Problem 193 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(\omega\) is an imaginary cube root of unity, then (1 \(\left.-\omega^{2}\right)^{7}\) equals: \(\quad[20\) (A) \(128 \omega\) (B) \(-128 \omega\) (C) \(128 \omega^{2}\) (D) \(-128 \omega^{2}\) Tot 7

Step-by-Step Solution

Answer

The result is 0 (none of the options provided).

Step 1: Identify Properties of Cube Roots of Unity

The imaginary cube roots of unity are solutions to the equation \(x^3 = 1\), apart from \(x = 1\). These roots satisfy the equation \(x^2 + x + 1 = 0\). The roots are usually denoted as \(\omega\) and \(\omega^2\), where \(\omega^3 = 1\), \(\omega^2 = -1 - \omega\), and \(\omega + \omega^2 + 1 = 0\).

Step 2: Simplify \(-\omega^2\)

Substitute \(-\omega^2\) using the identity \(\omega^2 = -1 - \omega\), thus \(-\omega^2 = 1 + \omega\).

Step 3: Calculate \((1 + \omega)^7\) Using Binomial Theorem

To find \((1 + \omega)^7\), apply the binomial theorem: \((1 + \omega)^7 = \sum_{k=0}^{7} \binom{7}{k} (1)^{7-k} (\omega)^k\).Note that when considering terms with \(\omega^3 = 1\), the powers of \(\omega\) can be reduced modulo 3: \(\omega^3 = 1\), \(\omega^4 = \omega\), and \(\omega^5 = \omega^2\), and so on.

Step 4: Evaluate Binomial Theorem Expansion Terms

Calculate each relevant term: - For powers of \(\omega^3\), \((\omega^3)^{n} = 1\).- The cycle \(\omega, \omega^2, 1\) repeats every three terms, simplifying the expansion.After evaluating coefficients with powers reduced, the effective terms simplify to: \((1 + \omega)^7 = 1 + 7\omega + 21\omega^2 + 35 \equiv 0 + 0 + 0 + 0 = 0\).

Step 5: Verify Cube Root Properties

Revalidate using the identity \((1 + \omega)^3 = 0\) (as observed from the cube roots' sum property), then \((1 + \omega)^6 = ((1 + \omega)^3)^2 = 0^2 = 0\).Thus, \((1 + \omega)^7 = 0 + 0 = 0\).

Key Concepts

Cube Roots of Unity

The concept of cube roots of unity revolves around the solutions to the equation \(x^3 = 1\). Naturally, one solution is \(x = 1\), and the other two are complex numbers. These complex roots are termed as \( \omega \) and \( \omega^2 \), where they possess some fascinating properties.
An essential property of these roots is the identity \( \omega^3 = 1 \). Additionally, these roots fulfill the equation \( x^2 + x + 1 = 0 \). This equation remains central in solving many problems, simplifying calculations due to the cyclical nature of multiplication with these terms.
A few vital properties you should remember are:

• \( \omega^2 = -1 - \omega \)
• \( \omega + \omega^2 + 1 = 0 \)
• The cycle follows \( \omega, \omega^2, 1 \) and repeats every three powers.

Knowing these properties allows us to manage expressions like \( (1 - \omega^2)^7 \) with improved insight, using algebraic simplification and cyclic properties of \( \omega \).
These key properties make working with cube roots an elegant process, providing a foundation for more complex mathematical explorations.

Binomial Theorem

The Binomial Theorem is a powerful tool in algebra that helps expand expressions of the form \((a + b)^n\). It states that \((a + b)^n\) can be expanded into a sum: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]where \( \binom{n}{k} \) is a binomial coefficient, calculated as \( \frac{n!}{k!(n-k)!} \). Each term in the expansion includes a combination of powers of \( a \) and \( b \) that adds up to \( n \).
In our exercise, we applied the Binomial Theorem to find \((1 + \omega)^7\). With the cube roots' property, \( \omega^3 = 1 \), we reduced higher powers of \( \omega \) effectively. For instance, \( \omega^4 = \omega \), \( \omega^5 = \omega^2 \), which simplifies the involved calculations.
Using this theorem not only makes such expansions mechanical and structured but also exploits properties like those of roots of unity to reduce complexity.

Roots of Unity Properties

Roots of Unity are fascinating elements in complex numbers. They denote numbers which, when raised to a specific power (indicative of 'unity'), yield one. For cube roots, as discussed, the equation \( x^3 = 1 \) results in unique cyclic properties.
A few core properties include:

• The sum of the roots, including the real root, equals zero. For cube roots, \( 1 + \omega + \omega^2 = 0 \).
• They can be visualized on a complex plane, forming an equilateral triangle for cube roots, perfectly spaced around the unit circle.
• Each root's cyclic nature allows sequences of calculations to simplify, as showcased by reducing high powers by substituting \( \omega^3 = 1 \).

This cyclical nature leads to expressions and calculations, such as our example of \((1 + \omega)^7\), becoming more efficient. By grasping these properties, tackling complex problems that initially seem daunting becomes more straightforward, allowing deeper insights into the symmetry and elegance of mathematics.