Problem 27 - A Complete Resource Book in Mathematics for JEE Main

Question

The roots of the equation \(z^{4}+1=0\) are (A) \((\pm 1 \pm i)\) (B) \((\pm 2 \pm 2 i)\) (C) \(\frac{1}{\sqrt{3}}(\pm 1 \pm i)\) (D) None of these

Step-by-Step Solution

Answer

(D) None of these

Step 1: Understanding the Problem

We are given the equation \( z^4 + 1 = 0 \). This can be rewritten as \( z^4 = -1 \). We need to find the roots of this equation.

Step 2: Expressing as Exponential

The complex number \(-1\) can be represented as \( e^{i\pi} \). Thus, we express \( z^4 = e^{i\pi} \).

Step 3: Finding the Roots Using De Moivre's Theorem

According to De Moivre's Theorem, to find the \( n^{th} \) roots of \( e^{i\pi} \), we use the formula: \( z = e^{i(\pi + 2k\pi)/4} \) for \( k = 0, 1, 2, 3 \).

Step 4: Calculate the Roots

- For \( k = 0 \): \( z = e^{i\pi/4} = \frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} \)- For \( k = 1 \): \( z = e^{i3\pi/4} = -\frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}} = -\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} \)- For \( k = 2 \): \( z = e^{i5\pi/4} = -\frac{1}{\sqrt{2}} - i\frac{1}{\sqrt{2}} = -\frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2} \)- For \( k = 3 \): \( z = e^{i7\pi/4} = \frac{1}{\sqrt{2}} - i\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2} \)

Step 5: Compare and Choose the Correct Option

The roots we found do not match any of the options given (A), (B), or (C). The closest form is \((\pm\frac{\sqrt{2}}{2} \pm i\frac{\sqrt{2}}{2})\), which after calculation does not equal none of the options. Thus, the answer is (D) None of these.

Key Concepts

De Moivre's Theorem

De Moivre's Theorem is a powerful tool in complex number calculations, specifically when dealing with powers and roots of complex numbers. It provides a straightforward method to calculate powers of complex numbers expressed in polar form.

The theorem states that for any complex number represented in polar form as \( r(\cos \theta + i \sin \theta) \), its \( n^{th} \) power is given by:

• \( [r (\cos \theta + i \sin \theta)]^n = r^n (\cos(n\theta) + i \sin(n\theta)) \)

This simplifies the process of calculating powers by transforming multiplication into addition within the trigonometric components.

In the context of solving equations like \( z^4 = -1 \), De Moivre's Theorem helps in interpreting the roots in complex exponential form \( e^{i(\pi+2k\pi)/4} \).
Here, the angle \( \pi \) is adjusted by \( 2k\pi \) for each root, where \( k = 0, 1, 2, 3 \) represents the distinct roots derived from De Moivre's theorem.

nth roots of unity

The concept of the \( n^{th} \) roots of unity plays a significant role in understanding complex numbers. These roots are the solutions to the equation \( z^n = 1 \). They are evenly spaced around the unit circle in the complex plane.

For a given integer \( n \), the \( n^{th} \) roots of unity are given by:

• \( 1, \omega, \omega^2, \ldots, \omega^{n-1} \)

Where \( \omega = e^{2\pi i/n} \) is the primitive root of unity.

These roots are essential for simplifying powers of complex numbers and play a part when working with polynomials, such as finding roots of equations like \( z^4 = -1 \). Even though \( \omega \) usually relates to the roots of \( 1 \), similar principles guide finding roots of different complex constants, for example, \( -1 = e^{i\pi} \), when raised to an even power.

Complex exponentials

Complex exponentials provide a neat and efficient way to represent and manipulate complex numbers. They are deeply intertwined with Euler's formula:

• \( e^{i\theta} = \cos \theta + i \sin \theta \)

This formula shows that any complex number can be expressed in an exponential form, greatly simplifying complex multiplication and exponentiation.

In the context of our exercise, expressing \(-1\) as a complex exponential \( e^{i\pi} \) allows us to utilize the rotational symmetry of complex numbers on the unit circle.

This makes it easier to apply operations like De Moivre's Theorem. By using complex exponentials, we can effortlessly calculate the required angle adjustments (e.g., \( \pi + 2k\pi \)) needed to find the distinct roots of complex equations, as seen with the roots of \( z^4 = e^{i\pi} \).