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CBSE 10th Class Mathematics

Past year question paper

Year 2020

SET 1, Code Number – 30/5/1

Question Number – 22 [a]

irrational number

under root number

prime number

# Show that 5 + 2 √7 is an irrational number, where √7 is given to be an irrational number

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Let, 5 + 2√7 be rational.

So 5 + 2√7 =𝑎/𝑏,

𝑤ℎ𝑒𝑟𝑒′𝑎′𝑎𝑛𝑑′𝑏′𝑎𝑟𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 𝑎𝑛𝑑 𝑏 ≠ 0

2√7 = [a/b] – 5

√7 = [a – (5b)] / 2b

Since ‘a’ and ‘b’ are integers a – 5b is also an integer.[a – (5b)] / 2b is rational. So RHS is rational.

LHS should be rational. but it is given that √7 is irrational .Our assumption is wrong.

So 5 + 2√7 is an irrational number.