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Cube roots of unity - (Concept)
Let $z$ be the cube root of unity (1)
So, $z^3=1$
$
\begin{aligned}
& \Rightarrow z^3-1=0 \\
& \Rightarrow(z-1)\left(z^2+z+1\right)=0
\end{aligned}
$
$
\Rightarrow z-1=0 \text { or } z^2+z+1=0
$
$
\therefore \mathrm{z}=1 \text { or } \mathrm{z}=\frac{-1 \pm \sqrt{(1-4)}}{2}=\frac{-1 \pm \mathrm{i} \sqrt{3}}{2}
$
Therefore, $z=1, z=\frac{-1+i \sqrt{3}}{2}$ and $z=\frac{-1-i \sqrt{3}}{2}$
If the second root is represented by $\omega$, then the third root will be represented by $\omega^2$ (we can check that by squaring the second root, we get the third root)
$
\omega=\frac{-1+\mathrm{i} \sqrt{3}}{2}, \omega^2=\frac{-1-\mathrm{i} \sqrt{3}}{2}
$
So, $1, \omega, \omega^2$ are cube roots of unity and $\omega, \omega^2$ are the non-real complex root of unity.
Properties of Cube roots of unity
i) $1+\omega+\omega^2=0$ and $\omega^3=1$ (Using sum and product of roots relations for the equation $z^3-1=0$ )
ii) To find $\omega^n$, first we write $\omega$ in multiple of 3 with the remainder being 0 or 1 or 2 .
Now $\omega^{\mathrm{n}}=\omega^{3 \mathrm{q}}+\mathrm{r}=\left(\omega^3\right)^{\mathrm{q}} \cdot \omega^{\mathrm{r}}=\omega^{\mathrm{r}}($ Where r is from $0,1,2)$
$\mathrm{Eg}, \omega^{121}=\omega^{3.40+1}=\left(\omega^3\right)^{40} \cdot \omega^1=\omega$
iii) $|\omega|=\left|\omega^2\right|=1, \arg (\omega)=2 \pi / 3, \arg \left(\omega^2\right)=4 \pi / 3$ or $-2 \pi / 3$
iv) We can see that $\omega$ and $\omega^2$ differ by the minus sign of the imaginary part hence $\bar{\omega}=\omega^2$
v) Cube roots of -1 are $-1,-\omega,-\omega^2$
vi) The cube roots of unity when represented on the complex plane have their point on vertices of a triangle circumscribed by a unit circle whose one vertices lies on the +ve X-axis.
| Exam | Chapter |
| JEE MAIN | Complex numbers and quadratic equations |
equals
| A. |
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If are the roots of the equation
then
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Let be a root of the quadratic equation,
. If
then arg z is equal to :
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Let . If
and
respectively denote the real and imaginary parts of
, then:
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Let If
and
then a and b are the roots of the quadratic equation:
| A. |
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| D. |
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If a and b are real numbers such that , where
, then a+b is equal to.
| A. |
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| D. |
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The sum of 162th power of the roots of the equation is ________.
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If f(x) and g(x) are two polynomials such that the polynomials is divisible by
, then P(1) is equal to ______
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equals
| A. |
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If is a cube root of unity, and
| A. |
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| B. |
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| C. |
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| D. |
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If then
is equal to:
| A. |
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If , then
is equal to __________.
| A. |
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Let be a root of equation
. Then the value of
is equal to:
| A. |
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| B. |
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| C. |
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| D. |
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If and
are the distinct roots of the equation
,then the value of
is equal to?
| A. |
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| B. |
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| D. |
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Then the value of
is _____
| A. |
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| D. |
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If then p and q are roots of the equation:
| A. |
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| B. |
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| C. |
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| D. |
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Let $\omega$ be a complex number such that $2 w+1=z$ where $z=\sqrt{-3}$ if
$
\left|\begin{array}{rrr}
1 & 1 & 1 \\
1 & -\omega^2-1 & \omega^2 \\
1 & \omega^2 & \omega^7
\end{array}\right|=3 \mathrm{k},
$
then k is equal to :
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If α, β C are the distinct roots, of the equation x2−x+1=0, then α101+β107 is
equal to:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The least positive integer n for which
is
| A. |
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| B. |
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| D. |
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Let $\alpha, \beta$ be the roots of the equation $x^2-\sqrt{2} x+2=0$, Then $\alpha^{14}+\beta^{14}$ is equal to
| A. |
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| B. |
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| C. |
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| D. |
