KCET · Maths · Complex Number
The amplitude of \((1+i)^{5}\) is
- A \(\frac{3 \pi}{4}\)
- B \(\frac{-3 \pi}{4}\)
- C \(\frac{-5 \pi}{4}\)
- D \(\frac{5 \pi}{4}\)
Answer & Solution
Correct Answer
(D) \(\frac{5 \pi}{4}\)
Step-by-step Solution
Detailed explanation
Given, \((1+i)^{5}\)
\[
\begin{aligned}
&=(\sqrt{2})^{5}\left(\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{2}}\right)^{5} \\
&=(\sqrt{2})^{5}\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)^{5} \\
&=(\sqrt{2})^{5}\left(\cos \frac{5 \pi}{4}+i \sin \frac{5 \pi}{4}\right)
\end{aligned}
\]
[by De-Moivre's theorem]
Now, amplitude \(=\tan ^{-1}\left(\frac{y}{x}\right)\)
\[
=\tan ^{-1}\left(\frac{\sin \frac{5 \pi}{4}}{\cos \frac{5 \pi}{4}}\right)=\frac{5 \pi}{4}
\]
\[
\begin{aligned}
&=(\sqrt{2})^{5}\left(\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{2}}\right)^{5} \\
&=(\sqrt{2})^{5}\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)^{5} \\
&=(\sqrt{2})^{5}\left(\cos \frac{5 \pi}{4}+i \sin \frac{5 \pi}{4}\right)
\end{aligned}
\]
[by De-Moivre's theorem]
Now, amplitude \(=\tan ^{-1}\left(\frac{y}{x}\right)\)
\[
=\tan ^{-1}\left(\frac{\sin \frac{5 \pi}{4}}{\cos \frac{5 \pi}{4}}\right)=\frac{5 \pi}{4}
\]
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- Given that, \(A\) and \(B\) are two events such that \(P(B)=\frac{3}{5}, P\left(\frac{A}{B}\right)=\frac{1}{2}\) and \(P(A \cup B)=\frac{4}{5}\), then \(P(A)\) is equal toKCET 2021 Easy
- If \(y=a \sin ^3 t, x=a \cos ^3 t\), then \(\frac{d y}{d x}\) at \(t=\frac{3 \pi}{4}\) isKCET 2025 Medium
- For constant \(a, \frac{d}{d x}\left(x^{x}+x^{a}+a^{x}+a^{a}\right)\) isKCET 2021 Medium
- \(G=\left\{\left[\begin{array}{ll}x & x \\ x & x\end{array}\right], x\right.\) is a non-zero real number \(\}\) is a group with respect to matrix multiplication.
In this group, the inverse of \(\left[\begin{array}{ll}\frac{1}{3} & \frac{1}{3} \\ \frac{1}{3} & \frac{1}{3}\end{array}\right]\) isKCET 2011 Medium - If \(A=\left|\begin{array}{ccc}a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3}\end{array}\right|\) and \(B=\left|\begin{array}{ccc}c_{1} & c_{2} & c_{3} \\ a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3}\end{array}\right|\), thenKCET 2008 Medium
- If \( x \) is real, then the minimum value of \( x^{2}-8 x+17 \) isKCET 2015 Easy
More PYQs from KCET
- The compound which forms acetaldehyde when heated with dilute \(\mathrm{NaOH}\), isKCET 2009 Medium
- The length of the subtangent to the curve \(x^{2} y^{2}=a^{4}\) at \((-a, a)\) isKCET 2007 Hard
- \( \int_{-2}^{2}|x \cos \pi x| d x \) is equal toKCET 2018 Medium
- Two tiny spheres carrying charges \(1.8 \mu \mathrm{C}\) and \(2.8 \mu \mathrm{C}\) arc located at \(40 \mathrm{~cm}\) apart. The potential at the mid - point of the line joining the two charges isKCET 2022 Medium
- The total number of terms in the expansion of \( (x+a)^{47}-(x-a)^{47} \) after simplification isKCET 2017 Easy
- Injection of an antidote against snake-bite is an example ofKCET 2020 Hard