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## Andhra Pradesh Board Class 10th Maths Chapter 1 Real Numbers InText Questions Textbooks Solutions PDF

Andhra Pradesh State Board STD 10th Maths Chapter 1 Real Numbers InText Questions Books Solutions with Answers are prepared and published by the Andhra Pradesh Board Publishers. It is an autonomous organization to advise and assist qualitative improvements in school education. If you are in search of AP Board Class 10th Maths Chapter 1 Real Numbers InText Questions Books Answers Solutions, then you are in the right place. Here is a complete hub of Andhra Pradesh State Board Class 10th Maths Chapter 1 Real Numbers InText Questions solutions that are available here for free PDF downloads to help students for their adequate preparation. You can find all the subjects of Andhra Pradesh Board STD 10th Maths Chapter 1 Real Numbers InText Questions Textbooks. These Andhra Pradesh State Board Class 10th Maths Chapter 1 Real Numbers InText Questions Textbooks Solutions English PDF will be helpful for effective education, and a maximum number of questions in exams are chosen from Andhra Pradesh Board.## Andhra Pradesh State Board Class 10th Maths Chapter 1 Real Numbers InText Questions Books Solutions

Board | AP Board |

Materials | Textbook Solutions/Guide |

Format | DOC/PDF |

Class | 10th |

Subject | Maths |

Chapters | Maths Chapter 1 Real Numbers InText Questions |

Provider | Hsslive |

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## AP Board Class 10th Maths Chapter 1 Real Numbers InText Questions Textbooks Solutions with Answer PDF Download

Find below the list of all AP Board Class 10th Maths Chapter 1 Real Numbers InText Questions Textbook Solutions for PDF’s for you to download and prepare for the upcoming exams:Question 1.

Find q and r for the following pairs of positive integers a and b, satisfying a = bq + r. (Page No. 3)

i) a = 13, b = 3

Answer:

13 = 3 × 4 + 1

here q = 4 ; r = 1

ii) a = 8, b = 80

Answer:

Take a = 80, b = 8

80 = 8 × 10 + 0 here q = 10 ; r = 0

iii) a = 125, b = 5

Answer:

125 = 5 × 25 + 0

here q = 25 ; r = 0

iv) a = 132, b = 11

Answer:

132 = 11 × 12 + 0

here q = 12 ; r = 0

Question 2.

Find the HCF of the following by using Euclid division lemma,

i) 50 and 70 (Page No. 4)

Answer:

For given two positive integers a > b;

there exists unique pair of integers q and r satisfying a = bq + r; 0≤r<b.

∴ 70 = 50 × 1 + 20

Here a = 70, b = 50, q = 1, r = 20.

Now consider 50, 20

50 = 20 × 2 + 10

Here a = 50, b = 20, q = 2, r = 10.

Now taking 20 and 10.

20 = 10 × 2 + 0

Here the remainder is zero.

∴ 10 is the HCF of 70 and 50.

ii) 96 and 72

Answer:

96 = 72 × 1 + 24

72 = 24 × 3 + 0

∴ HCF = 24

iii) 300 and 550

Answer:

550 = 300 × 1 + 250

300 = 250 × 1 + 50

250 = 50 × 5 + 0

∴ HCF = 50

iv) 1860 and 2015

Answer:

2015 = 1860 × 1 + 155

1860 = 155 × 12 + 0

∴ HCF = 155

Think & Discuss

Question 1.

From the above questions in ‘DO THIS’, what is the nature of q and r? (Page No. 3)

Answer:

Given: a = bq + r

q > 0 and r lies in between 0 and b

i.e. q > 0 and 0 ≤ r < b

Question 2.

Can you find the HCF of 1.2 and 0.12? Justify your answer. (Page No. 4)

Answer:

Given: 1.2 and 0.12

we have 1.2 = 1210 = 120100

0.12 = 12100

Now considering the numerators 12 and 120, their HCF is 12.

∴ HCF of 1.2 and 0.12 is 12100 = 0.12

i.e., if x is a factor of y then x is the HCF of x and y.

Question 3.

