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STD 11 - 4.1 complex nubers — JEE STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceJEESTD 11 - 4.1 complex nubers4 Marks
MCQ
The least positive integer $\mathrm{n}$ such that $\frac{(2 \mathrm{i})^{\mathrm{n}}}{(1-\mathrm{i})^{\mathrm{n}-2}}, \mathrm{i}=\sqrt{-1}$ is a positive integer, is ..... .
  • A
    $2$
  • B
    $4$
  • ✓
    $6$
  • D
    $8$
✓

Answer

Correct option: C.
$6$
c
$\frac{(2 \mathrm{i})^{\mathrm{n}}}{(1-\mathrm{i})^{\mathrm{n}-2}}=\frac{(2 \mathrm{i})^{\mathrm{n}}}{(-2 \mathrm{i})^{\frac{\mathrm{n}-2}{2}}}$

$=\frac{(2 \mathrm{i})^{\frac{\mathrm{n}+2}{2}}}{(-1)^{\frac{\mathrm{n}-2}{2}}}=\frac{2^{\frac{n+2}{2}} ; \mathrm{i}^{\frac{n+2}{2}}}{(-1)^{\frac{n-2}{2}}}$

This is positive integer for $\mathrm{n}=6$

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