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R S AGGARWAL AND V AGGARWAL Solutions Mathematics Class 10 Chapter 1 Real Numbers

Chapter 1 – Real Numbers Exercise Ex. 1A


Question 1

What do you mean by Euclid’s division lemma?

Solution 1

For any two given positive integers a and b there exist unique whole numbers q and r such that  

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers 

Here, we call ‘a’ as dividend, b as divisor, q as quotient and r as remainder.

Dividend = (divisor R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbersquotient) + remainder

 


Question 2

A number when divided by 61 gives 27 as quotient and 32 as remainder. Find the number.

Solution 2

By Euclid’s Division algorithm we have:

Dividend = (divisor × quotient) + remainder

= (61 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers27) + 32 = 1647 + 32 = 1679

 


Question 4
By what number should 1365 be divided to get 31 as quotient and 32 as remainder?
Solution 3

By Euclid’s Division Algorithm, we have:

Dividend = (divisor R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers quotient) + remainder

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 4

Using Euclid’s algorithm, find the HCF of:

(i) 405 and 2520

(ii) 504 and 1188

(iii) 960 and 1575

Solution 4

(i)

On dividing 2520 by 405, we get

Quotient = 6, remainder = 90

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers2520 = (405 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers6) + 90

Dividing 405 by 90, we get

Quotient = 4,

Remainder = 45

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers405 = 90 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers4 + 45

Dividing 90 by 45

Quotient = 2, remainder = 0

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers90 = 45 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers2

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real NumbersH.C.F. of 405 and 2520 is 45

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

(ii) Dividing 1188 by 504, we get

Quotient = 2, remainder = 180

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers1188 = 504 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers2+ 180

Dividing 504 by 180

Quotient = 2, remainder = 144

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers504 = 180 × 2 + 144

Dividing 180 by 144, we get

Quotient = 1, remainder = 36

Dividing 144 by 36

Quotient = 4, remainder = 0

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real NumbersH.C.F. of 1188 and 504 is 36

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

(iii) Dividing 1575 by 960, we get

Quotient = 1, remainder = 615

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers1575 = 960 × 1 + 615

Dividing 960 by 615, we get

Quotient = 1, remainder = 345

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers960 = 615 × 1 + 345

Dividing 615 by 345

Quotient = 1, remainder = 270

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers615 = 345 × 1 + 270

Dividing 345 by 270, we get

Quotient = 1, remainder = 75

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers345 = 270 × 1 + 75

Dividing 270 by 75, we get

Quotient = 3, remainder =45

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers270 = 75 × 3 + 45

Dividing 75 by 45, we get

Quotient = 1, remainder = 30

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers75 = 45 × 1 + 30

Dividing 45 by 30, we get

Remainder = 15, quotient = 1

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers45 = 30 × 1 + 15

Dividing 30 by 15, we get

Quotient = 2, remainder = 0

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real NumbersH.C.F. of 1575 and 960 is 15

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 5

Show
that every positive integer is either even or odd.

Solution 5

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 6

Show
that any positive odd integer is of the form (6m + 1) or (6m + 3) or (6m +
5), where m is some integer.

Solution 6

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 7

Show
that any positive odd integer is of the form (4m + 1) or (4m + 3), where in
is some integer.

Solution 7

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Chapter 1 – Real Numbers Exercise Ex. 1B

Question 1

Using
prime factorization, find the HCF and LCM of:

36,
84

In
each case, verify that:

HCF
x LCM = product of given numbers.

Solution 1

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 2

Using
prime factorization, find the HCF and LCM of:

23,
31

In
each case, verify that:

HCF
x LCM = product of given numbers.

Solution 2

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 3

Using
prime factorization, find the HCF and LCM of:

96,
404

In
each case, verify that:

HCF
x LCM = product of given numbers.

Solution 3

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 4

Using
prime factorization, find the HCF and LCM of:

144,198

In
each case, verify that:

HCF
x LCM = product of given numbers.