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If $\alpha$ satisfies the equation $\mathrm{x}^2+\mathrm{x}+1=0$ and $(1+\alpha)^7=\mathrm{A}+\mathrm{B} \alpha+\mathrm{C} \alpha^2, \mathrm{~A}, \mathrm{~B}, \mathrm{C} \geq 0$, then $5(3 \mathrm{~A}-2 \mathrm{~B}-\mathrm{C})$ is equal to ______
| A. |
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| B. |
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| D. |
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A is a square matrix of order 3. If $\omega$ is the cube root of unity. Then $\left(w^{12} A\right)^{\theta{\text { }}}$ is
| A. |
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| B. |
|
| C. |
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| D. |
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If $w$ is the complex cube root of unity $\left(1+w^2\right)^{11}=a+b w$, Then ordered pair $(a, b)$ is
| A. |
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| B. |
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| C. |
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| D. |
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If $\omega$ is the cube root of unity, then $\left(3+5 \omega+3 \omega^2\right)^2+\left(3+3 \omega+5 \omega^2\right)^2=$
| A. |
|
| B. |
|
| C. |
|
| D. |
|
$
\text { What is the value of } w+w^2 \text {, if these are Cube roots of unity, other than } 1 \text {. }
$
| A. |
|
| B. |
|
| C. |
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| D. |
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The cube roats of -27 are
| A. |
|
| B. |
|
| C. |
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| D. |
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If $=\sum_{\mathrm{r}=1}^{24}\left(\omega^{\mathrm{r}}+\frac{1}{\omega^{\mathrm{r}}}\right)^2$, where $\omega$ is complex cube root of unity, then value of $\frac{k}{6}$ is
| A. |
|
| B. |
|
| C. |
|
| D. |
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The value of $\sum_{k=1}^6\left(\cos \left(\frac{2 k \pi}{7}\right)+i \sin \left(\frac{2 k \pi}{7}\right)\right)$ is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Let $\omega$ be a cube root of unity not equal to 1 . Then the maximum possible value of $\left|a+b w+c w^2\right|$ where $a, b, c \in\{+1,-1\}$ is:
| A. |
|
| B. |
|
| C. |
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| D. |
|
$\begin{aligned} & \text { Let } \omega=-\frac{1}{2}+i \frac{\sqrt{3}}{2} . \text { Then the value of the determinant } \\ & \left|\begin{array}{ccc}1 & 1 & 1 \\ 1 & -1-\omega^2 & \omega^2 \\ 1 & \omega^2 & \omega^4\end{array}\right|_{\text {is }}\end{aligned}$
| A. |
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| B. |
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| C. |
|
| D. |
|
What is the value of $\left(x+\frac{1}{x}\right)^2+\left(x^2+\frac{1}{x^2}\right)^2+\ldots+\left(x^9+\frac{1}{x^9}\right)^2$ when $1+x+x^2=0$ is:
| A. |
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| B. |
|
| C. |
|
| D. |
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The value of $\omega^5+\omega^6+\omega^7+\ldots\ldots+\omega^{12}$ is
| A. |
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| B. |
|
| C. |
|
| D. |
|
Let $\left(1+x+x^2\right)^{2014}=a_0+a_1 x+a_2 x^2+a_3 x^3+\ldots \ldots . .+a_{4028} x^{4028}$, and let
$
\mathrm{A}=\mathrm{a}_0-\mathrm{a}_3+\mathrm{a}_6-\ldots \ldots \ldots+\mathrm{a}_{4026}
$
$
\mathrm{B}=\mathrm{a}_1-\mathrm{a}_4+\mathrm{a}_7-\ldots \ldots . . \mathrm{a}_{4027}
$
$
\mathrm{c}=\mathrm{a}_2-\mathrm{a}_5+\mathrm{a}_8-\ldots \ldots . .+\mathrm{a}_{4028}
$
Then
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $n$ is a positive integer and $\omega \neq 1$ is a cube root of unity the number of possible values of
$
\left|e^{\sum_{k=0}^n\binom{n}{k} \omega^k}\right|
$
is
| A. |
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| B. |
|
| C. |
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| D. |
|
Let $\mathrm{I}, \omega$ and $\omega^2$ be the cube roots of unity. The least possible degree of a polynomial, with real coefficients, having $2 \omega^2, 3+4$ $\omega, 3+4 \omega^2$, and $5-\omega-\omega^2$ as roots are:
| A. |
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| B. |
|
| C. |
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| D. |
|
Let $a$ be a fixed non-zero complex number with $|a|<1$ and where $z$ is a complex number. Then $w=\left(\frac{z-a}{1-\bar{a} z}\right)$
| A. |
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| B. |
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| C. |
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| D. |
|
Let integers $a, b \in[-3,3]$ be such that $a+b \neq 0$. Then the number of all possible ordered pairs (a, b), for which $\left|\frac{z-a}{z+b}\right|=1$ and $\left|\begin{array}{ccc}z+1 & \omega & \omega^2 \\ \omega & z+\omega^2 & 1 \\ \omega^2 & 1 & z+\omega\end{array}\right|$
$=1, z \in C$, where $\omega$ and $\omega^2$ are the roots of $x^2+x+$ $1=0$, is equal to $\qquad$ _______.
| A. |
|
| B. |
|
| C. |
|
| D. |
|
If $\alpha$ is a root of the equation $x^2+x+1=0$ and $\sum_{\mathrm{k}=1}^{\mathrm{n}}\left(\alpha^{\mathrm{k}}+\frac{1}{\alpha^{\mathrm{k}}}\right)^2=20$, then n is equal to ______
| A. |
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| B. |
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| C. |
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| D. |
|
