Problem 28 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The integral solution of the equation \((1-i)^{n}=2^{n}\) is (A) \(n=0\) (B) \(n=1\) (C) \(n=-1\) (D) None of these

Step-by-Step Solution

Answer

The solution is \(n = 0\).

Step 1: Recognize the given equation

The equation given is \((1-i)^{n} = 2^{n}\). We need to find the integer value of \(n\) for which this equation holds true.

Step 2: Write in polar form

Convert \(1-i\) into polar form. The complex number \(1-i\) can be written as \(\sqrt{2} \text{cis} \left(-\frac{\pi}{4}\right)\) where \(\text{cis}(\theta) = \cos(\theta) + i\sin(\theta)\).

Step 3: Equate magnitudes

For the magnitudes to be equal, we have \(\sqrt{2}^{n} = 2^{n}\). This implies \((2^{1/2})^{n} = 2^{n}\), which simplifies to \(2^{n/2} = 2^n\).

Step 4: Compare exponents

Since the bases are the same, compare the exponents: \(\frac{n}{2} = n\). Solving for \(n\), we get \(n = 0\).

Step 5: Check angle consistency

For angle consistency, \(-\frac{n\pi}{4}\) must be equivalent to a multiple of \(2\pi k\), where \(k\) is an integer. When \(n = 0\), both sides are zero, satisfying the consistency condition.

Key Concepts

Integrals

The integral of a function is a fundamental concept in calculus, representing the area under a curve described by the function's graph. While the original exercise focuses on complex numbers, integrating a function often plays a crucial role in solving more extensive mathematical problems that involve calculating continuous sums or areas. In the context of complex numbers, integrals might not be immediately applicable unless you're working on a problem that requires integration of functions involving complex variables.

Key points to remember about integrals include:

• An integral can be definite or indefinite.
• Definite integrals provide the total value accumulated between two bounds.
• Indefinite integrals find the antiderivative of a function, which is a broader concept.

It's essential to understand whether a problem requires integral calculus. In this exercise, our primary concern wasn't solving an integral, but understanding the manipulation of complex numbers.

Polar Form

Converting complex numbers to polar form simplifies multiplication and exponentiation. Polar form expresses a complex number in terms of a magnitude and an angle. The equation For a complex number like \(1-i\), converting to polar form involves finding its magnitude and direction (or angle). To convert, follow these steps:

• Calculate the magnitude: \(|1-i| = \sqrt{1^2 + (-1)^2} = \sqrt{2}\).
• Determine the angle: Use the arctangent function, \(\tan^{-1}\left(\frac{-1}{1}\right)\), which gives an angle of \(-\frac{\pi}{4}\).

Polar form becomes \( \sqrt{2} \text{cis} \left(-\frac{\pi}{4}\right)\), where \( \text{cis}(\theta) = \cos(\theta) + i\sin(\theta) \).

This conversion is useful in simplifying the exponentiation of complex numbers, as seen in our solution, where it allowed comparison between the magnitudes and angles of two complex expressions.

Exponents

Exponents are a crucial tool in mathematics, allowing us to express and manipulate numbers that are repeatedly multiplied by themselves. In the realm of complex numbers, using exponents becomes interesting as it requires understanding both the magnitude and the angle of the number.

In the exercise, we dealt with expressing a complex number and its powers by equating it with another using exponents. Here's how the exponents conceptually worked:

• We compared \((1-i)^n\) with \(2^n\) by breaking them down to their core components.
• This involved equating \( (\sqrt{2} \text{ cis}(-\pi/4) )^n \) with both magnitude \((\sqrt{2}^n = 2^n)\) and angles.

The exponent properties helped arrive at equality through both magnitude and angle: achieving a solution by considering both factors ensures correctness. When dealing with exponents, pay attention to how they affect not just the size of the number but also its direction, especially for complex numbers in polar form.