is rational . But this contradicts the fact that is irrational . So , is irrational .

Question: Given that is an irrational , prove that is an irrational number . [CBSE 2018]

Solution: Let us assume , to the contrary that is rational .

We can find co-prime and ( ) such that,

Since 3 , and are integers , is rational and so, is rational .

Given is irrational . So, is irrational .

Question: Prove that is irrational and hence show that is also irrational .

Solution: Let us assume , to the contrary that is rational .

So, we can find integers and () such that [ and are co-prime]

Therefore , is divisible by 5 and so is also divisible by 5 .

let , , for some integer .

From and , we get

Therefore , is divisible by 5 and so is also divisible by 5 .

So, and have at least 5 as a common factor .

But this contradicts the fact that and are co-prime . So, is irrational .

Since is an irrational number . Therefore , is also irrational .

Question: Find HCF and LCM of 404 and 96 and verify that HCF × LCM Product of the two given numbers . [SEBA2019 , CBSE 2018]

Solution: We have, and

HCF (96 , 404)

LCM (96 , 404)

HCF (96 , 404) × LCM (96 , 404)

Verified.

Question: Rajesh has two vessels containing 720 ml and 405 ml of milk respectively . Milk from these containers is poured into glasses of equal capacity to their brim . Find the minimum number of glasses that can be filled .

Solution: We have ,

and

Therefore, the number of glasses is 45 .

Question: Find the LCM and HCF of 510 and 92 . Verify that LCM × HCF = product of the two numbers .

Solution: We have , and

HCF(510 , 92) and LCM(510 , 92)

Now ,

Verified .

Question: Prove that is irrational .

Solution: let us assume , to the contrary that is rational .

We can find co-prime and ( ) such that ,

Since, 2 , and are integers , is rational and so, is rational . But this contradicts the fact that is irrational . So, is irrational .

Posted 5 years ago

Chapter 1: Real Numbers – Chapter-Wise Important Questions and Answers | Class 10 Mathematics CBSE Solutions

CBSE Class 10 Mathematics Chapter 1: Real Numbers Important Questions with Solutions and Answers

Chapter 1. Real Numbers

SECTION = A

Answers Following The Question :

Section = II

SECTION = B

SECTION = C

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