If r = 0, then what is the relationship between a, b and q in a = bq + r of Euclid divison lemma? (Page No. 6)

Answer:

Given: r = 0 in a = bq + r then a = bq

i.e., b divides a completely.

i.e., b is a factor of a.

Do this

Question 1.

Express 2310 as a product of prime factors. Also see how your friends have factorized the number. Have they done it as you ? Verify your final product with your friend’s result. Try this for 3 or 4 more numbers. What do you conclude? (Page No. 7)

Answer:

Given: 2310

2310 = 2 × 1155

= 2 × 3 × 385

= 2 × 3 × 5 × 77

2310 = 2 × 3 × 5 × 7 × 11

We notice that this prime factorization is unique.

And also notice that prime factorization of any number is unique i.e., every composite number can be expressed as a product of primes and this factorization is unique.

E.g: 144 = 2 × 72

= 2 × 2 × 36

= 2 × 2 × 2 × 18

= 2 × 2 × 2 × 2 × 9

= 2 × 2 × 2 × 2 × 3 × 3

= 24 × 32

320 = 2 × 160

= 2 × 2 × 80

= 2 × 2 × 2 × 40

= 2 × 2 × 2 × 2 × 20

= 2 × 2 × 2 × 2 × 2 × 10

= 2 × 2 × 2 × 2 × 2 × 2 × 5

= 26 × 5

125 = 5 × 25

= 5 × 5 × 5

= 5^{3}

Question 2.

Find the HCF and LCM of the following given pairs of numbers by prime factorization, (Page No. 8)

i) 120, 90

Answer:

We have 120 = 2 × 2 × 2 × 3 × 5

= 2^{3} × 3 × 5

90 = 2 × 3 × 3 × 5

= 2 × 3^{2} × 5

∴ HCF = 2 × 3 × 5 = 30

LCM = 2<sup>3</sup> × 3<sup>2</sup> × 5 = 360

ii) 50, 60

Answer:

We have

50 = 2 × 5 × 5 = 2 × 5^{2}

60 = 2 × 2 × 3 × 5 = 2^{2} × 3 × 5

∴ HCF = 2 × 5 = 10

LCM = 2^{2} × 3 × 5^{2} = 300

iii) 37, 49

Answer:

We have

37 = 1 × 37

49 = 7 × 7 = 7^{2}

∴ HCF = 1

LCM = 37 × 7^{2}

Note: H.C.F. of two relatively prime numbers is 1 and LCM is equal to product of the numbers.

Try this

Question 1.

Show that 3^{n} × 4^{m} cannot end with the digit 0 or 5 for any natural numbers ‘n’ and’m’. (Page No. 8)

Answer:

Given number is 3^{4} × 4^{m}.

So the prime factors to it are 3 and 2 only.

I: but if a number want to be end with zero it should have 2 and 5 as its prime factors, but the given hasn’t ‘5’ as its prime factor.

So it cannot be end with zero.

II : now if a number went to be end with 5 it should have ‘5’ as its one of prime factors. But given 3^{n} × 4^{m} do not have 5 as a factor.

So it cannot be end with 5.

Hence proved.

Do this

Question 1.

Write the following terminating decimals in the form of p/q, q ≠ 0 and p, q are co-primes.

i) 15.265

ii) 0.1255

iii) 0.4

iv) 23.34

v) 1215.8

What can you conclude about the denominators through this process? (Page No. 10)

Answer:

i) 15.265

ii) 0.1255

iii) 0.4

0.4 = 410 = 25

iv) 23.34

v) 1215.8

Two and five are the factors for the denominator.

Question 2.

Write the following rational numbers in the form of p/q, where q is of the form 2^{n}.5^{m} where n, m are non-negative integers and then write the numbers in their decimal form. (Page No. 11)

i) 34

ii) 725

iii) 5164

iv) 1425

v) 80100

Answer:

i) 34

34 = 32×2 = 322

Decimal form of 34 = 0.75

ii) 725

725 = 75×5 = 752

Decimal form of 725 = 0.28

iii) 5164

5164 = 5126

[∵ 64 = 2 × 32

= 2^{2} × 16

= 2^{3} × 8

= 2^{4} × 4 = 2^{5} × 2 = 2^{6}]

Decimal form of 5164 = 0.796875

iv) 1425

v) 80100

80100 = 8022×52 = 80102 = 0.80

Question 3.

