Problem 210 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\omega(\neq 1)\) is a cube root of unity, and \((1+\omega)^{7}=A+\) \(B \omega\). Then \((A, B)\) equals [2011] (A) \((1,1)\) (B) \((1,0)\) (C) \((-1,1)\) (D) \((0,1)\)

Step-by-Step Solution

Answer

The pair \((A, B)\) is \((1, 0)\).

Step 1: Understand Cube Root of Unity

The cube roots of unity are solutions to the equation \[ x^3 = 1. \]These roots are 1, \( \omega, \) and \( \omega^2 \), where \( \omega eq 1 \). Additionally, they satisfy the relation \[ 1 + \omega + \omega^2 = 0. \]

Step 2: Simplify Expression Using Relations

Given the equation, we know that \[ \omega^2 = -1 - \omega. \]We need to raise \( 1+\omega \) to the 7th power, but let's first simplify the expression using its properties.

Step 3: Expand the Power

Use the binomial theorem to expand \[(1 + \omega)^7.\]The expansion involves terms like \[ \binom{7}{k} (\omega)^k. \]

Step 4: Use Properties of \( \omega \)

Given the properties \( \omega^3 = 1 \), any power of \( \omega \) that is a multiple of 3 will return to 1 (e.g., \( \omega^6 = 1, \omega^5 = \omega^2, \omega^4 = \omega \)). Use these to reduce higher powers of \( \omega \).

Step 5: Simplify the Expression

Calculate and simplify each term by reducing powers of \( \omega \):\((1+\omega)^7 = 1 + 7(\omega) + 21 (\omega^2) + 35(1) + 35(\omega) + 21(\omega^2) + 7(1) \).Combine terms with similar powers of \( \omega \).

Step 6: Combine Terms

Utilizing \( \omega^2 = -1 - \omega \), we get:\(1 + 35 + 7 = 43,\)\(7\omega + 35\omega = 42\omega,\)\(21(-1 - \omega) + 21(-1 - \omega) = -42 - 42\omega.\)Combine these to get:\[ (43 - 42) + (42\omega - 42\omega) = 1. \]

Step 7: Conclude the Values of A and B

Combine all terms:\[ 1 + 42\omega - 42\omega - 42 = 1 \]Thus, \(A = 1 \) and \( B = 0\).

Key Concepts

Binomial Theorem

The binomial theorem is a powerful tool that helps us expand expressions raised to a power. It is especially useful when dealing with algebraic expressions like \((a + b)^n\). The theorem provides a way to express this as a sum of terms involving binomial coefficients. The general formula for the binomial expansion is:
\[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]where \(\binom{n}{k}\) is a binomial coefficient, representing the number of ways to choose \(k\) elements from \(n\) and is calculated using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\).
Understanding and applying the binomial theorem involves:

• Identifying the values for \(a\), \(b\), and \(n\).
• Calculating each term in the expanded expression by substituting into the formula.
• Combining like terms when necessary.

The theorem provides a structured pathway to expanding powers, simplifying computations, and makes working with higher powers of expressions more manageable.

Complex Numbers

Complex numbers are essential in modern mathematics and they form the backbone of understanding equations like \(x^3 = 1\). A complex number is expressed as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, satisfying \(i^2 = -1\).
In dealing with the roots of unity, complex numbers allow us to work with non-real solutions of polynomial equations. These numbers can be represented on the complex plane, which is a two-dimensional plane where the x-axis is the real axis and the y-axis is the imaginary axis.
Key characteristics and operations involving complex numbers include:

• Addition and Subtraction: Combine like terms, keeping \(a\) and \(b\) separate.
• Multiplication: Use distributive properties, remembering that \(i^2 = -1\).
• Conjugates: For a complex number \(a+bi\), the conjugate is \(a-bi\) which helps in simplification and division.

Understanding complex numbers is crucial when exploring higher-order polynomials and their roots.

Properties of Roots of Unity

The properties of the roots of unity are fascinating and play a pivotal role in various areas of mathematics, including solving polynomial equations and understanding symmetry. For any integer \(n\), the \(n\)-th roots of unity are the solutions to the equation \(x^n = 1\), and they are evenly spaced on the unit circle in the complex plane.
When \(n = 3\), we get the cube roots of unity: \(1\), \(\omega\), and \(\omega^2\). These roots have unique properties:

• The sum of the roots: \(1 + \omega + \omega^2 = 0\).
• The relations: \(\omega^3 = 1\) and \(\omega^2 + \omega + 1 = 0\).
• Any power of \(\omega\) that is a multiple of 3, like \(\omega^3, \omega^6, \omega^9,\) and so on, equals 1.

These properties mean that when raising expressions involving \(\omega\) to a power or simplifying complex expressions, recognizing these patterns significantly reduces computational complexity. The mathematical beauty of the roots of unity lies in their symmetry and their ability to simplify calculations, particularly in polynomial algebra and the study of complex numbers.