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RD Sharma solutions for Class 10 Mathematics chapter 1 - Real Numbers

10 Mathematics

RD Sharma 10 Mathematics Chapter 1: Real Numbers

Ex. 1.10Ex. 1.20Ex. 1.30Ex. 1.40Ex. 1.50Ex. 1.60Others

Chapter 1: Real Numbers Exercise 1.10 solutions [Page 10]

Ex. 1.10 | Q 1 | Page 10

If a and b are two odd positive integers such that a > b, then prove that one of the two numbers (a+b)/2and(a-b)/2 is odd and the other is even.

Ex. 1.10 | Q 2 | Page 10

Prove that the product of two consecutive positive integers is divisible by 2.

Ex. 1.10 | Q 3 | Page 10

Prove that the product of three consecutive positive integer is divisible by 6.

Ex. 1.10 | Q 4 | Page 10

For any positive integer n , prove that n3 − n divisible by 6.

Ex. 1.10 | Q 5 | Page 10

Prove that if a positive integer is of the form 6q + 5, then it is of the form 3q + 2 for some integer q, but not conversely.

Ex. 1.10 | Q 6 | Page 10

Prove that the square of any positive integer of the form 5q + 1 is of the same form.

Ex. 1.10 | Q 7 | Page 10

Prove that the square of any positive integer is of the form 3m or, 3m + 1 but not of the form 3m +2.

Ex. 1.10 | Q 8 | Page 10

Prove that the square of any positive integer is of the form 4q or 4q + 1 for some integer q.

Ex. 1.10 | Q 9 | Page 10

Prove that the square of any positive integer is of the form 5q, 5q + 1, 5q + 4 for some integer q.

Ex. 1.10 | Q 10 | Page 10

Show that the square of an odd positive integer is of the form 8q + 1, for some integer q.

Ex. 1.10 | Q 11 | Page 10

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

Chapter 1: Real Numbers Exercise 1.20 solutions [Pages 27 - 28]

Ex. 1.20 | Q 1.01 | Page 27

Define HOE of two positive integers and find the HCF of the following pair of numbers:

32 and 54

Ex. 1.20 | Q 1.02 | Page 27

Define HOE of two positive integers and find the HCF of the following pair of numbers:

18 and 24

Ex. 1.20 | Q 1.03 | Page 27

Define HOE of two positive integers and find the HCF of the following pair of numbers:

70 and 30

Ex. 1.20 | Q 1.04 | Page 27

Define HOE of two positive integers and find the HCF of the following pair of numbers:

56 and 88

Ex. 1.20 | Q 1.05 | Page 27

Define HOE of two positive integers and find the HCF of the following pair of numbers:

475 and 495

Ex. 1.20 | Q 1.06 | Page 27

Define HOE of two positive integers and find the HCF of the following pair of numbers:

75 and 243

Ex. 1.20 | Q 1.07 | Page 27

Define HOE of two positive integers and find the HCF of the following pair of numbers:

240 and 6552

Ex. 1.20 | Q 1.08 | Page 27

Define HOE of two positive integers and find the HCF of the following pair of numbers:

155 and 1385

Ex. 1.20 | Q 1.09 | Page 27

Define HOE of two positive integers and find the HCF of the following pair of numbers:

100 and 190

Ex. 1.20 | Q 1.1 | Page 27

Define HOE of two positive integers and find the HCF of the following pair of numbers:

105 and 120

Ex. 1.20 | Q 2.1 | Page 27

Using Euclid's division algorithm, find the H.C.F. of 135 and 225

Ex. 1.20 | Q 2.2 | Page 27

Using Euclid's division algorithm, find the H.C.F. of 196 and 38220

Ex. 1.20 | Q 3.1 | Page 27

Find the HCF of the following pairs of integers and express it as a linear combination of 963 and 657.

Ex. 1.20 | Q 3.2 | Page 27

Find the HCF of the following pairs of integers and express it as a linear combination of 592 and 252.

Ex. 1.20 | Q 3.3 | Page 27

Find the HCF of the following pairs of integers and express it as a linear combination of 506 and 1155.

