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4.1 complex nubers — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHS4.1 complex nubers1 Mark
MCQ
If $z = \frac{{7 - i}}{{3 - 4i}}$ then ${z^{14}} = $
  • A
    ${2^7}$
  • B
    ${2^7}i$
  • C
    ${2^{14}}i$
  • ✓
    $ - {2^7}i$
✓

Answer

Correct option: D.
$ - {2^7}i$
d
(d) $z = \frac{{7 - i}}{{3 - 4i}} \times \frac{{3 + 4i}}{{3 + 4i}}$=$\frac{{21 + 25i + 4}}{{16 + 9}} = \frac{{25\,(1 + i)}}{{25}}$ = $(1 + i)$
${z^{14}} = {(1 + i)^{14}} = {[{(1 + i)^2}]^7}$= ${(2i)^7} = {2^7}{i^7} = - {2^7}i$.
 

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