Question

# Let the four roots of unity be z1,z2,z3, and z4, respectively.Statement 1: z21+z22+z23+z24=0Statement 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=1x2=±1x2=1 and x2=−1x=±1 and x=±iHence the four roots are1,−1,i,−i.Now z1=1=−z2z3=i=−z4Hencez21+z22+z23+z24=1+1+(i)2+(−i)2=2−2=0 ...(i)And alsoz1+z2+z3+z4=1−1+i−i=0 ...(ii)Both the statements are true, but Statement 2 is not the correct explanation for Statement 1.

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