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

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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

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  1. 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.

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