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NCERT solutions Mathematics Class 10 chapter 1 Real Numbers

Chapters

NCERT Mathematics Class 10

Mathematics Textbook for Class 10

Chapter 1 - Real Numbers

Page 7

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

Q 1.1 | Page 7

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

Q 1.2 | Page 7

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

Q 1.3 | Page 7

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.

Q 2 | Page 7

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?

Q 3 | Page 7

Page 11

Express each number as a product of its prime factors:

(i) 140

(ii) 156

(iii) 3825

(iv) 5005

(v) 7429

Q 1 | Page 11

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

Q 2 | Page 11

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

Q 3 | Page 11

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

Q 4 | Page 11

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

Q 5 | Page 11

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

Q 6 | Page 11

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?

Q 7 | Page 11

Page 14

Prove that `sqrt5` is irrational.

Q 1 | Page 14

Prove that 3 + 2`sqrt5` is irrational

Q 2 | Page 14

Prove that the following are irrationals

(i)`1/sqrt2`

(ii)` 7/sqrt5`

(iii)` 6+sqrt2`

Q 3 | Page 14

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`

Q 1 | Page 17

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)`

Q 3 | Page 18

Extra questions

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

Find two irrational numbers between 2 and 2.5.

Find 3 irrational numbers between 3 and 5

Prove that `2sqrt7` is irrational.

Prove that `sqrt5/3` is irrational.

Prove that is `sqrt2` irrational number.

 Find two irrational numbers between 0.12 and 0.13

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

Prove that 7-`sqrt3` is irrational.

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.

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

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

Prove that is `sqrt3` irrational number.

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`

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

 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.

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

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

NCERT Mathematics Class 10

Mathematics Textbook for Class 10
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