Ex. 1.20 | Q 3.4 | Page 27

Find the HCF of the following pairs of integers and express it as a linear combination of 1288 and 575.

Ex. 1.20 | Q 4 | Page 27

Find the largest number which divides 615 and 963 leaving remainder 6 in each case.

Ex. 1.20 | Q 5 | Page 27

If the HCF of 408 and 1032 is expressible in the form 1032 m − 408 × 5, find m.

Ex. 1.20 | Q 6 | Page 27

If the HCF of 657 and 963 is expressible in the form 657x + 963y − 15, find x.

Ex. 1.20 | Q 7 | Page 27

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?

Ex. 1.20 | Q 8 | Page 27

A merchant has 120 liters of oil of one kind, 180 liters of another kind and 240 liters of third kind. He wants to sell the oil by filling the three kinds of oil in tins of equal capacity. What should be the greatest capacity of such a tin?

Ex. 1.20 | Q 9 | Page 27

During a sale, colour pencils were being sold in packs of 24 each and crayons in packs of 32 each. If you want full packs of both and the same number of pencils and crayons, how many of each would you need to buy?

Ex. 1.20 | Q 10 | Page 28

144 cartons of Coke Cans and 90 cartons of Pepsi Cans are to be stacked in a Canteen. If each stack is of the same height and is to contain cartons of the same drink, what would be the greatest number of cartons each stack would have?

Ex. 1.20 | Q 11 | Page 28

Find the greatest number which divides 285 and 1249 leaving remainders 9 and 7 respectively.

Ex. 1.20 | Q 12 | Page 28

Find the largest number which exactly divides 280 and 1245 leaving remainders 4 and 3, respectively.

Ex. 1.20 | Q 13 | Page 28

What is the largest number that divides 626, 3127 and 15628 and leaves remainders of 1, 2 and 3 respectively.

Ex. 1.20 | Q 14 | Page 28

Find the greatest number that will divide 445, 572 and 699 leaving remainders 4, 5 and 6 respectively.

Ex. 1.20 | Q 15 | Page 28

Find the greatest number which divides 2011 and 2623 leaving remainders 9 and 5 respectively.

Ex. 1.20 | Q 17 | Page 28

Two brands of chocolates are available in packs of 24 and 15 respectively. If I need to buy an equal number of chocolates of both kinds, what is the least number of boxes of each kind I would need to buy?

Ex. 1.20 | Q 18 | Page 28

A mason has to fit a bathroom with square marble tiles of the largest possible size. The size of the bathroom is 10 ft. by 8 ft. What would be the size in inches of the tile required that has to be cut and how many such tiles are required?

Ex. 1.20 | Q 19 | Page 28

15 pastries and 12 biscuit packets have been donated for a school fete. These are to be packed in several smaller identical boxes with the same number of pastries and biscuit packets in each. How many biscuit packets and how many pastries will each box contain?

Ex. 1.20 | Q 20 | Page 28

105 goats, 140 donkeys and 175 cows 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. Can you tell how many animals went in each trip?

Ex. 1.20 | Q 21 | Page 28

The length, breadth and height of a room are 8 m 25 cm, 6 m 75 cm and 4 m 50 cm, respectively. Determine the longest rod which can measure the three dimensions of the room exactly.

Ex. 1.20 | Q 22 | Page 28

Express the HCF of 468 and 222 as 468x + 222y where x, y are integers in two different ways.

Chapter 1: Real Numbers Exercise 1.30 solutions [Page 35]

Ex. 1.30 | Q 1.1 | Page 35

Express each of the following integers as a product of its prime factors:

420

Ex. 1.30 | Q 1.2 | Page 35

Express each of the following integers as a product of its prime factors:

468

Ex. 1.30 | Q 1.3 | Page 35

Express each of the following integers as a product of its prime factors:

945

Ex. 1.30 | Q 1.4 | Page 35

Express each of the following integers as a product of its prime factors:

7325

Ex. 1.30 | Q 2.1 | Page 35

Determine the prime factorisation of each of the following positive integer:

20570

Ex. 1.30 | Q 2.2 | Page 35

Determine the prime factorisation of each of the following positive integer:

58500

Ex. 1.30 | Q 2.3 | Page 35

Determine the prime factorisation of each of the following positive integer:

45470971

Ex. 1.30 | Q 3 | Page 35

Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.