Write the following rational numbers as decimal form and find out the block of repeating digits in the quotient. (Page No. 11)

i) 13

ii) 27

iii) 511

iv) 1013

Answer:

i) 13

13 = 0.3333…. = 0.3⎯⎯⎯

Block of digits, repeating in the quotient = period = 3.

ii) 27

Decimal form of 27 = 0.285714….

Repeating part/period = 285714

∴ 27 = 0.285714⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

iii) 511

Period = 45

Decimal form of 511 = 0.454545.

= 0.45⎯⎯⎯⎯⎯⎯

iv) 1013

Decimal form of 1013 = 0.769230.

= 0.769230⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯

Period = 769230

Do this

Question 1.

Verify the statement proved above for p = 2, p = 5 and for a^{2} = 1, 4, 9, 25, 36, 49, 64 and 81. (Page No. 14)

Answer:

From the above we can conclude that if a prime number ‘p’ divides a^{2}, then it also divides a.

Think and Discuss

Question 1.

Write the nature of y, a and x in y = a^{x}. Can you determine the value of x for a given y? Justify your answer. (Page No. 17)

Answer:

y = a^{x} here a ≠ 0

We can determine the value of ‘x’ for a given y.

for example y = 5, a = 2

We cannot express y = a^{x} for y = 5, a = 2 and for y = 7, a = 3, we cannot express seven (7) as a power of 3.

Question 2.

You know that 2^{1} = 2, 4^{1} = 4, 8^{1} = 8 and 10^{1} = 10. What do you notice about the values of log_{2} 2, log_{4} 4, log_{8} 8 and log_{10} 10? What can you generalise from this? (Page No. 18)

Answer:

From the graph log_{2} 2 = log_{4} 4 = log_{8} 8 = log_{10} 10 = 1

We conclude that log_{a} a = 1 where a is a natural number.

Question 3.

Does log_{10} 0 exist? (Page No. 18)

Answer:

No, log_{10} 0 doesn’t exist, i.e a^{x} ≠ 0 ∀ a, x ∈ N.

Question 4.

We know that, if 7 = 2^{x} then x = log_{2} 7. Then what is the value of 2log27? Justify your answer. Generalise the above by taking some more examples for 𝐚loga𝐍. (Page No. 21)

Answer:

We know that if 7 = 2^{x} then x = log_{2} 7

We want to find the value of 2log27;

Now put log_{2} 7 = x in the given

∴ 2log27 = 2^{x} = 7 (given)

∴ 2log27 = 7

Thus 𝐚loga𝐍 = N

a) 3log38

Answer:

If x = 3log38 then

log_{3} x = log_{3} 8

⇒ x = 8

b) 5log510

Answer:

If y = 5log510

then log_{5} y = log_{5} 10

⇒ y = 10

Do this

Question 1.

Write the powers to which the bases to be raised in the following. (Page No. 18)

i) 64 = 2^{x}

Answer:

64 = 2^{x}

We know that

64 = 2 × 32

= 2 × 2 × 16

= 2 × 2 × 2 × 8

= 2 × 2 × 2 × 2 × 4

= 2 × 2 × 2 × 2 × 2 × 2

64 = 2^{6}

⇒ x = 6

ii) 100 = 5^{b}

Answer:

Here also 100 cannot be written as any power of 5.

i.e., there exists no integer for b such that 5^{b} = 100

iii) 181 = 3^{c}

Answer:

We know that 81 = 3 x 27

= 3 × 3 × 9

= 3 × 3 × 3 × 3

= 34

∴ 181 = 3^{-4 } [∵ a^{-m} = 1am]

∴ c = – 4

iv) 100 = 10^{z}

Answer:

100 = 10^{2}

z = 2

v) 1256 = 4^{a}

Answer:

We know that 256 = 4 × 64

= 4 × 4 × 16

= 4 × 4 × 4 × 4

∴ 1256 = 4^{-4}

∴ a = – 4

Question 2.