Solution 4

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 5

Using
prime factorization, find the HCF and LCM of:

396,
1080

In
each case, verify that:

HCF
x LCM = product of given numbers.

Solution 5

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 6

Using
prime factorization, find the HCF and LCM of:

1152,
1664

In
each case, verify that:

HCF
x LCM = product of given numbers.

Solution 6

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 7

Using
prime factorization, find the HCF and LCM of:

8,
9, 25

Solution 7

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 8

Using
prime factorization, find the HCF and LCM of:

12,
15, 21

Solution 8

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 9

Using
prime factorization, find the HCF and LCM of:

17,
23, 29

Solution 9

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 10

Using
prime factorization, find the HCF and LCM of:

24,
36, 40

Solution 10

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 11

Using
prime factorization, find the HCF and LCM of:

30,
72, 432

Solution 11

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 12

Using
prime factorization, find the HCF and LCM of:

21,
28, 36, 45

Solution 12

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 13

The HCF of two numbers is 23 and their LCM is 1449. If one of the numbers is 161, find the other.

Solution 13

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 14

The
HCF of two numbers is 145 and their LCM is 2175. If one of the numbers is
725, find the other.

Solution 14

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 15

The
HCF of two numbers is 18 and their product is 12960. Find their LCM.

Solution 15

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 16

Is
it possible to have two numbers whose HCF is 18 and LCM is 760? Give reason.

Solution 16

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 17

Find the simplest form of:

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

(iv)R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 17

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

(iv)

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 18

Find
the largest number which divides 438 and 606, leaving remainder 6 in each
case.

Solution 18

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 19

Find the largest number which divides 320 and 457 leaving remainders 5 and 7 respectively.

Solution 19

Subtracting 5 and 7 from 320 and 457 respectively:

320 – 5 = 315,

457 – 7 = 450

Let us now find the HCF of 315 and 405 through prime factorization:

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real NumbersThe required number is 45.

Question 20

Find
the least number which when divided by 35, 56 and 91 leaves the same
remainder 7 in each case.

Solution 20

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 21

Find
the smallest number which when divided by 28 and 32 leaves remainders 8 and
12 respectively.

Solution 21

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 22

Find
the smallest number which when increased by 17 is exactly divisible by both
468 and 520.

Solution 22

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 23

Find
the greatest number of four digits which is exactly divisible by 15, 24 and
36.

Solution 23

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 24

In
a seminar, 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.

Solution 24

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 25

Three sets of English, Mathematics and Science books containing 336, 240 and 96 books respectively have to be stacked in such a way that all the books are stored subjectwise and the height of each stack is the same. How many stacks will be there?

Solution 25

Let us find the HCF of 336, 240 and 96 through prime factorization:

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Each stack of book will contain 48 books

Number of stacks of the same height

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 26

Three pieces of timber 42 m, 49 m and 63 m long have to be divided into planks of the same length. What is the greatest possible length of each plank?

Solution 26

The prime factorization of 42, 49 and 63 are:

42 = 2 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers3 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers7, 49 = 7 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers7, 63 = 3 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers3 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers7

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real NumbersH.C.F. of 42, 49, 63 is 7

Hence, greatest possible length of each plank = 7 m

Question 27

Find the greatest possible length which can be used to measure exactly the lengths 7 m, 3 m 85 cm and 12 m 95 cm.

Solution 27

7 m = 700cm, 3m 85cm = 385 cm

12 m 95 cm = 1295 cm

Let us find the prime factorization of 700, 385 and 1295:

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real NumbersR-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Greatest possible length = 35cm

Question 28

Find the maximum number of students among whom 1001 pens and 910 pencils can be distributed in such a way that each student gets the same number of pens and the same number of pencils.

Solution 28

Let us find the prime factorization of 1001 and 910:

1001 = 11 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers7 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers13

910 = 2 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers5 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers7 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers13

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

H.C.F. of 1001 and 910 is 7 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers13 = 91

Maximum number of students = 91

Question 29

Find the least number of square tiles required to pave the ceiling of a room 15 m 17 cm long and 9 m 2 cm broad.

