Let -1 be a cube root of unity- Then the minimum of the set - a - b - c -2-2- a - b - c- distinct non-zero integers equals

Name: JEE- Grade 11- Mathematics- Complex Numbers- Session 09- D10
Uploaded: 2022-07-04T10:36:42+05:30
Description: |a+bω +cω^2|^2 nbsp; =(a+bω +cω^2) (a+bω +cω^2) nbsp; =(a+bω +cω^2) (a+bω^2 +cω) =(a^2+abω^2+acω+abω+b^2ω^3+ bcω^2+acω^2 + bcω^4+c^2ω^3) =(a^2+ω^3(b^ ...

|a+bω +cω^2|^2 nbsp; =(a+bω +cω^2) (a+bω +cω^2) nbsp; =(a+bω +cω^2) (a+bω^2 +cω) =(a^2+abω^2+acω+abω+b^2ω^3+ bcω^2+acω^2 + bcω^4+c^2ω^3) =(a^2+ω^3(b^ ...

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Byju's Answer

Standard XII

Mathematics

Cube Root of a Complex Number

Let ω'=1 be a...

Question

Let $ω \neq 1$ be a cube root of unity. Then the minimum of the set
${| a + b ω + c ω^{2} |^{2} : a, b, c$ distinct non-zero integers} equals

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Solution

${| a + b ω + c ω^{2} |^{2}$
$= (a + b ω + c ω^{2}) ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ (a + b ω + c ω^{2})$
$= (a + b ω + c ω^{2}) (a + b ω^{2} + c ω)$
$= (a^{2} + a b ω^{2} + a c ω + a b ω + b^{2} ω^{3} + b c ω^{2} + a c ω^{2} + b c ω^{4} + c^{2} ω^{3})$
$= (a^{2} + ω^{3} (b^{2} + c^{2}) + a b (ω^{2} + ω) + a c (ω^{2} + ω) + b c (ω^{2} + ω^{4}))$
$= a^{2} + b^{2} + c^{2} - a b - a c - b c$
$= \frac{1}{2} [(a - b)^{2} + (b - c)^{2} + (c - a)^{2}]$
$L e t a > b > c \Rightarrow | a - b | \geq 1, | b - c | \geq 1, | a - c | \geq 2$
$\geq \frac{1}{2} [1 + 1 + 4]$
so, minimum of set ${| a + b ω + c ω^{2} |^{2}$ $= 3$

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