( - 3 + i)^ 53 where i^2 = - 1 is equal to - Vidyadip

MCQ

${( - \sqrt 3 + i)^{53}}$ where ${i^2} = - 1$ is equal to

A
${2^{53}}(\sqrt 3 + 2i)$
B
${2^{52}}(\sqrt 3 - i)$
✓
${2^{53}}\,\left( {\frac{{\sqrt 3 }}{2} + \frac{1}{2}i} \right)$
D
${2^{53}}(\sqrt 3 - i)$

✓

Answer

Correct option: C.

${2^{53}}\,\left( {\frac{{\sqrt 3 }}{2} + \frac{1}{2}i} \right)$

c
(c) ${( - \sqrt 3 + i)^{53}}$$ = {2^{53}}{\left( {\frac{{ - \sqrt 3 }}{2} + \frac{i}{2}} \right)^{53}}$
= ${2^{53}}{(\cos {150^o} + i\sin {150^o})^{53}}$
$ = {2^{53}}[\cos ({150^o} \times 53) + i\sin ({150^o} \times 53)]$
$ = {2^{53}}[\cos (22\pi + {30^o}) + i\sin (22\pi + {30^o})]$
$ = {2^{53}}[\cos {30^o} + i\sin {30^o}]$$ = {2^{53}}\left[ {\frac{{\sqrt 3 }}{2} + i\frac{1}{2}} \right]$.

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

More "MCQ" from 4.1 complex nubers→4.1 complex nubers — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

If $\tan \alpha $ equals the integral solution of the inequality $4{x^2} - 16x + 15 < 0$ and $\cos \beta $ equals to the slope of the bisector of first quadrant, then $\sin (\alpha + \beta )\sin (\alpha - \beta )$ is equal to

Which of the following is a statement.

There are two candles of same length and same size. Both of them burn at uniform rate. The first one burns in $5\,hr$ and the second one burns in $3\,h$. Both the candles are lit together. After how many minutes the length of the first candle is $3$ times that of the other?

Which axis is known as imaginary axis in argand plane?

If a $cos^3 \alpha + 3a \,cos\, \alpha \, sin^2\, \alpha = m$ and $asin^3\, \alpha + 3a \, cos^2\, \alpha \,sin\, \alpha = n$ . Then $(m + n)^{2/3} + (m - n)^{2/3}$ is equal to :

If $\sin (A + B) =1 $ and $\cos (A - B) = \frac{{\sqrt 3 }}{2},$ then the smallest positive values of $A$ and $ B$ are

One card is drawn from a pack of $52$ cards.The probability of getting a $10$ of black suit is:

A line through $A( - 5, - \;4)$ meets the lines $x + 3y + 2 = 0,$ $2x + y + 4 = 0$ and $x - y - 5 = 0$ at $B, \,C$ and $D$ respectively. If ${\left( {\frac{{15}}{{AB}}} \right)^2} + {\left( {\frac{{10}}{{AC}}} \right)^2} = {\left( {\frac{6}{{AD}}} \right)^2},$ then the equation of the line is

If $\sin x=-\frac{3}{5}$, where $\pi < x < \frac{3 \pi}{2}$ then $80\left(\tan ^2 x-\cos x\right)$ is equal to :

If $\cos ec\,\theta = \frac{{p + q}}{{p - q}}$ $\left( {p \ne q \ne 0} \right)$, then $\left| {\cot \left( {\frac{\pi }{4} + \frac{\theta }{2}} \right)} \right|$ is equal to