AP Board Class 10 Maths Chapter 1 Real Numbers Ex 1.4 Textbook Solutions PDF: Download Andhra Pradesh Board STD 10th Maths Chapter 1 Real Numbers Ex 1.4 Book Answers ~ HSSlive: Plus One & Plus Two Notes & Solutions for Kerala State Board

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AP Board Class 10 Maths Chapter 1 Real Numbers Ex 1.4 Textbook Solutions PDF: Download Andhra Pradesh Board STD 10th Maths Chapter 1 Real Numbers Ex 1.4 Book Answers

AP Board Class 10th Maths Chapter 1 Real Numbers Ex 1.4 Textbooks Solutions and answers for students are now available in pdf format. Andhra Pradesh Board Class 10th Maths Chapter 1 Real Numbers Ex 1.4 Book answers and solutions are one of the most important study materials for any student. The Andhra Pradesh State Board Class 10th Maths Chapter 1 Real Numbers Ex 1.4 books are published by the Andhra Pradesh Board Publishers. These Andhra Pradesh Board Class 10th Maths Chapter 1 Real Numbers Ex 1.4 textbooks are prepared by a group of expert faculty members. Students can download these AP Board STD 10th Maths Chapter 1 Real Numbers Ex 1.4 book solutions pdf online from this page.

Andhra Pradesh Board Class 10th Maths Chapter 1 Real Numbers Ex 1.4 Textbooks Solutions PDF

Andhra Pradesh State Board STD 10th Maths Chapter 1 Real Numbers Ex 1.4 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 Ex 1.4 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 Ex 1.4 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 Ex 1.4 Textbooks. These Andhra Pradesh State Board Class 10th Maths Chapter 1 Real Numbers Ex 1.4 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 Ex 1.4 Books Solutions

Board	AP Board
Materials	Textbook Solutions/Guide
Format	DOC/PDF
Class	10th
Subject	Maths
Chapters	Maths Chapter 1 Real Numbers Ex 1.4
Provider	Hsslive

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AP Board Class 10th Maths Chapter 1 Real Numbers Ex 1.4 Textbooks Solutions with Answer PDF Download

Find below the list of all AP Board Class 10th Maths Chapter 1 Real Numbers Ex 1.4 Textbook Solutions for PDF’s for you to download and prepare for the upcoming exams:

Question 1.
Prove that the following are irrational,
i) 12√
ii) √3 + √5
iii) 6 + √2
iv) √5
v) 3 + 2√5
Answer:
i) 12√
On the contrary, suppose that is a 12√ rational number;
then 12√ is of the form where 𝑝𝑞 and q are integers.
∴ 12√ = 𝑝𝑞
⇒ 2√1 = 𝑞𝑝
(i.e.,) √2 is a rational number and it is a contradiction. This contradiction arised due to our supposition that 12√ is a rational number.
Hence 12√ is an irrational number.

ii) Suppose √3 + √5 is not an irrational number.
Then √3 + √5 must be a rational number.
√3 + √5 = 𝑝𝑞, q ≠ 0 and p, q ∈ Z
Squaring on both sides
but √15 is an irrational number.
𝑝2−8𝑞22𝑞2 is a rational number
(p² – 8q², 2q² ∈ Z, 2q² ≠ 0)
but an irrational number can’t be equal to a rational number, so our supposition that √3 + √5 is not an irrational number is false.
∴ √3 + √5 is an irrational number.

iii) 6 + √2
To prove: 6 + √2 is an irrational number.
Let us suppose that 6 + √2 is a rational number.
∴ 6 + √2 = 𝑝𝑞, q ≠ 0
⇒ √2 = 𝑝𝑞 – 6
⇒ √2 = Difference of two rational numbers
⇒ √2 = rational number But this contradicts the fact that √2 is an irrational number.
∴ Our supposition is wrong.
Hence the given statement is true.
6 + √2 is an irrational number.

iv) √5
To prove: √5 is an irrational number.
On the contrary, let us assume that √5 is a rational number.
∴ √5 = 𝑝𝑞, q ≠ 0
If p, q have a common factor, on cancelling the common factor let it be
reduces to 𝑎𝑏 where a, b are co-primes.
Now √5 = 𝑎𝑏, where HCF (a, b) = 1
Squaring on both sides we get
⇒ (√5)² = (𝑎𝑏)2
⇒ 5 = a2 b2
⇒ 5b² = a²
⇒ 5 divides a² and thereby 5 divides 5
Now, take a = 5c
then, a² = 25c²
i.e., 5b² = 25c²
⇒ b² = 5c²
⇒ 5 divides b² and thereby b.
⇒ 5 divides both b and a.
This contradicts that a and b are co-primes.
This contradiction arised due to our assumption that √5 is a rational number.
Hence our assumption is wrong and the given statement is true, i.e., √5 is an irrational number,

v) 3 + 2√5
To Prove: 3 + 2√5 is an irrational.
On the contrary, let us assume that 3 + 2√5 is a rational number.

Here p, q being integers we can say that 𝑝−3𝑞2𝑞 is a rational number.
This contradicts the fact that √5 is an irrational number. This is due to our assumption “3 + 2√5 is a rational number”.
Hence our assumption is wrong.
∴ 3 + 2√5 is an irrational number.

Question 2.
Prove that √p + √q is an irrational, where p, q are primes.
Answer:
Given that p, q are primes.
Hence fp and fq are irrationals.
[∵ p, q have no factors other than 1 ∵ they are primes.]
Now √p + √q = sum of two irrational numbers = an irrational number
Hence proved.

AP Board Textbook Solutions PDF for Class 10th Maths

AP Board Class 10 Maths Chapter 1 Real Numbers Textbook Solutions PDF

AP Board Class 10 Maths Chapter 2 Sets Textbook Solutions PDF

AP Board Class 10 Maths Chapter 3 Polynomials Textbook Solutions PDF

AP Board Class 10 Maths Chapter 4 Pair of Linear Equations in Two Variables Textbook Solutions PDF

AP Board Class 10 Maths Chapter 5 Quadratic Equations Textbook Solutions PDF

AP Board Class 10 Maths Chapter 6 Progressions Textbook Solutions PDF

AP Board Class 10 Maths Chapter 7 Coordinate Geometry Textbook Solutions PDF

AP Board Class 10 Maths Chapter 8 Similar Triangles Textbook Solutions PDF

AP Board Class 10 Maths Chapter 9 Tangents and Secants to a Circle Textbook Solutions PDF

AP Board Class 10 Maths Chapter 10 Mensuration Textbook Solutions PDF

AP Board Class 10 Maths Chapter 11 Trigonometry Textbook Solutions PDF

AP Board Class 10 Maths Chapter 12 Applications of Trigonometry Textbook Solutions PDF

AP Board Class 10 Maths Chapter 13 Probability Textbook Solutions PDF

AP Board Class 10 Maths Chapter 14 Statistics Textbook Solutions PDF

Andhra Pradesh Board Class 10th Maths Chapter 1 Real Numbers Ex 1.4 Textbooks for Exam Preparations

Andhra Pradesh Board Class 10th Maths Chapter 1 Real Numbers Ex 1.4 Textbook Solutions can be of great help in your Andhra Pradesh Board Class 10th Maths Chapter 1 Real Numbers Ex 1.4 exam preparation. The AP Board STD 10th Maths Chapter 1 Real Numbers Ex 1.4 Textbooks study material, used with the English medium textbooks, can help you complete the entire Class 10th Maths Chapter 1 Real Numbers Ex 1.4 Books State Board syllabus with maximum efficiency.

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Tuesday, June 7, 2022