Given that √3 is irrational, prove that 5 + 2√3 is irrational

This question is Similar to Question-27 CBSE Class 10 Sample Paper for 2021 Boards - Maths Standard

Slide70.JPG

Slide71.JPG
Slide72.JPG

Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


Transcript

Question 26 Given that √3 is irrational, prove that 5 + 2√3 is irrationalWe have to prove 5 + 2√3 is irrational Let us assume the opposite, i.e., 5 + 2√𝟑 is rational Hence, 5 + 2√3 can be written in the form 𝑎/𝑏 where a and b (b≠ 0) are co-prime (no common factor other than 1) Hence, 5 + 2√𝟑 = 𝒂/𝒃 2√3 = 𝑎/𝑏 − 5 √3 = 𝑎/2𝑏 − 5/2 √𝟑 = (𝒂 − 𝟓𝒃)/𝟐𝒃 Here, (𝒂 − 𝟓𝒃)/𝟐𝒃 is a rational number But √3 is irrational Since, Rational ≠ Irrational This is a contradiction ∴ Our assumption is incorrect Hence, 5 + 2√𝟑 is irrational Hence proved.

Ask a doubt
Davneet Singh's photo - Co-founder, Teachoo

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.