Ex. 1.30 | Q 4 | Page 35

Check whether 6n can end with the digit 0 for any natural number n.

Chapter 1: Real Numbers Exercise 1.40 solutions [Pages 39 - 40]

Ex. 1.40 | Q 1.1 | Page 39

Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = Product of the integers:

26 and 91

Ex. 1.40 | Q 1.2 | Page 39

Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = Product of the integers:

510 and 92

Ex. 1.40 | Q 1.3 | Page 39

Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = Product of the integers:

336 and 54

Ex. 1.40 | Q 2.1 | Page 39

Find the LCM and HCF of the following integers by applying the prime factorisation method:

12, 15 and 21

Ex. 1.40 | Q 2.2 | Page 39

Find the LCM and HCF of the following integers by applying the prime factorisation method:

17, 23 and 29

Ex. 1.40 | Q 2.3 | Page 39

Find the LCM and HCF of the following integers by applying the prime factorisation method:

8, 9 and 25

Ex. 1.40 | Q 2.4 | Page 39

Find the LCM and HCF of the following integers by applying the prime factorisation method:

40, 36 and 126

Ex. 1.40 | Q 2.5 | Page 39

Find the LCM and HCF of the following integers by applying the prime factorisation method:

84, 90 and 120

Ex. 1.40 | Q 2.6 | Page 39

Find the LCM and HCF of the following integers by applying the prime factorisation method:

24, 15 and 36

Ex. 1.40 | Q 3 | Page 39

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

Ex. 1.40 | Q 4 | Page 40

Can two numbers have 16 as their HCF and 380 as their LCM? Give reason.

Ex. 1.40 | Q 5 | Page 40

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

Ex. 1.40 | Q 6 | Page 40

The HCF to two numbers is 16 and their product is 3072. Find their LCM.

Ex. 1.40 | Q 7 | Page 40

The LCM and HCF of two numbers are 180 and 6 respectively. If one of the numbers is 30, find the other number.

Ex. 1.40 | Q 8 | Page 40

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

Ex. 1.40 | Q 9 | Page 40

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

Ex. 1.40 | Q 10 | Page 40

What is the smallest number that, when divided by 35, 56 and 91 leaves remainders of 7 in each case?

Ex. 1.40 | Q 11 | Page 40

A rectangular courtyard is 18 m 72 cm long and 13 m 20 cm broad. it is to be paved with square tiles of the same size. Find the least possible number of such tiles.

Ex. 1.40 | Q 12 | Page 40

Find the greatest number of 6 digits exactly divisible by 24, 15 and 36.

Ex. 1.40 | Q 13 | Page 40

Determine the number nearest to 110000 but greater than 100000 which is exactly divisible by each of 8, 15 and 21.

Ex. 1.40 | Q 14 | Page 40

Find the least number that is divisible by all the numbers between 1 and 10 (both inclusive).

Ex. 1.40 | Q 15 | Page 40

A circular field has a circumference of 360 km. Three cyclists start together and can cycle 48, 60 and 72 km a day, round the field. When will they meet again?

Ex. 1.40 | Q 16 | Page 40

In a morning walk three persons step off together, their steps measure 80 cm, 85 cm and 90 cm respectively. What is the minimum distance each should walk so that he can cover the distance in complete steps?

Chapter 1: Real Numbers Exercise 1.50 solutions [Page 49]

Ex. 1.50 | Q 1.1 | Page 49

Show that the following numbers are irrational.

$\frac{1}{\sqrt{2}}$
Ex. 1.50 | Q 1.2 | Page 49

Show that the following numbers are irrational.

$7\sqrt{5}$
Ex. 1.50 | Q 1.3 | Page 49

Show that the following numbers are irrational.

$6 + \sqrt{2}$
Ex. 1.50 | Q 1.4 | Page 49

Show that the following numbers are irrational.

