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Question

If z(1) is complex number such that z1z+1 is purely imaginary, then |z| is equal to

A
1
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B
2
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C
3
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D
5
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Solution

The correct option is A 1
Given z1z+1 is purely imaginary
Let z=x+iy
Now z1z+1=x+iy1x+iy+1=(x1)+iy(x+1)+iy
=(x1)+iy(x+1)+iy(x+1)iy(x+1)iy
=(x1)(x+1)i2y2+iy[x+1(x1)](x+1)2i2y2
=x21+y2+i2y(x+1)2+y2
z1z+1=x21+y2(x+1)2+y2+i2y(x+1)2y2
But z1z+1 is imaginary, Re (z1z+1)=0
x21+y2(x+1)2+y2=0
x2+y2=1
|z|=1.

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