Solution 29

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 30

Three measuring rods are 64 cm, 80 cm and 96 cm in length. Find the least length of cloth that can be measured an exact number of times, using any of the rods.

Solution 30

Let us find the LCM of 64, 80 and 96 through prime factorization:

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

L.C.M of 64, 80 and 96

= R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

 Therefore, the least length of the cloth that can be measured an exact number of times by the rods of 64cm, 80cm and 96cm = 9.6m

Question 31

An electronic device makes a beep after every 60 seconds. Another device makes a beep after every 62 seconds. They beeped together at 10 a.m. At what time will they beep together at the earliest?

Solution 31

Interval of beeping together = LCM (60 seconds, 62 seconds)

The prime factorization of 60 and 62:

60 = 30 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers2, 62 = 31 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers2

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real NumbersL.C.M of 60 and 62 is 30 31 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers2 = 1860 sec = 31min

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numberselectronic device will beep after every 31minutes

After 10 a.m., it will beep at 10 hrs 31 minutes

Question 32

The traffic lights at three different road crossings change after every 48 seconds, 72 seconds and 108 seconds respectively. If they all change simultaneously at 8 hours, then at what time will they again change simultaneously?

Solution 32

 

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

 

Question 33

Six bells commence tolling together and toll at intervals of 2, 4, 6, 8, 10, 12 minutes respectively. In 30 hours, how many times do they toll together?

Solution 33

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 34

Find the missing numbers in the following factorisation:

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 34

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

By going upward

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers5 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers11= 55

55 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers3= 165

165R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers2 = 330

330 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers2 = 660

Chapter 1 – Real Numbers Exercise Ex. 1C

Question 1

Without
actual division, show that each of the following rational number is a
terminating decimal. Express each in decimal form.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 1

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 2

Without
actual division, show that each of the following rational number is a
terminating decimal. Express each in decimal form.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 2

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 3

Without
actual division, show that each of the following rational number is a
terminating decimal. Express each in decimal form.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 3

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 4

Without
actual division, show that each of the following rational number is a
terminating decimal. Express each in decimal form.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 4

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 5

Without
actual division, show that each of the following rational number is a
terminating decimal. Express each in decimal form.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 5

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 6

Without
actual division, show that each of the following rational number is a
terminating decimal. Express each in decimal form.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 6

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 7

Without
actual division, show that each of the following rational number is a nonterminating repeating decimal.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 7

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 8

Without
actual division, show that each of the following rational number is a nonterminating repeating decimal.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 8

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 9

Without
actual division, show that each of the following rational number is a nonterminating repeating decimal.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 9

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 10

Without
actual division, show that each of the following rational number is a nonterminating repeating decimal.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 10

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 11

Without
actual division, show that each of the following rational number is a nonterminating repeating decimal.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 11

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 12

Without
actual division, show that each of the following rational number is a nonterminating repeating decimal.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 12

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 13

Without
actual division, show that each of the following rational number is a nonterminating repeating decimal.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 13

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 14

Without
actual division, show that each of the following rational number is a nonterminating repeating decimal.

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 14

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 15

Express each of the following as a fraction in simplest form:

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 15

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers
R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Chapter 1 – Real Numbers Exercise Ex. 1D

Question 1

Define (i) rational numbers, (ii) irrational numbers, (iii) real numbers.

Solution 1

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 2

Classify the following numbers as rational or irrational:

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 2

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 3

Prove that each of the following numbers is irrational:

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 3

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers
R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers
R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers
R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers
R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers
R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers
R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 4

Prove that R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbersis irrational.

Solution 4

 R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

 

Question 5

(i) Give an example of two irrationals whose sum is rational.

(ii) Give an example of two irrationals whose product is rational.

Solution 5

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

 

Question 6

State whether the given statement is true or false:

(i) The sum of two rationals is always rational.

(ii) The product of two rationals is always rational.

