isi-entrance 2011 Q21

isi-entrance · India · solved Complex numbers 2 Roots of Unity and Cyclotomic Properties

Let $a < b < c$ be three real numbers and $w$ denote a complex cube root of unity. If $\left( a + bw + cw ^ { 2 } \right) ^ { 3 } + \left( a + bw ^ { 2 } + cw \right) ^ { 3 } = 0$, then which of the following must be true?
(a) $a + b + c = 0$
(b) $abc = 0$
(c) $ab + bc + ca = 0$
(d) $b = ( c + a ) / 2$.

Let $a < b < c$ be three real numbers and $w$ denote a complex cube root of unity. If $\left( a + bw + cw ^ { 2 } \right) ^ { 3 } + \left( a + bw ^ { 2 } + cw \right) ^ { 3 } = 0$, then which of the following must be true?\\
(a) $a + b + c = 0$\\
(b) $abc = 0$\\
(c) $ab + bc + ca = 0$\\
(d) $b = ( c + a ) / 2$.

Paper Questions

Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20 Q21