(C) Let $P(z) = z^4 + z^3 + z^2 + z + 1 = (z - z_1)(z - z_2)(z - z_3)(z - z_4)$.We want to find the value of $\prod_{i=1}^{4} (z_i + 2)$.Note that $\p

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(C) Let $P(z) = z^4 + z^3 + z^2 + z + 1 = (z - z_1)(z - z_2)(z - z_3)(z - z_4)$.We want to find the value of $\prod_{i=1}^{4} (z_i + 2)$.Note that $\prod_{i=1}^{4} (z_i + 2) = (-1)^4 \prod_{i=1}^{4} (-2 - z_i) = P(-2)$.Substituting $z = -2$ into the polynomial $P(z)$:$P(-2) = (-2)^4 + (-2)^3 + (-2)^2 + (-2) + 1$$P(-2) = 16 - 8 + 4 - 2 + 1$$P(-2) = 11$.Therefore,the product is $11$.

Class 11 Mathematics 1 Complex Numbers and Quadratic Equations: Complex Numbers Mix Examples-Complex numbers

If $z_1, z_2, z_3, z_4$ are the roots of the equation $z^4 + z^3 + z^2 + z + 1 = 0$,then $\prod_{i=1}^{4} (z_i + 2)$ is equal to:

A
$1$
B
$8$
C
$11$
D
$17$

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Chapter

1 Complex Numbers and Quadratic Equations: Complex Numbers

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Mix Examples-Complex numbers

Let $z, w \in \mathbb{C}$ satisfy $z^2 + \bar{w} = z$ and $w^2 + \bar{z} = w$. Then,the number of ordered pairs of complex numbers $(z, w)$ is equal to

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If $\alpha+i \beta$ and $\gamma+i \delta$ are the roots of $x^2-(3-2 i) x-(2 i-2)=0$,where $i=\sqrt{-1}$,then $\alpha \gamma+\beta \delta$ is equal to :

MediumJEE MAIN 2025

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If $1+2i$ is a root of the equation $x^4-3x^3+8x^2-7x+5=0$,then the sum of the squares of the other roots is

DifficultTS EAMCET 2025

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Let $z_1$ and $z_2$ be any two non-zero complex numbers such that $3|z_1| = 4|z_2|$. If $z = \frac{3z_1}{2z_2} + \frac{2z_2}{3z_1}$,then:

DifficultJEE MAIN 2019

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If $(x-iy)^{1/3} = a-ib$,then the value of $\frac{x}{a} + \frac{y}{b}$ is

MediumAP EAMCET 2022

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If $z_1, z_2, z_3, z_4$ are the roots of the equation $z^4 + z^3 + z^2 + z + 1 = 0$,then $\prod_{i=1}^{4} (z_i + 2)$ is equal to:

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Let $z, w \in \mathbb{C}$ satisfy $z^2 + \bar{w} = z$ and $w^2 + \bar{z} = w$. Then,the number of ordered pairs of complex numbers $(z, w)$ is equal to

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