(iii) The sum of two irrationals is an irrational.

(iv) The product of two irrationals is an irrational.

(v) The sum of a rational and an irrational is irrational.

(vi) The product of a rational and an irrational is irrational.

Solution 6

(i) The sum of two rationals is always rational – True

(ii) The product of two rationals is always rational – True

(iii) The sum of two irrationals is an irrational – False

(iv) The product of two irrationals is an irrational – False

(v) The sum of a rational and an irrational is irrational – True

(vi) The product of a rational and an irrational is irrational – True

Question 7

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 7

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 8

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 8

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 9

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 9

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 10

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 10

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 11

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 11

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Chapter 1 – Real Numbers Exercise Ex. 1E

Question 1

State
Euclid’s division lemma.

Solution 1

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 2

State
fundamental theorem of arithmetic.

Solution 2

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 3

Express
360 as product of its prime factors.

Solution 3

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 4

If
a and b are two prime numbers then find HCF(a, b).

Solution 4

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 5

If
a and b are two prime numbers then find LCM(a, b).

Solution 5

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 6

If
the product of two numbers is 1050 and their HCF is 25, find their LCM.

Solution 6

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 7

What is a composite number?

Solution 7

A whole number that can be divided evenly by numbers other than 1 or itself.

Question 8

If
a and b are relatively prime then what is their HCF?

Solution 8

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 9

If
the rational number R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbershas a terminating decimal expansion, what is the condition
to be satisfied by b?

Solution 9

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 10

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 10

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 11

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 11

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 12

Show
that there is no value of n for which (2n x 5n) ends in
5.

Solution 12

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 13

Is
it possible to have two numbers whose HCF is 25 and LCM is 520?

Solution 13

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 14

Give
an example of two irrationals whose sum is rational.

Solution 14

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 15

Give
an example of two irrationals whose product is rational.

Solution 15

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 16

If
a and b are relatively prime, what is their LCM?

Solution 16

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 17

The
LCM of two numbers is 1200. Show that the HCF of these numbers cannot be 500.
Why?

Solution 17

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 18

Express
R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbersas a rational number in simplest form.

Solution 18

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 19

Express
R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbersas a rational number in simplest form

Solution 19

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 20

Explain
why 0.15015001500015 … is an irrational number.

Solution 20

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 21

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 21

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 22

Write
a rational number betweenR-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbersand 2.

Solution 22

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 23

Explain
whyR-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers is a rational number.

Solution 23

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Chapter 1 – Real Numbers Exercise FA

Question 1

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

(a) a terminating decimal

(b) a nonterminating,
repeating decimal

(c) a nonterminating
and nonrepeating decimal

(d)none of these

Solution 1

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 2

Which of the following has a terminating
decimal expansion?

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 2

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 3

On dividing a positive integer n by 9, we
get 7 as remainder. What will be the remainder if (3n – 1) is divided by 9?

(a) 1

(b) 2

(c) 3

(d)4

Solution 3

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 4

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 4

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 5

Show that any number of the form 4n,
n
N can never end with the digit 0.

Solution 5

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 6

The HCF of two numbers is 27 and their LCM
is 162. If one of the number is 81, find the other.

Solution 6

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 7

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 7

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 8

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 8

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 9

Which of the following numbers are
irrational?

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 9

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 10

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 10

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 11

Find the HCF and LCM of 12, 15, 18, 27.

Solution 11

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 12

Give an example of two irrationals whose
sum is rational.

Solution 12

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 13

Give prime factorization of 4620.

Solution 13

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 14

Find the HCF of 1008 and 1080 by prime
factorization method.

Solution 14

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 15

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 15

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 16

Find the largest number which divides 546
and 764, leaving remainders 6 and 8 respectively.

Solution 16

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 17

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 17

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 18

Show that every positive odd integer is of
the form (4q + 1) or (4q + 3) for some integer q.

Solution 18

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 19

Show that one and only one out of n, (n+2)
and (n+4) is divisible by 3, where n is any positive integer.