$3 - \sqrt{5}$
Ex. 1.50 | Q 2.1 | Page 49

Prove that following numbers are irrationals:

$\frac{2}{\sqrt{7}}$
Ex. 1.50 | Q 2.2 | Page 49

Prove that following numbers are irrationals:

$\frac{3}{2\sqrt{5}}$
Ex. 1.50 | Q 2.3 | Page 49

Prove that following numbers are irrationals:

$4 + \sqrt{2}$
Ex. 1.50 | Q 2.4 | Page 49

Prove that following numbers are irrationals:

$5\sqrt{2}$
Ex. 1.50 | Q 3 | Page 49

Show that $2 - \sqrt{3}$ is an irrational number.

Ex. 1.50 | Q 4 | Page 49

Show that $3 + \sqrt{2}$ is an irrational number.

Ex. 1.50 | Q 5 | Page 49

Prove that $4 - 5\sqrt{2}$ is an irrational number.

Ex. 1.50 | Q 6 | Page 49

Show that $5 - 2\sqrt{3}$ is an irrational number.

Ex. 1.50 | Q 7 | Page 49

Prove that $2\sqrt{3} - 1$ is an irrational number.

Ex. 1.50 | Q 8 | Page 49

Prove that $2 - 3\sqrt{5}$ is an irrational number.

Ex. 1.50 | Q 9 | Page 49

Prove that $\sqrt{5} + \sqrt{3}$ is irrational.

Ex. 1.50 | Q 11 | Page 49

Prove that for any prime positive integer p, $\sqrt{p}$

is an irrational number.

Ex. 1.50 | Q 12 | Page 49

If p, q are prime positive integers, prove that $\sqrt{p} + \sqrt{q}$ is an irrational number.

Chapter 1: Real Numbers Exercise 1.60 solutions [Pages 56 - 58]

Ex. 1.60 | Q 1.1 | Page 56

Without actually performing the long division, state whether state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion.

$\frac{23}{8}$
Ex. 1.60 | Q 1.2 | Page 56

Without actually performing the long division, state whether state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion.

$\frac{125}{441}$
Ex. 1.60 | Q 1.3 | Page 56

Without actually performing the long division, state whether state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion.

$\frac{35}{50}$
Ex. 1.60 | Q 1.4 | Page 57

Without actually performing the long division, state whether state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion.

$\frac{77}{210}$
Ex. 1.60 | Q 1.5 | Page 56

Without actually performing the long division, state whether state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion.

$\frac{129}{2^2 \times 5^7 \times 7^{17}}$
Ex. 1.60 | Q 1.6 | Page 56

Without actually performing the long division, state whether state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion.

$\frac{987}{10500}$
Ex. 1.60 | Q 2.1 | Page 56

Write down the decimal expansions of the following rational numbers by writing their denominators in the form 2m × 5n, where, mn are non-negative integers. $\frac{3}{8}$

Ex. 1.60 | Q 2.2 | Page 56

Write down the decimal expansions of the following rational numbers by writing their denominators in the form 2m × 5n, where, mn are non-negative integers.$\frac{13}{125}$

Ex. 1.60 | Q 2.3 | Page 56

Write down the decimal expansions of the following rational numbers by writing their denominators in the form 2m × 5n, where, mn are non-negative integers.$\frac{7}{80}$

Ex. 1.60 | Q 2.4 | Page 56

Write down the decimal expansions of the following rational numbers by writing their denominators in the form 2m × 5n, where, mn are non-negative integers.$\frac{14588}{625}$

Ex. 1.60 | Q 2.5 | Page 56

Write down the decimal expansions of the following rational numbers by writing their denominators in the form 2m × 5n, where, mn are non-negative integers.$\frac{129}{2^2 \times 5^7}$

Ex. 1.60 | Q 4.1 | Page 57

What can you say about the prime factorisations of the denominators of the following rationals: 43.123456789

Ex. 1.60 | Q 4.2 | Page 57

What can you say about the prime factorisations of the denominators of the following rationals: $43 . 123456789$

Ex. 1.60 | Q 4.3 | Page 57

​What can you say about the prime factorisations of the denominators of the following rationals: $27 . \bar{{142857}}$

Ex. 1.60 | Q 4.4 | Page 57

​What can you say about the prime factorisations of the denominators of the following rationals: 0.120120012000120000 ...

