Problem 107 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(i=\sqrt{-1}\), then \(4+5\left(-\frac{1}{2}+\frac{i \sqrt{3}}{2}\right)^{334}+3\left(-\frac{1}{2}+\frac{i \sqrt{3}}{2}\right)^{365}\) is equal to (A) \(1-i \sqrt{3}\) (B) \(-1+i \sqrt{3}\) (C) \(i \sqrt{3}\) (D) \(-\sqrt{3} i\)

Step-by-Step Solution

Answer

The expression evaluates to (D) \\(-\sqrt{3}i\\).

Step 1: Identify the complex number

Recognize that \(-\frac{1}{2} + \frac{i\sqrt{3}}{2}\) is a root of unity. In fact, it is one of the cube roots of unity. This complex number can be represented as \(\omega\), where \(\omega^3 = 1\) and \(1 + \omega + \omega^2 = 0\).

Step 2: Simplify powers using properties of cube roots of unity

For \(\omega\) being a primitive cube root of unity: \(\omega^3 = 1\). Therefore, to simplify \(\omega^{334}\) and \(\omega^{365}\), we find the remainders of 334 and 365 when divided by 3. - \((334 \, \text{mod} \, 3) = 2\), so \(\omega^{334} = \omega^2\).- \((365 \, \text{mod} \, 3) = 2\), so \(\omega^{365} = \omega^2\).

Step 3: Substitute and simplify the expression

Replace the powers in the original expression: \[4 + 5\omega^{334} + 3\omega^{365} = 4 + 5\omega^2 + 3\omega^2.\] Since both powers equal \(\omega^2\), combine like terms:\[4 + (5 + 3)\omega^2 = 4 + 8\omega^2.\]

Step 4: Substitute the value of \(\omega^2\)

Since \(\omega^2 = -\frac{1}{2} - \frac{i\sqrt{3}}{2}\), substitute back:\[4 + 8\left(-\frac{1}{2} - \frac{i\sqrt{3}}{2}\right) = 4 - 4 - 4i\sqrt{3} = -4i\sqrt{3}.\]

Step 5: Match the result to the available options

The value computed is \(-i\sqrt{3}\), which corresponds to the option (D) \(-\sqrt{3}i\). The answer given matches option (D).

Key Concepts

Cube Roots of Unity

The cube roots of unity are a special set of complex numbers, which when raised to the power of three, equals one. These roots are solutions to the equation:\[x^3 = 1\]The cube roots of unity are denoted as 1, \( \omega \), and \( \omega^2 \).
The simplest root is 1. The other two roots, \( \omega \) and \( \omega^2 \), are complex and display a particular pattern.
For example:

• \( \omega^3 = 1 \)
• \( 1 + \omega + \omega^2 = 0 \)

This observation is critical in simplifying expressions with powers of cube roots, as demonstrated in the exercise you are studying. It allows the breakdown of large exponents into manageable terms by reducing them using modulo operation (dividing by 3).
So for inputs like \( \omega^{334} \) and \( \omega^{365} \), we find these correspond to \( \omega^2 \), saving us the effort of complex calculations.

Properties of Complex Numbers

Complex numbers are made up of a real part and an imaginary part and are expressed in the form \( a + bi \), where \( i \) is the imaginary unit (\( i = \sqrt{-1} \)).
Properties of complex numbers often come in handy when solving moreadvanced problems.Some key properties include:

• Conjugate: The conjugate of \( a + bi \) is \( a - bi \).
• Magnitude: The magnitude or modulus is \( \sqrt{a^2 + b^2} \).
• Addition/Subtraction: Combine like terms: real with real, imaginary with imaginary.
• Multiplication: Expand using distributive property and simplify \( i^2 \) to \(-1\).

Understanding the conjugate and modulus is particularly useful when simplifying expressions or finding the inverse of complex numbers.
In problems involving roots of unity, multiplying and adding complex numbers often leads us to employ these properties in clever ways.

Simplifying Complex Expressions

Simplifying complex expressions involves more than just arithmetic; it takes advantage of special properties of complex numbers and their roots. Let's take an expression like your exercise as an example.
You'll encounter terms that rely on powers of complex numbers, specifically rooted in unity, such as the cube roots.Here's how you approach these:

• Identify and use properties: Recognize when a complex number is a known root of unity. Use its cyclical properties like \( \omega^3 = 1 \) to simplify exponents.
• Combine like terms: Reduce expressions by adding or subtracting like coefficients, especially when terms share the same powers of the root.
• Substitute effectively: Know the equivalent value of terms like \( \omega^2 \) and replace them.
• Final simplification: Ensure the final complex expression matches one of the basic options provided. This exercise resulted in \(-i\sqrt{3}\), matched to a given answer choice.

Simplifying involves a mix of recognizing patterns and applying algebraic rules, essential in both complex number problems and many fields of higher mathematics.