Solution 19

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 20

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 20

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Chapter 1 – Real Numbers Exercise MCQ

Question 1

Which of the following is a pair of co-primes?

(a) (14, 35)

(b) (18, 25)

(c) (31,93)

(d)(32, 62)

Solution 1

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 2

If a = (22×33×54) and b = (23×32×5) then HCF (a, b) = ?

(a) 90

(b) 180

(c) 360

(d)540

Solution 2

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 3

HCF of (23×32×5), (22×33×52) and (24×3×53×7) is

(a) 30

(b) 48

(c) 60

(d)105

Solution 3

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 4

LCM of (23×3×5) and (24×5×7) is

(a) 40

(b) 560

(c) 1120

(d)1680

Solution 4

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 5

The HCF of two numbers is 27 and their LCM is 162. If one of the numbers is 54, what is the other number?

(a) 36

(b) 45

(c) 9

(d)81

Solution 5

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 6

The product of two numbers is 1600 and their HCF is 5. The LCM of the numbers is

(a) 8000

(b) 1600

(c) 320

(d)1605

Solution 6

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 7

What is the largest number that divides each one of 1152 and 1664 exactly?

(a) 32

(b) 64

(c) 128

(d)256

Solution 7

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 8

What is the largest number that divides 70 and 125, leaving remainders 5 and 8 respectively?

(a) 13

(b) 9

(c) 3

(d)585

Solution 8

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 9

What is the largest number that divides 245 and 1029, leaving remainder 5 in each case?

(a) 15

(b) 16

(c) 9

(d)5

Solution 9

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 10

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 10

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 11

Euclid’s division lemma states that for any positive integers a and b, there exist unique integers q and r such that a = bq + r, where r must satisfy

(a) 1 < r < b

(b) 0 < r  b

(c) 0 r  < b

(d)0 < r < b

Solution 11

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 12

A number when divided by 143 leaves 31 as remainder. What will be the remainder when the same number is divided by 13?

(a) 0

(b) 1

(c) 3

(d)5

Solution 12

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 13

Which of the following is an irrational number?

(a) R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

(b) 3.1416

(c) R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

(d) 3.141141114 …

Solution 13

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 14

𝜋 is

(a) an integer

(b) a rational number

(c) an irrational number

(d)none of these

Solution 14

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 15

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

(a) an integer

(b) a rational number

(c) an irrational number

(d) none of these

Solution 15

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 16

2.13113111311113… is

(a) an integer

(b) a rational number

(c) an irrational number

(d)none of these

Solution 16

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 17

The number 3.24636363 … is

(a) an integer

(b) a rational number

(c) an irrational number

(d)none of these

Solution 17

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 18

Which of the following rational numbers is expressible as a terminating decimal?

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Solution 18

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 19

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

(a) one decimal place

(b) two decimal places

(c) three decimal places

(d) four decimal places

Solution 19

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 20

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

(a) one decimal place

(b) two decimal places

(c) three decimal places

(d)four decimal places

Solution 20

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 21

The number 1.732 is

(a) an irrational number

(b) a rational number

(c) an integer

(d)a whole number

Solution 21

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 22

a and b are two positive integers such that the least prime factor of a is 3 and the least prime factor of b is 5. Then, the least prime factor of (a+b) is

(a) 2

(b) 3

(c) 5

(d)8

Solution 22

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 23

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

(a) a rational number

(b) an irrational number

(c) a terminating decimal

(d)a nonterminating repeating decimal

Solution 23

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 24

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

(a) a fraction

(b) a rational number

(c) an irrational number

(d)none of these

Solution 24

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 25

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

(a) an integer

(b) a rational number

(c) an irrational number

(a) none of these

Solution 25

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

Question 26

What is the least number that is divisible by all the natural numbers from 1 to 10 (both inclusive)

(a) 100

(b) 1260

(c) 2520

(d) 5040

Solution 26

R-s-aggarwal-and-v-aggarwal Solutions Cbse Class 10 Mathematics Chapter - Real Numbers

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