Problem 169 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\alpha\) is the fifth root of unity, then (A) \(\left|1+\alpha+\alpha^{2}+\alpha^{3}+\alpha^{4}\right|=0\) (B) \(\left|1+\alpha+\alpha^{2}+\alpha^{3}\right|=1\) (C) \(\left|1+\alpha+\alpha^{2}\right|=2 \cos \frac{\pi}{5}\) (D) \(\mid 1+\alpha=2 \cos \frac{\pi}{10}\)

Step-by-Step Solution

Answer

Option (A) is correct.

Step 1: Understanding Roots of Unity

The fifth roots of unity are the solutions to the equation \(x^5 = 1\). They are given by \(1, \alpha, \alpha^2, \alpha^3,\) and \(\alpha^4\), where \(\alpha = e^{2\pi i / 5}\). These roots satisfy the equation \(1 + \alpha + \alpha^2 + \alpha^3 + \alpha^4 = 0\).

Step 2: Evaluate Option (A)

The expression \(1 + \alpha + \alpha^2 + \alpha^3 + \alpha^4 = 0\) implies that the modulus \(\left|1 + \alpha + \alpha^2 + \alpha^3 + \alpha^4\right| = 0\) as the sum is zero. Therefore, option (A) is correct.

Step 3: Evaluate Option (B)

We are given the complex number \(1 + \alpha + \alpha^2 + \alpha^3\). The fifth root of unity, \(\alpha\), is such that \(\alpha^5 = 1\) and their sum up to four is related to root magnitudes. However, since this sum is converted by setting \(\alpha^4 = -1 - \alpha - \alpha^2 - \alpha^3\), the modulus is not equal to 1. It does not satisfy the condition \(|1 + \alpha + \alpha^2 + \alpha^3| = 0\).

Step 4: Evaluate Option (C)

Evaluate the modulus of \(1 + \alpha + \alpha^2\). We know \(|1 + x + x^2| = |1 + 2 \cos\theta|\) for \(\theta = 2\pi/5\). Hence, \(\left|1 + \alpha + \alpha^2\right| = 2 \cos\left(\frac{\pi}{5}\right)\), which matches option (C).

Step 5: Evaluate Option (D)

Consider the expression \(1 + \alpha\). For the fifth root of unity, this does not simplify to \(2 \cos\left(\frac{\pi}{10}\right)\). There is no direct transformation for fifth roots that yields this value. Hence, this option is incorrect.

Key Concepts

Complex Numbers

Complex numbers extend the concept of one-dimensional number lines to two dimensions. They have a real part and an imaginary part, typically expressed in the form:

• \( z = a + bi \)

where \( a \) is the real part and \( bi \) is the imaginary part, with \( i \) being the imaginary unit satisfying \( i^2 = -1 \).
The beauty of complex numbers is evident in their ability to solve equations like \( x^2 + 1 = 0 \), for which no real number solution exists.
Complex numbers are represented graphically on the complex plane, with the x-axis serving as the real axis and the y-axis as the imaginary axis.
The concept becomes particularly handy when dealing with polynomial roots, as is the case with roots of unity like the fifth roots: \( 1, \alpha, \alpha^2, \alpha^3, \) and \( \alpha^4 \). Here, \( \alpha = e^{2\pi i / 5} \), showing their connection with Euler's formula, integrating trigonometric identities through a polar representation:

• \( \alpha = \cos(\frac{2\pi}{5}) + i\sin(\frac{2\pi}{5}) \)

Modulus of a Complex Number

The modulus of a complex number is a measure of its distance from the origin in the complex plane. For a complex number \( z = a + bi \), the modulus is noted as \( |z| \) and calculated by the formula:

• \( |z| = \sqrt{a^2 + b^2} \)

This is akin to using the Pythagorean theorem to determine the distance, representing the absolute magnitude irrespective of direction.
The modulus is especially useful in dealing with roots of unity. For instance, the fifth roots of unity are equidistant on the unit circle, both real and imaginary components contributing to a modulus of 1 for each root.
When evaluating complex expressions like \( 1 + \alpha + \alpha^2 \), it's crucial to understand that the modulus captures the resultant magnitude, not the individual sum of its components. Thus, in some formula transformations, the modulus equates to simpler trigonometric expressions, leveraging identities like \( |1 + 2\cos(\theta)| \).

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are valid for all variable values within a domain. These are crucial in simplifying complex number expressions and analyzing periodic phenomena like roots of unity.
Some of the most frequently used trigonometric identities include:

• Pythagorean identities: \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Sum and difference formulas: \( \cos(\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta \)
• Double angle formulas: \( \cos 2\theta = 2\cos^2\theta - 1 \)

These identities facilitate the transition between complex exponential forms and trigonometric forms of complex numbers. For example, in the context of roots of unity, each root on the unit circle can be represented using trigonometric forms, where \( \alpha = e^{2\pi i / 5} = \cos(\frac{2\pi}{5}) + i\sin(\frac{2\pi}{5}) \).
These transformations simplify the calculation of expressions involving sums and powers of roots of unity, crucial for understanding options such as the modulus computations found in exercises analyzing different scenarios of their summation and verification.