Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 2: Problem 101

If \(z=x+i y, x, y \in R\) and \(3 x+(3 x-y) i=4-6 i\) then \(z=\) (a) \((4 / 3)+\mathrm{i} 10\) (b) \((4 / 3)-\mathrm{i} 10\) (c) \([(-4) / 3]+\mathrm{i} 10\) (d) \([(-4) / 3]-\mathrm{i} 10\)

Short Answer

Expert verified
\( z = (\frac{4}{3})+ i 10 \)

Step by step solution

01

Equate Real and Imaginary Parts

Since we are given that \(3x+(3x-y)i=4-6i\), we can equate the real and imaginary parts. Real Parts: \(3x = 4\) Imaginary Parts: \(3x - y = -6\)
02

Solve for Real Part, x

To solve for the real part, \(x\), we have the following equation: \(3x = 4\) Divide both sides by 3: \( x = \frac{4}{3} \)
03

Solve for Imaginary Part, y

Now we know that \(x= \frac{4}{3}\). Let's use this value to find \(y\): \(3x - y = -6\) Substitute \(x = \frac{4}{3}\): \( 3 \left( \frac{4}{3} \right) - y = -6 \) Now, simplify the equation and isolate \(y\): \( 4 - y = -6 \) Add 6 to both sides: \(y = 4 + 6\) \(y = 10\)
04

Find the Correct Value for z

Now that we have the values for \(x\) and \(y\), we can write the complex number \(z\): \( z = \frac{4}{3} + 10i \) Now, let's compare this result with the given options: (a) \((\frac{4}{3})+ i 10\) (b) \((\frac{4}{3})- i 10\) (c) \([-(\frac{4}{3})]+i 10\) (d) \([-(frac{4}{3})]-i 10\) It matches option (a), therefore the correct solution is: \( z = (\frac{4}{3})+ i 10 \)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

If \(\mathrm{x}>1\) then the square roots of \(1-\sqrt{\left(\mathrm{x}^{2}-1\right) \mathrm{i}}\) (a) \(\pm[\sqrt{\\{}(\mathrm{x}+1) / 2\\}-\mathrm{i} \sqrt{\\{}(\mathrm{x}-1) / 2\\}]\) (b) \(\pm[\sqrt{\\{}(\mathrm{x}+1) / 2\\}+\mathrm{i} \sqrt{\\{}(\mathrm{x}-1) / 2\\}]\) (c) \(\pm[\sqrt{\\{}(\mathrm{x}-1) / 2\\}-\mathrm{i} \sqrt{\\{}(\mathrm{x}+1) / 2\\}]\) (d) \(\pm[\sqrt{\\{}(\mathrm{x}-1) / 2\\}+\mathrm{i} \sqrt{\\{}(\mathrm{x}+1) / 2\\}]\)

The value of \((-i)^{(-1)}=\) (a) \(-(\pi / 2)\) (b) \((\pi / 2)\) (c) \(\mathrm{e}^{-(\pi / 2)}\) (b) \(\mathrm{e}^{(\pi / 2)}\)

It \(\mathrm{z}^{2}+\mathrm{z}+1=0\) where \(\mathrm{z}\) is a complex number, then the value of \([z+(1 / z)]^{2}+\left[z^{2}+\left(1 / z^{2}\right)\right]^{2}+\left[z^{3}+\left(1 / z^{3}\right)\right]^{2}+\ldots\) \(+\left[z^{6}+\left(1 / z^{6}\right)\right]^{2}\) is (a) 18 (b) 54 (c) 6 (d) 12

The area of the triangle in the Arg and diagram formed by the Complex number \(\mathrm{z}\), iz and \(\mathrm{z}+\mathrm{iz}\) is (a) \(|z|^{2}\) (b) \((\sqrt{3} / 2)|z|^{2}\) (c) \((1 / 2)|\mathrm{z}|^{2}\) (d) \((3 / 2)|\mathrm{z}|^{2}\)

For any integer \(\mathrm{n}, \arg \left[(\sqrt{3}+\mathrm{i})^{4 n+1} /(1-\mathrm{i} \sqrt{3})^{4 \mathrm{n}}\right]=\) (a) \((\pi / 3)\) (b) \((\pi / 6)\) (c) \((2 \pi / 3)\) (d) \((5 \pi / 6)\)
See all solutions

Recommended explanations on English Textbooks

Lexis and Semantics Summary

Read Explanation

Rhetoric

Read Explanation

Textual Analysis

Read Explanation

Graphology

Read Explanation

Phonology

Read Explanation

Global English

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free