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Question

Let the four roots of unity be z1,z2,z3, and z4, respectively.
Statement 1: z21+z22+z23+z24=0
Statement 2: z1+z2+z3+z4=0.

A
Both the statements are true, and Statement 2 is the correct explanation for Statement 1.
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B
Both the statements are true, but Statement 2 is not the correct explanation for Statement 1.
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C
Statement 1 is true and Statement 2 is false.
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D
Statement 1 is false and Statement 2 is true.
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Solution

The correct option is B Both the statements are true, but Statement 2 is not the correct explanation for Statement 1.
x4=1
x2=±1
x2=1 and x2=1
x=±1 and x=±i
Hence the four roots are
1,1,i,i.
Now
z1=1=z2
z3=i=z4
Hence
z21+z22+z23+z24
=1+1+(i)2+(i)2
=22
=0 ...(i)
And also
z1+z2+z3+z4
=11+ii
=0 ...(ii)
Both the statements are true, but Statement 2 is not the correct explanation for Statement 1.

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