Q 16 | Page 58

If p and q are two prime number, then what is their HCF?

Chapter 1: Real Numbers solutions [Pages 50 - 58]

Q 1 | Page 57

State Euclid's division lemma.

Q 2 | Page 57

State Fundamental Theorem of Arithmetic.

Q 3 | Page 57

Write 98 as product of its prime factors.

Q 4 | Page 57

Write the exponent of 2 in the price factorization of 144.

Q 5 | Page 57

Write the sum of the exponents of prime factors in the prime factorization of 98.

Q 6 | Page 57

If the prime factorization of a natural number n is 23 ✕ 32 ✕ 52 ✕ 7, write the number of consecutive zeros in n.

Q 7 | Page 57

If the product of two numbers is 1080 and their HCF is 30, find their LCM.

Q 8 | Page 58

Write the condition to be satisfied by q so that a rational number$\frac{p}{q}$has a terminating decimal expansions.

Q 9 | Page 58

Write the condition to be satisfied by q so that a rational number$\frac{p}{q}$has a terminating decimal expansion.

Q 10 | Page 58

Complete the missing entries in the following factor tree. Q 11 | Page 58

The decimal expansion of the rational number $\frac{43}{2^4 \times 5^3}$ will terminate after how many places of decimals?

Q 12 | Page 58

Has the rational number $\frac{441}{2^2 \times 5^7 \times 7^2}$a terminating or a nonterminating decimal representation?

Q 13 | Page 58

Write whether $\frac{2\sqrt{45} + 3\sqrt{20}}{2\sqrt{5}}$

on simplification gives a rational or an irrational number.

Q 14 | Page 58

What is an algorithm?

Q 15 | Page 58

What is a lemma?

Q 17 | Page 58

If p and q are two prime number, then what is their LCM?

Q 18 | Page 58

What is the total number of factors of a prime number?

Q 19 | Page 58

What is a composite number?

Q 20 | Page 58

What is the HCF of the smallest composite number and the smallest prime number?

Q 21 | Page 58

HCF of two numbers is always a factor of their LCM (True/False).

Q 22 | Page 58

π is an irrational number (True/False).

Q 23 | Page 58

The sum of two prime number is always a prime number (True/ False).

Q 24 | Page 58

The product of any three consecutive natural number is divisible by 6 (True/False).

Q 25 | Page 58

Every even integer is of the form 2m, where m is an integer (True/False).

Q 26 | Page 58

Every odd integer is of the form 2m − 1, where m is an integer (True/False).

Q 27 | Page 58

The product of two irrational numbers is an irrational number (True/False).

Q 28 | Page 58

The sum of two irrational number is an irrational number (True/False).

Q 29 | Page 50

For what value of n, 2n ✕ 5n ends in 5.

Q 30 | Page 58

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

Q 31 | Page 58

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

Q 32 | Page 58

Two numbers have 12 as their HCF and 350 as their LCM (True/False).

Chapter 1: Real Numbers solutions [Pages 10 - 61]

Q 1 | Page 59

The exponent of 2 in the prime factorisation of 144, is

• 4

• 5

• 6

• 3

Q 2 | Page 59

The LCM of two numbers is 1200. Which of the following cannot be their HCF?

• 600

• 500

• 400

•  200

Q 3 | Page 59

If n = 23 ✕ 34 ✕ 54 ✕ 7, then the number of consecutive zeros in n, where n is a natural number, is

• 2

• 3

• 4

• 7

Q 4 | Page 59

The sum of the exponents of the prime factors in the prime factorisation of 196, is

• 1

• 2

• 4

• 6

Q 5 | Page 59

The number of decimal place after which the decimal expansion of the rational number $\frac{23}{2^2 \times 5}$ will terminate, is

