If 1, ω and ω^2 are the cube roots of unity, then the value of (1-ω+ω^2 ) (1+ω-ω^2 ) is equal to

Correct answer: (A) 4. (1-ω .+ω^2 ) (1+ω-ω^2 ) = (1+ω^2-ω ) (1+ω-ω^2 ) =(-ω-ω) (-ω^2-ω^2 ) (∵ 1+ω+ω^2=0 ) =(-2 ω) (-2 ω^2 ) =4 (ω^3 )=4 × 1 (∵ ω^3=1 ) =4

Q77 / 170 · Manipal (MET) 2020 · Mathematics

If $1, \omega$ and $\omega^2$ are the cube roots of unity, then the value of $\left(1-\omega+\omega^2\right)\left(1+\omega-\omega^2\right)$ is equal to

Correct answer
A
$4$
B
$0$
C
$2$
D
$3$

Solution

$\begin{aligned}(1-\omega & \left.+\omega^2\right)\left(1+\omega-\omega^2\right) \\ & =\left(1+\omega^2-\omega\right)\left(1+\omega-\omega^2\right) \\ & =(-\omega-\omega)\left(-\omega^2-\omega^2\right)\left(\because 1+\omega+\omega^2=0\right) \\ & =(-2 \omega)\left(-2 \omega^2\right)\end{aligned}$ $=4\left(\omega^3\right)=4 \times 1 \quad\left(\because \omega^3=1\right)$ $=4$

Hence, the correct answer is option: 'A'.

If $1, \omega$ and $\omega^2$ are the cube roots of unity, then the value of $\left(1-\omega+\omega^2\right)\left(1+\omega-\omega^2\right)$ is equal to

Solution

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