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Solutions ⇒ Class 10th ⇒ Mathematics ⇒ 1. Real Numbers

Exercise 1.1 Class 10 maths 1. Real Numbers - ncert solutions - Toppers Study

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Exercise 1.1 Class 10 maths 1. Real Numbers - ncert solutions - Toppers Study

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Exercise 1.1 class 10 Mathematics Chapter 1. Real Numbers

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  • Exercise 1.1 Class 10 Maths 1. Real Numbers - Ncert Solutions - Toppers Study
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  • Chapter 1. Real Numbers Exercise 1.1 Class 10

1. Real Numbers

| Exercise 1.1 |

Exercise 1.1 Class 10 maths 1. Real Numbers - ncert solutions - Toppers Study


Exercise 1.1 class 10 maths Chapter 1. Real Numbers


1. Use Euclid’s division algorithm to find the HCF of :

(i) 135 and 225 (ii) 196 and 38220 (iii) 867 and 255

Sol:

(1)         135 and 225

  a = 225, b = 135               {Greatest number is ‘a’ and smallest number is ‘b’}

 Using Euclid’s division algorithm

  a = bq + r (then)

  225 = 135 ×1 + 90

 135 = 90 ×1 + 45

 90 = 45 × 2 + 0                  {when we get r=0, our computing get stopped}

 b = 45 {b is HCF}

Hence:  HCF = 45

 

Sol:

 (ii)        196 and 38220

 a = 38220, b = 196          {Greatest number is ‘a’ and smallest number is ‘b’}

 Using Euclid’s division algorithm

 a = bq + r (then)

 38220= 196 ×195 + 0 {when we get r=0, our computing get stopped}

 b = 196 {b is HCF}

Hence:  HCF = 196

Sol:

(iii)        867 and 255

a = 867, b = 255               {Greatest number is ‘a’ and smallest number is ‘b’}

Using Euclid’s division algorithm

a = bq + r (then)

38220= 196 ×195 + 0 {when we get r=0, our computing get stopped}

b = 196 {b is HCF}

Hence:  HCF = 196

 

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

 

Sol:

Let a is the positive odd integer

Where b = 6,

When we divide a by 6 we get reminder 0, 1, 2, 3, 4 and 5,          {r < b}

Here a is odd number then reminder will be also odd one.

We get reminders 1, 3, 5 

Using Euclid’s division algorithm

So we get

a = 6q + 1, 6q+3 and 6q+5

 

3.   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?

Sol:

      Maximum number of columns = HCF (616, 32)

      a = 616, b = 32  {Greatest number is ‘a’ and smallest number is ‘b’}

      Using Euclid’s division algorithm

      a = bq + r (then)

      616 = 32 ×19 + 8 {when we get r=0, our computing get stopped}

      32 = 8 × 4 + 0

       b = 8 {b is HCF}

       HCF = 8

       Hence: Maximum number of columns = 8

 

4. Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3m or 3m + 1 for some integer m.

[Hint : Let x be any positive integer then it is of the form 3q, 3q + 1 or 3q + 2. Now square each of these and show that they can be rewritten in the form 3m or 3m + 1.]

 

Solution :

To Show :

a2 = 3m or 3m + 1

a = bq + r

Let a be any positive integer, where b = 3 and r = 0, 1, 2 because 0 ≤ r < 3

Then a = 3q + r for some integer q ≥ 0

Therefore, a = 3q + 0 or 3q + 1 or 3q + 2

Now we have;

a2 = (3q + 0)2 or (3q + 1)2 or (3q +2)2

a2 = 9q2 or 9q2 + 6q + 1 or 9q2 + 12q + 4

a2 = 9q2 or 9q2 + 6q + 1 or 9q2 + 12q + 3 + 1

a2 = 3(3q2) or 3(3q2 + 2q) + 1 or 3(3q2 + 4q + 1) + 1

Let m = (3q2) or (3q2 + 2q)  or (3q2 + 4q + 1)

Then we get;

a2 = 3m or 3m + 1 or 3m + 1

 

5. Use Euclid’s division lemma to show that the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.

Solution:

Let , a is any positive integer

By using Euclid’s division lemma;

a = bq + r             where; 0 ≤ r < b

Putting b = 9

a = 9q + r             where; 0 ≤ r < 9

when r = 0

a = 9q + 0 = 9q

a3  = (9q)3 = 9(81q3) or 9m where m = 81q3

when r = 1

a = 9q + 1 

a3  = (9q + 1)3 = 9(81q3 + 27q2 + 3q) + 1

      = 9m + 1  where m = 81q3 + 27q2 + 3q

when r = 2

a = 9q + 2 

a3  = (9q + 2)3 = 9(81q3 + 54q2 + 12q) + 8

      = 9m + 2  where m = 81q3 + 54q2 + 12q

⇒ The End

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Study Materials List:

Solutions ⇒ Class 10th ⇒ Mathematics
1. Real Numbers
2. Polynomials
3. Pair of Linear Equations in Two Variables
4. Quadratic Equations
5. Arithmetic Progressions
6. Triangles
7. Coordinate Geometry
8. Introduction to Trigonometry
9. Some Applications of Trigonometry
10. Circles
11. Constructions
12. Areas Related to Circles
13. Surface Areas and Volumes
14. Statistics
15. Probability

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