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Question

If x+iy=a+ibc+id, then x4+y4+2x2y2 is equal to

A
(a+b)2(c+d)2
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B
a2+b2c2+d2
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C
(ab)2(cd)2
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D
a2b2c2d2
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Solution

The correct option is B a2+b2c2+d2
We have,
x+iy=a+ibc+id(x+iy)2=a+ibc+id¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(x+iy)2=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(a+ibc+id)(xiy)2=aibcid[¯¯¯¯¯¯¯¯¯(z)n=(¯¯¯z)n, ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯(z1z2)=¯¯¯¯¯z1¯¯¯¯¯z2]

Therefore,
x4+y4+2x2y2=(x2+y2)2=[(x+iy)(xiy)]2=(x+iy)2×(xiy)2=a+ibc+id×aibcid=a2+b2c2+d2

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