Express the logarithms of the following into sum of the logarithms. (Page No. 19)

i) 35 × 46

Answer:

log xy = log x + log y

log_{10}35 × 46 = log_{10}35 + log_{10}46

ii) 235 × 437

Answer:

log_{10}235 × 437 = log_{10}235 + log_{10}437 [∵ log xy = log x + log y]

iii) 2437 × 3568

Answer:

log_{10} 2437 × 3568 = log_{10}2437 + log_{10}3568 [∵ log xy = log x + log y]

Question 3.

Express the logarithms of the follow¬ing into difference of the logarithms. (Page No. 20)

i) 2334

Answer:

log_{10} = 2334 = log_{10} 23 – log_{10} 34

[∵ log 𝑥𝑦 = log x – log y]

ii) 373275

Answer:

log_{10} = 373275 = log_{10} 373 – log_{10} 275

[∵ log 𝑥𝑦 = log x – log y]

iii) 45253734

Answer:

log_{10} = 45253734 = log_{10} 4525 – log_{10} 3734

[∵ log 𝑥𝑦 = log x – log y]

iv) 50553303

Answer:

log_{10} = 50553303 = log_{10} 5055 – log_{10} 3303

[∵ log 𝑥𝑦 = log x – log y]

Question 4.

By using the formula log_{a}x^{n} = n log_{a} x, convert the following. (Page No. 21)

i) log_{2} 7^{25}

Answer:

log_{2} 7^{25} = 25 log_{2} 7

ii) log_{5} 8^{50}

Answer:

log_{5} 8^{50} = 50 log_{5} 8 = 50 log_{5} 23

= 3 × 50 log_{5}2 = 150 log_{5}2

iii) log 5^{23}

Answer:

log 5^{23} = 23 log 5

iv) log 1024

Answer:

log 1024 = log 2^{10} [∵ 1024 = 2^{10}]

= 10 log 2

Try this

Question 1.

Write the following relation in exponential form and find the values of respective variables. (Page No. 18)

i) log_{2}32 = x

Answer:

log_{2}32 = x

⇒ log_{2}2^{5} = x [∵ 32 = 2^{5}]

⇒ 5 log_{2}2 = x [∵ log a^{m} = m log a]

⇒ 5 × 1 = x [∵ log_{a} a = 1]

∴ x = 5

ii) log_{5}625 = y

Answer:

log_{5}625 = y

⇒ log_{5}4 = y [∵ 625 = 5^{4}]

⇒ 4 log_{5} 5 = y [∵ log a^{m} = m log a]

⇒ 4 × 1 = y [∵ log_{a} a = 1]

∴ y = 4

iii) log_{10}10000 = z

Answer:

log_{10}10000 = z

=> log_{10}10^{4} = z [∵ 10000 = 10 × 10 × 10 × 10 = 10^{4}]

=> 4 log_{10}10 = z [∵ log a^{m} = m log a]

=> 4 × 1 = z [∵ log_{a} a = 1]

∴ z = 4

iv) log71343 = -a

Answer:

Question 2.

i) Find the value of log_{2}32. (Page No. 21)

Answer:

log_{2} 32 = log_{2} 2^{5}

[∵ 32 = 2 × 2 × 2 × 2 × 2 = 2^{5}]

= 5 log_{2} 2 [∵ log a^{m} = m log a]

= 5 × 1 [∵ log_{a} a = 1]

= 5

ii) Find the value of log_{c} √c.

Answer:

= 12 × 1 [∵ log_{a} a = 1]

= 12

iii) Find the value of log_{10}0.001

Answer:

iv) Find the value of log23827

Answer:

## AP Board Textbook Solutions PDF for Class 10th Maths

- AP Board Class 10 Maths Chapter 1 Real Numbers Ex 1.1 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 1 Real Numbers Ex 1.2 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 1 Real Numbers Ex 1.3 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 1 Real Numbers Ex 1.4 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 1 Real Numbers Ex 1.5 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 1 Real Numbers Optional Exercise Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 1 Real Numbers InText Questions Textbook Solutions PDF

- AP Board Class 10 Maths Chapter 2 Sets Ex 2.1 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 2 Sets Ex 2.2 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 2 Sets Ex 2.3 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 2 Sets Ex 2.4 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 2 Sets InText Questions Textbook Solutions PDF

