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NCERT solutions Mathematics Textbook for Class 10 chapter 1 Real Numbers

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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 | view solution

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

Q 1.2 | Page 7 | view solution

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

Q 1.3 | Page 7 | view solution

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 | view solution

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 | view solution

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 | view solution

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 | view solution

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 | view solution

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

Q 4 | Page 11 | view solution

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

Q 5 | Page 11 | view solution

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

Q 6 | Page 11 | view solution

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 | view solution

Page 14

Prove that `sqrt5` is irrational.

Q 1 | Page 14 | view solution

Prove that 3 + 2`sqrt5` is irrational

Q 2 | Page 14 | view solution

Prove that the following are irrationals

(i)`1/sqrt2`

(ii)` 7/sqrt5`

(iii)` 6+sqrt2`

Q 3 | Page 14 | view solution

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 | view solution

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 | view solution

Extra questions

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

view solution

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.

view solution

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.

view solution

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

view solution

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

view solution

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.

view solution

 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.

view solution

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`

view solution

Prove that is `sqrt2` irrational number.

view solution

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

view solution

Find two irrational numbers between 2 and 2.5.

view solution

Prove that is `sqrt3` irrational number.

view solution

Prove that 7-`sqrt3` is irrational.

view solution

Prove that `sqrt5/3` is irrational.

view solution

Find 3 irrational numbers between 3 and 5

view solution

 Find two irrational numbers between 0.12 and 0.13

view solution

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

view solution

Prove that `2sqrt7` is irrational.

view solution

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