#### Chapters

Chapter 2 - Polynomials

Chapter 3 - Pair of Linear Equations in Two Variables

Chapter 4 - Quadratic Equations

Chapter 5 - Arithmetic Progressions

Chapter 6 - Triangles

Chapter 7 - Coordinate Geometry

Chapter 8 - Introduction to Trigonometry

Chapter 9 - Some Applications of Trigonometry

Chapter 10 - Circles

Chapter 11 - Constructions

Chapter 12 - Areas Related to Circles

Chapter 13 - Surface Areas and Volumes

Chapter 14 - Statistics

Chapter 15 - Probability

## Chapter 1 - Real Numbers

#### Page 7

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

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

Using Euclid's division algorithm, find the H.C.F. of (iii) 867 and 255

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

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?

#### Page 11

Express each number as a product of its prime factors:

(i) 140

(ii) 156

(iii) 3825

(iv) 5005

(v) 7429

Find the LCM and HCF of the following pairs of integers and verify that LCM × HCF = product of the two numbers

(i) 26 and 91

(ii) 510 and 92

(iii) 336 and 54

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

(i) 12, 15 and 21

(ii) 17, 23 and 29

(iii) 8, 9 and 25

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

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

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

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?

#### Page 14

Prove that `sqrt5` is irrational.

Prove that 3 + 2`sqrt5` is irrational

Prove that the following are irrationals

(i)`1/sqrt2`

(ii)` 7/sqrt5`

(iii)` 6+sqrt2`

#### Pages 17 - 18

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

(i)`13/3125`

(ii) `17/8`

(iii) `64/455`

(iv) `15/1600`

(v) `29/343`

(vi) `23/(2^3`

(vii) `129/(2^2`

(viii) `6/15`

(ix) `35/50`

(x) `77/210`

The following real numbers have decimal expansions as given below. In each case, decide whether they are rational or not. If they are rational, and of the form p /q what can you say about the prime factors of q?

(1) 43.123456789

(2) 0.120120012000120000. . .

(3) `43.bar(123456789)`

#### Extra questions

Write down the decimal expansions of those rational numbers which have terminating decimal expansions.

(a) `13/3125`

(b) `17/8`

(c)`15/1600`

(d)`23/(2^2`

(e)`6/15`

(f)`35/50`

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

** **Use Euclid's Division Algorithm to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.

Consider the number 6^{n} where n is a natural number. Check whether there is any value of n ∈ N for which 6^{n} is divisible by 7.

Consider the number 12^{n} where n is a natural number. Check whether there is any value of n ∈ N for which 12^{n} ends with the digital zero.

** **Insert a rational and an irrational number between 2 and 3.

Prove that `2sqrt7` is irrational.

Find two irrational numbers lying between `sqrt2" and "sqrt3`

Prove that is `sqrt3` irrational number.

Find two irrational numbers between 2 and 2.5.

Find 3 irrational numbers between 3 and 5

Prove that is `sqrt2` irrational number.

Prove that `sqrt5/3` is irrational.

Prove that 7-`sqrt3` is irrational.

** **Find two irrational numbers between 0.12 and 0.13

** **Show that any positive odd integer is of the form 4q + 1 or 4q + 3, where q is some integer.

Show that every positive integer is of the form 2q and that every positive odd integer is of the from 2q + 1, where q is some integer.

Use Euclid's Division Algorithm to show that the cube of any positive integer is either of the 9m, 9m + 1 or 9m + 8 for some integer m

#### Textbook solutions for Class 10

## NCERT solutions for Class 10 Mathematics chapter 1 - Real Numbers

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

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