Problem 180 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The roots of the equation \((2+z)^{6}+(2-z)^{6}=0\) are (A) \(\pm 2 i \tan \pi / 12\) (B) \(\pm 2 i \tan 5 \pi^{\prime} 12\) (C) \(\pm 2 i\) (D) \(\pm 2\)

Step-by-Step Solution

Answer

The roots are \(\pm 2i \tan(\frac{\pi}{12})\) from option (A).

Step 1: Recognize the Trigonometric Identity

Recognize that the expression \((2+z)^6 + (2-z)^6 = 0\) is symmetrical and can be expanded using the identity for sum of sixth powers, particularly using trigonometric expressions involving cosine and sine.

Step 2: Set the Rational Expression

Set \(u = 2 + z\) and \(v = 2 - z\), noting that the sum of sixth powers, \(u^6 + v^6 = 0\), means that one is the negative of the other.

Step 3: Use Polar Coordinates and Symmetry

Write each term in polar form. In polar form, using identities, \((2+z)^6\) and \((2-z)^6\) symmetrically reflect around the imaginary axis. Recognize that their symmetry implies these complexes annihilate each other.

Step 4: Simplify Using Complex Roots

Recognizing that sum leads to zero, implies roots lie on purely imaginary line. Utilize Euler's formula and polynomial properties on symmetry; \(z\) can be found as only imaginary entries solving \((2+z)^{6} = -(2-z)^{6}\).

Step 5: Solve Using Trigonometric Identities

Apply identities from step 3, letting \(z = 2i \tan(\theta)\), simplifying to \(\tan(2\theta)\) roots for known values at principal angles gives roots as \(\tan(\frac{\pi}{12})\) and equivalently for \(\tan(\frac{5\pi}{12})\).

Step 6: Verify Roots

Verify the solution. For each potential "angle," calculate the trigonometric output, and ensure it satisfies the setup equation \((2 + z)^6 + (2 - z)^6 = 0\) algebraically.

Key Concepts

Trigonometric Identities

Trigonometric identities play a vital role in simplifying complex equations. For the equation \((2+z)^{6}+(2-z)^{6}=0\), recognizing the symmetry in the expression is crucial. This involves using identities that connect powers and angles, such as the sum of sixth powers. These identities allow us to replace complex trigonometric expressions with simpler values involving sine and cosine. This simplification is especially helpful when dealing with complex roots.

• The identity is especially powerful when expressions appear identical but with different variable signs, such as \((2+z)\) and \((2-z)\).
• These expressions can be expanded or simplified using known trigonometric properties, like cosine or sine rules.

Understanding and identifying when and how to apply these identities can make solving even intricate equations more manageable.

Symmetry in Complex Numbers

Complex numbers often exhibit symmetry, which is a useful property in simplifying equations. For the given problem, recognizing that \((2+z)^6 + (2-z)^6=0\) has symmetry can guide us to an elegant solution. Symmetry in complex numbers usually implies that parts of an equation can reflect or cancel each other out.

• In this case, the parts \((2+z)^6\) and \((2-z)^6\) reflect around the imaginary axis.
• This reflection hints that these terms can potentially cancel out, leading to their annihilation.

Understanding symmetry is crucial as it can drastically reduce the complexity of solving complex equations, ensuring that solutions are not overlooked.

Polar Coordinates

Utilizing polar coordinates is a powerful technique in analyzing complex numbers, especially when it involves roots. Each complex number can be represented in polar form, which expresses a number in terms of its modulus and angle. For \((2+z)\) and \((2-z)\), writing these expressions in polar coordinates makes it easier to perceive the symmetry and reflectivity around the imaginary axis.

• Polar coordinates enable us to use Euler's formula effectively, relating complex numbers to exponential forms.
• This approach simplifies calculations and visualizes the rotation and angular displacement involved.

By employing polar coordinates, complex roots can be more easily understood and derived, exploiting the natural periodicity and angles associated with trigonometric functions.

Complex Roots Analysis

Analyzing complex roots involves multiple steps and techniques, including the understanding of symmetry, trigonometric identities, and polar form transformations. Once recognizing the expression's symmetry in \((2+z)^6+(2-z)^6=0\), the task becomes finding roots that satisfy this condition. Trigonometric identities hint at potential angles, such as \(\frac{\pi}{12}\) and its supplementary forms, which lead to the complex roots.

• Using properties of trigonometric functions helps determine angles like \(\tan(\theta)\) involved in finding the roots.
• Finding angles where these tangent values hold true gives the imaginary components of the roots.

Through a clear analysis, valid complex roots like \(\pm 2i \tan(\frac{\pi}{12})\) offer insightful solutions ensured by verifying their validity in the original equation.