• 1

• 2

• 3

• 4

Q 6 | Page 59

If p1 and p2 are two odd prime numbers such that p1 > p2, then

$p_1^2 - p_2^2$  is
•  an even number

• an odd number

• an odd prime number

• a prime number

Q 7 | Page 59

If two positive ingeters a and b are expressible in the form a = pq2 and b = p3qpq being prime number, then LCM (ab) is

• pq

• p3q3

• p3q2

•  p2q2

Q 8 | Page 59

In Q.No. 7, HCF (ab) is

• pq

•  p3q3

• p3q2

• p2q2

Q 9 | Page 59

If two positive integers m and n are expressible in the form m = pq3 and n = p3q2, where pq are prime numbers, then HCF (mn) =

• pq

• pq2

• p3q2

• p2q2

Q 10 | Page 60

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

• 2

• 3

• 4

• 1

Q 11 | Page 60

The HCF of 95 and 152, is

• 57

• 1

• 19

• 38

Q 12 | Page 60

If HCF (26, 169) = 13, then LCM (26, 169) =

• 26

•  2

• 3

• 4

Q 13 | Page 10

If a = 23 ✕ 3, = 2 ✕ 3 ✕ 5, c = 3n ✕ 5 and LCM (abc) = 23 ✕ 32 ✕ 5, then n =

• 1

• 2

• 3

• 4

Q 14 | Page 10

The decimal expansion of the rational number $\frac{14587}{1250}$  will terminate after

• one decimal place

• two decimal place

• three decimal place

• four decimal place

Q 15 | Page 60

If p and q are co-prime numbers, then p2 and q2 are

•  coprime

• not coprime

• even

• odd

Q 16 | Page 60

Which of the following rational numbers have terminating decimal?

• $\frac{16}{225}$
• $\frac{5}{18}$
• $\frac{2}{21}$
• $\frac{7}{250}$

• Non of the above

Q 17 | Page 60

If 3 is the least prime factor of number a and 7 is the least prime factor of number b, then the least prime factor of a + b, is

• 2

• 3

• 5

• 10

Q 18 | Page 60
$3 . 27$  is
• an integer

• a rational number

• a natural number

• an irrational number

Q 19 | Page 60

The smallest number by which $\sqrt{27}$  should be multiplied so as to get a rational number is

• $\sqrt{27}$
• $3\sqrt{3}$

• $\sqrt{3}$
• 3

Q 20 | Page 60

The smallest rational number by which $\frac{1}{3}$ should be multiplied so that its decimal expansion terminates after one place of decimal, is

• $\frac{3}{10}$
• $\frac{1}{10}$
• 3

• $\frac{3}{100}$
Q 21 | Page 60

If n is a natural number, then 92n − 42n is always divisible by

•  5

• 13

• both 5 and 13

• None of these

Q 22 | Page 60

If n is any natural number, then 6n − 5n always ends with

• 1

• 3

• 5

• 7

Q 23 | Page 61

The LCM and HCF of two rational numbers are equal, then the numbers must be

• prime

• co-prime

• composite

• equal

Q 24 | Page 61

If the sum of LCM and HCF of two numbers is 1260 and their LCM is 900 more than their HCF, then the product of two numbers is

• 203400

•  194400

• 198400

• 205400

Q 25 | Page 61

The remainder when the square of any prime number greater than 3 is divided by 6, is

• 1

• 3

• 2

• 4

Chapter 1: Real Numbers

Ex. 1.10Ex. 1.20Ex. 1.30Ex. 1.40Ex. 1.50Ex. 1.60Others

RD Sharma 10 Mathematics RD Sharma solutions for Class 10 Mathematics chapter 1 - Real Numbers

RD Sharma solutions for Class 10 Maths chapter 1 (Real Numbers) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE 10 Mathematics solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 10 Mathematics chapter 1 Real Numbers are Introduction of Real Numbers, Real Numbers Examples and Solutions, Euclid’s Division Lemma, Fundamental Theorem of Arithmetic, Fundamental Theorem of Arithmetic Motivating Through Examples, Proofs of Irrationality, Revisiting Irrational Numbers, Revisiting Rational Numbers and Their Decimal Expansions.

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