If for z as real or complex,1+z2+z48=C0+C1z2+C2z4+⋯+C16z32,then

1+z2+z48=C0+C1z2+C2z4+…+C16z32 (1)Putting z = i, where i=−1,(1−1+1)8=C0−C1+C2−C3+…+C16or C0−C1+C2−C3+…+C16=1Also, putting z=ω,1+ω2+ω48=C0+C1ω2+C2ω4+…+C16ω32or C0+C1ω2+C2ω+C3+…+C16ω2=0 (2)Putting x=ω2,1+ω4+ω88=C0+C1ω4+C2ω8+…+C16ω64or C0+C1ω+C2ω2+…+C16ω=0 (3)Putting x = 1 38=C0+C1+C2+…+C16 (4)Adding (2), (3) and (4), we have 3C0+C3+…+C15=38or C0+C3+…+C15=37Similarly, first multiplying (1) by z and then putting 1,ω,ω2and adding, we getC1+C4+C7+C10+C13+C16=37Multiplying (1) z2 and then putting 1, ω,ω2and adding,We get C2+C5+C8+C11+C14=37

If for z as real or complex,1+z2+z48=C0+C1z2+C2z4+⋯+C16z32,then

1+z2+z48=C0+C1z2+C2z4+…+C16z32 (1)Putting z = i, where i=−1,(1−1+1)8=C0−C1+C2−C3+…+C16or C0−C1+C2−C3+…+C16=1Also, putting z=ω,1+ω2+ω48=C0+C1ω2+C2ω4+…+C16ω32or C0+C1ω2+C2ω+C3+…+C16ω2=0 (2)Putting x=ω2,1+ω4+ω88=C0+C1ω4+C2ω8+…+C16ω64or C0+C1ω+C2ω2+…+C16ω=0 (3)Putting x = 1 38=C0+C1+C2+…+C16 (4)Adding (2), (3) and (4), we have 3C0+C3+…+C15=38or C0+C3+…+C15=37Similarly, first multiplying (1) by z and then putting 1,ω,ω2and adding, we getC1+C4+C7+C10+C13+C16=37Multiplying (1) z2 and then putting 1, ω,ω2and adding,We get C2+C5+C8+C11+C14=37

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If for z as real or complex, ${(1 + z^{2} + z^{4})}^{8} = C_{0} + C_{1} z^{2} + C_{2} z^{4} + \dots + C_{16} z^{32}$ ,then

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$C_{0} - C_{1} + C_{2} - C_{3} + \dots + C_{16} = 1$

$C_{0} + C_{3} + C_{6} + C_{9} + C_{12} + C_{15} = 3^{7}$

$C_{2} + C_{5} + C_{8} + C_{11} + C_{14} = 3^{6}$

$C_{1} + C_{4} + C_{7} + C_{10} + C_{13} + C_{16} = 3^{7}$

answer is A, B, D.

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Detailed Solution

${(1 + z^{2} + z^{4})}^{8} = C_{0} + C_{1} z^{2} + C_{2} z^{4} + \dots + C_{16} z^{32} (1)$
Putting z = i, where $i = \sqrt{- 1}$ ,
$(1 - 1 + 1)^{8} = C_{0} - C_{1} + C_{2} - C_{3} + \dots + C_{16}$
or    $C_{0} - C_{1} + C_{2} - C_{3} + \dots + C_{16} = 1$
Also, putting $z = ω,$
${(1 + ω^{2} + ω^{4})}^{8} = C_{0} + C_{1} ω^{2} + C_{2} ω^{4} + \dots + C_{16} ω^{32}$
or $C_{0} + C_{1} ω^{2} + C_{2} ω + C_{3} + \dots + C_{16} ω^{2} = 0 (2)$
Putting $x = ω^{2}$ ,
${(1 + ω^{4} + ω^{8})}^{8} = C_{0} + C_{1} ω^{4} + C_{2} ω^{8} + \dots + C_{16} ω^{64}$
or $C_{0} + C_{1} ω + C_{2} ω^{2} + \dots + C_{16} ω = 0 (3)$
Putting x = 1
   $3^{8} = C_{0} + C_{1} + C_{2} + \dots + C_{16} (4)$
Adding (2), (3) and (4), we have
   $3 (C_{0} + C_{3} + \dots + C_{15}) = 3^{8}$
or $C_{0} + C_{3} + \dots + C_{15} = 3^{7}$
Similarly, first multiplying (1) by z and then putting 1, $ω, ω^{2}$ and adding, we get
$C_{1} + C_{4} + C_{7} + C_{10} + C_{13} + C_{16} = 3^{7}$
Multiplying (1) z²and then putting 1, $ω, ω^{2}$ and adding,
We get $C_{2} + C_{5} + C_{8} + C_{11} + C_{14} = 3^{7}$