सम्मिश्र संख्या \(\rm i+2\over1-3i\) का संयुग्म क्या है?

  1. \(\rm {-1\over10}-{7\over10}i\)
  2. \(\rm {1\over10}-{7\over10}i\)
  3. \(\rm {1\over5}-{7\over5}i\)
  4. \(\rm {-1\over5}-{7\over5}i\)

Answer (Detailed Solution Below)

Option 1 : \(\rm {-1\over10}-{7\over10}i\)
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Detailed Solution

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संकल्पना:

माना कि z = x + iy एक सम्मिश्र संख्या है।

  • z का मापांक = \(\left| {\rm{z}} \right| = {\rm{}}\sqrt {{{\rm{x}}^2} + {{\rm{y}}^2}} = {\rm{}}\sqrt {{\rm{Re}}{{\left( {\rm{z}} \right)}^2} + {\rm{Im\;}}{{\left( {\rm{z}} \right)}^2}}\)
  • arg (z) = arg (x + iy) = \({\tan ^{ - 1}}\left( {\frac{y}{x}} \right)\)
  • z का संयुग्म = z̅ = x – iy

गणना:

दी गयी सम्मिश्र संख्या z = \(\rm i+2\over1-3i\) है। 

z = \(\rm {i+2\over1-3i}\times{1+3i\over1+3i}\)

z = \(\rm i+2+6i+3i^2\over1^2-(3i)^2\)

z = \(\rm 7i-1\over10\)

z = \(\rm {-1\over10}+{7\over10}i\)

z का संयुग्म = (z̅) = \(\rm {-1\over10}-{7\over10}i\)

नोट:

सम्मिश्र संयुग्म में समान वास्तविक भाग होता है जो z और समान काल्पनिक भाग होता है लेकिन विपरीत चिन्ह के साथ।

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