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Question

The least positive integer n such that (2i1+i)n is a positive integer, is 



A

16

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B

8

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C

4

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D

2

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Solution

The correct option is C

8


Let z=(2i1+i)n z=2i1+i×1i1i z=2i(1i)1i2 z=2i(1i)1+1    [ i2=1] z=2i(1i)2 z=ii2 z=i+1Now, zn=(1+i)nFor n=2,z2=(1+i)2=1+i2+2i=11+2i=2i                ...(1)

Since this is not a positive integer, 

For n =4,z4=(1+i)4=[(1+i)2]2=(2i)2         [Using (1)]=4i2=4             ...(2)

This is a negative integer,

For n = 8, 

z8=(1+i)8=[(1+i)4]2=(4)2    [Using (2)]=16

This is a positive integer,

This, z=(2i1+i)n is positive for n = 8

Therefore, 8 is the least positive integer such that (2i1+i)n is a positive integer.


Mathematics
RD Sharma
Standard XI

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