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Misc 18 - |1 - i|x = 2x, find non integral solutions - Miscellaneous

  1. Chapter 5 Class 11 Complex Numbers
  2. Serial order wise
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Misc, 18 Find the number of non-zero integral solutions of the equation |1 โ€“ ๐‘–|๐‘ฅ = 2๐‘ฅ . We need to find the value of x which should be an integer but not 0 Lets first find the value of |1 โ€“ ๐‘–| 1 โ€“ ๐‘– Complex number is of the form x + iy Where ๐‘ฅ = 1 ๐‘ฆ = โˆ’1 |1 โ€“ ๐‘–| = โˆš(๐‘ฅ^2+๐‘ฆ2) = โˆš((1)2+(โˆ’1)2) = โˆš(1+1) = โˆš2 Hence, |1 โ€“ ๐‘–| = โˆš2 Given |1 โ€“ ๐‘–|๐‘ฅ = 2๐‘ฅ Putting |1 โ€“ ๐‘–| = โˆš2 (โˆš2)๐‘ฅ = 2๐‘ฅ (โˆš2)๐‘ฅ = (โˆš2 ร— โˆš2)๐‘ฅ (โˆš2)๐‘ฅ = (โˆš2)^(2 ร— ๐‘ฅ) (โˆš2)๐‘ฅ = ใ€–โˆš2ใ€—^( 2๐‘ฅ) Comparing Powers ๐‘ฅ = 2๐‘ฅ ๐‘ฅ โˆ’2๐‘ฅ = 0 โˆ’ ๐‘ฅ = 0 ๐‘ฅ = 0 Hence, There is no non-zero integral solution

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