If (1 + i 3 )^9 = a + ib, then b is equal to - Vidyadip

Question

If ${(1 + i\sqrt 3 )^9} = a + ib,$ then $b$ is equal to

✓

Answer

c
(c) $1 + i\sqrt 3 = 2\left( {\frac{1}{2} + i\frac{{\sqrt 3 }}{2}} \right) = 2\left[ {\cos \frac{\pi }{3} + i\sin \frac{\pi }{3}} \right] = 2{e^{i\pi /3}}$
${(1 + i\sqrt 3 )^9} = {(2{e^{i\pi /3}})^9} = {2^9}.{e^{i(3\pi )}}$
$ = {2^9}(\cos 3\pi + i\sin 3\pi ) = - {2^9}$
$a + ib = {(1 + i\sqrt 3 )^9} = - {2^9}$;

$b = 0$.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

If $\left| {\,\begin{array}{*{20}{c}}{y + z}&{x - z}&{x - y}\\{y - z}&{z - x}&{y - x}\\{z - y}&{z - x}&{x + y}\end{array}\,} \right| = k\,xyz$, then the value of $k $ is

The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is

Let $S=\left\{\sin ^2 2 \theta:\left(\sin ^4 \theta+\cos ^4 \theta\right) x^2+(\sin 2 \theta) x+\right.$
$\left(\sin ^6 \theta+\cos ^6 \theta\right)=0$ has real roots$\}$. If $\alpha$ and $\beta$ be the smallest and largest elements of the set $S ,$ respectively, then $3\left((\alpha-2)^2+(\beta-1)^2\right)$ equals $.....$

The derivative of ${\tan ^{ - 1}}\left( {\frac{{\sin \,x - \cos \,x}}{{\sin \,x + \cos \,x}}} \right)$, with respect to $\frac{x}{2}$, where $\left( {x \in \left( {0,\frac{\pi }{2}} \right)} \right)$ is

Let $\overrightarrow{ a }=\alpha \hat{ i }+2 \hat{ j }-\hat{ k }$ and $\overrightarrow{ b }=-2 \hat{ i }+\alpha \hat{ j }+\hat{ k }$, where $\alpha \in R$. If the area of the parallelogram whose adjacent sides are represented by the vectors $\vec{a}$ and $\vec{b}$ is $\sqrt{15\left(\alpha^{2}+4\right)}$, then the value of $2|\vec{a}|^{2}+(\vec{a} \cdot \vec{b})|\vec{b}|^{2}$ is equal to

Let $f : R \rightarrow R$ be a differentiable function such that $f^{\prime}(x)+f(x)=\int \limits_0^2 f(t) d t$. If $f(0)=e^{-2}$, then $2 f (0)- f (2)$ is equal to $.........$.

If $|a|\,\, = 3,\,\,\,|b\,\,|\,\, = 4$ and the angle between $ a$ and $b$ be ${120^o}$, then $|4a + 3b|\,\, = $

If the sum and product of four positive consecutive terms of a $G.P.$, are $126$ and $1296$, respectively, then the sum of common ratios of all such $GPs$ is $.........$.

If $\vec a = 2 \hat i + 3 \hat j+ \hat k ,\vec b = \hat i - \hat j+ \hat k , \vec c = \hat i + \hat j+ \hat k$ and let $\vec d$ be such that $\vec a \times \vec b = \vec d \times \vec b, \vec d. \vec c = 8$, then value of $ \vec d. \vec b$ is -

If the sum of the first $11$ terms of the series ${\left( {1\frac{4}{7}} \right)^2} + {\left( {1\frac{5}{7}} \right)^2} + {\left( {1\frac{6}{7}} \right)^2} + {2^2} + {\left( {2\frac{1}{7}} \right)^2} + ......$ is $\frac{{11}}{7}\lambda $ then $\lambda $ is equal to