SATHEE: Complex Numbers 5 Question 13

Complex Numbers 5 Question 13

14. Let $ω$ be the complex number $\cos \frac{2 π}{3} + i \sin \frac{2 π}{3}$ . Then the number of distinct complex number $z$ satisfying $| \begin{array}{ccc} z + 1 & ω & ω^{2} ω & z + ω^{2} & 1 ω^{2} & 1 & z + ω \end{array} | = 0$ is equal to … .

(2010)

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Answer:

Correct Answer: 14. (1)

Solution:

$A = \begin{array}{ccc} ω & ω^{2} & 1 \\ ω^{2} & 1 & ω \end{array}$

$\begin{array}{lll} 0 & 0 & 0 \end{array}$

Now, $A^{2} = \begin{array}{llll} 0 & 0 & 0 \end{array}$ and $Tr (A) = 0, | A | = 0$

$\begin{array}{lll} 0 & 0 & 0 \end{array}$

$A^{3} = 0$

$\begin{aligned} \Rightarrow | \begin{array}{ccc} z + 1 & ω & ω^{2} \\ ω & z + ω^{2} & 1 \\ ω^{2} & 1 & z + ω \end{array} | = [A + z l] = 0 \\ \Rightarrow z^{3} = 0 \end{aligned}$

$\Rightarrow z = 0$ , the number of $z$ satisfying the given equation is 1 .

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Complex Numbers 5 Question 13

14. Let ω be the complex number cos⁡2π3+isin⁡2π3. Then the number of distinct complex number z satisfying |z+1ωω2 ωz+ω21 ω21z+ω|=0 is equal to … .

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14. Let $ω$ be the complex number $\cos \frac{2 π}{3} + i \sin \frac{2 π}{3}$ . Then the number of distinct complex number $z$ satisfying $| \begin{array}{ccc} z + 1 & ω & ω^{2} ω & z + ω^{2} & 1 ω^{2} & 1 & z + ω \end{array} | = 0$ is equal to … .