- AP Board Class 10 Maths Chapter 3 Polynomials Ex 3.1 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 3 Polynomials Ex 3.2 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 3 Polynomials Ex 3.3 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 3 Polynomials Ex 3.4 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 3 Polynomials Optional Exercise Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 3 Polynomials InText Questions Textbook Solutions PDF

- AP Board Class 10 Maths Chapter 4 Pair of Linear Equations in Two Variables Ex 4.1 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 4 Pair of Linear Equations in Two Variables Ex 4.2 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 4 Pair of Linear Equations in Two Variables Ex 4.3 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 4 Pair of Linear Equations in Two Variables Optional Exercise Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 4 Pair of Linear Equations in Two Variables InText Questions Textbook Solutions PDF

- AP Board Class 10 Maths Chapter 5 Quadratic Equations Ex 5.1 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 5 Quadratic Equations Ex 5.2 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 5 Quadratic Equations Ex 5.3 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 5 Quadratic Equations Ex 5.4 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 5 Quadratic Equations Optional Exercise Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 5 Quadratic Equations InText Questions Textbook Solutions PDF

- AP Board Class 10 Maths Chapter 6 Progressions Ex 6.1 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 6 Progressions Ex 6.2 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 6 Progressions Ex 6.3 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 6 Progressions Ex 6.4 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 6 Progressions Ex 6.5 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 6 Progressions Optional Exercise Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 6 Progressions InText Questions Textbook Solutions PDF

- AP Board Class 10 Maths Chapter 7 Coordinate Geometry Ex 7.1 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 7 Coordinate Geometry Ex 7.2 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 7 Coordinate Geometry Ex 7.3 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 7 Coordinate Geometry Ex 7.4 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 7 Coordinate Geometry Optional Exercise Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 7 Coordinate Geometry InText Questions Textbook Solutions PDF

- AP Board Class 10 Maths Chapter 8 Similar Triangles Ex 8.1 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 8 Similar Triangles Ex 8.2 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 8 Similar Triangles Ex 8.3 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 8 Similar Triangles Ex 8.4 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 8 Similar Triangles Optional Exercise Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 8 Similar Triangles InText Questions Textbook Solutions PDF

- AP Board Class 10 Maths Chapter 9 Tangents and Secants to a Circle Ex 9.1 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 9 Tangents and Secants to a Circle Ex 9.2 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 9 Tangents and Secants to a Circle Ex 9.3 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 9 Tangents and Secants to a Circle Optional Exercise Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 9 Tangents and Secants to a Circle InText Questions Textbook Solutions PDF

- AP Board Class 10 Maths Chapter 10 Mensuration Ex 10.1 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 10 Mensuration Ex 10.2 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 10 Mensuration Ex 10.3 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 10 Mensuration Ex 10.4 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 10 Mensuration Optional Exercise Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 10 Mensuration InText Questions Textbook Solutions PDF

- AP Board Class 10 Maths Chapter 11 Trigonometry Ex 11.1 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 11 Trigonometry Ex 11.2 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 11 Trigonometry Ex 11.3 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 11 Trigonometry Ex 11.4 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 11 Trigonometry Optional Exercise Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 11 Trigonometry InText Questions Textbook Solutions PDF

- AP Board Class 10 Maths Chapter 12 Applications of Trigonometry Ex 12.1 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 12 Applications of Trigonometry Ex 12.2 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 12 Applications of Trigonometry Optional Exercise Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 12 Applications of Trigonometry InText Questions Textbook Solutions PDF

- AP Board Class 10 Maths Chapter 13 Probability Ex 13.1 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 13 Probability Ex 13.2 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 13 Probability Optional Exercise Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 13 Probability InText Questions Textbook Solutions PDF

- AP Board Class 10 Maths Chapter 14 Statistics Ex 14.1 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 14 Statistics Ex 14.2 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 14 Statistics Ex 14.3 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 14 Statistics Ex 14.4 Textbook Solutions PDF
- AP Board Class 10 Maths Chapter 14 Statistics InText Questions Textbook Solutions PDF

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