Problem 85 - A Complete Resource Book in Mathematics for JEE Main

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 (A) \(n=0\).

Step 1: Examine the Given Equation

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

Step 2: Consider the Magnitude of Complex Numbers

First, let's compare magnitudes. The magnitude of \(1-i\) is calculated as \(|1-i| = \sqrt{1^2 + (-1)^2} = \sqrt{2}\). The magnitude of \(2\) is \(2\). For the equation to hold for any integer \(n\), the magnitudes should match.\(|(1-i)^n| = |1-i|^n = (\sqrt{2})^n\) and \(|2^n| = 2^n\).

Step 3: Equate Magnitudes for Comparison

For equality, \((\sqrt{2})^n = 2^n\). This simplifies to \(2^{n/2} = 2^n\) which implies \(n/2 = n\). This condition is only satisfied by \(n=0\).

Step 4: Check by Substitution

Substitute \(n=0\) to check if both sides of the original equation are equal:\((1-i)^0 = 1\) and \(2^0 = 1\). Both sides are indeed equal for \(n=0\).

Step 5: Verify Other Options

Substitute \(n=1\) and \(n=-1\). For \(n=1\),\((1-i)^1 = 1-i\) and \(2^1 = 2\); they are not equal. For \(n=-1\), \((1-i)^{-1} = \frac{1}{1-i}\) and \(2^{-1} = \frac{1}{2}\); they are not equal.

Key Concepts

Complex Numbers

Complex numbers are numbers that include both a real part and an imaginary part. They are generally expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, satisfying \(i^2 = -1\).
For example, in the expression \(1-i\), the real part is \(1\) and the imaginary part is \(-1\). Complex numbers are a fundamental part of mathematics, often used in fields such as engineering and physics. They allow us to solve equations that don't have solutions in the realm of real numbers.
Polar forms of complex numbers can also be useful. A complex number \(z = a + bi\) can be represented as \(r(\cos \theta + i\sin \theta)\), where \(r = \sqrt{a^2 + b^2}\) is the magnitude and \(\theta\) is the argument of the complex number. This form is beneficial when performing multiplication or exponentiation of complex numbers, as multiplying becomes a matter of multiplying magnitudes and adding angles.

Magnitude Comparison

The magnitude, or modulus, of a complex number \(z = a + bi\) is calculated as \(|z| = \sqrt{a^2 + b^2}\). The magnitude represents the distance of the complex number from the origin on the complex plane, similar to the absolute value for real numbers.

• In our exercise, to compare the magnitudes of \((1-i)^n\) and \(2^n\), we calculate the magnitude of \(1-i\): \(|1-i| = \sqrt{1^2 + (-1)^2} = \sqrt{2}\).
• The comparison involves finding \((\sqrt{2})^n\) and \(2^n\) because \(|(1-i)^n| = (\sqrt{2})^n\) and \(|2^n| = 2^n\).

The requirement for these magnitudes to be equal leads us to the equation \((\sqrt{2})^n = 2^n\), simplifying to \(2^{n/2} = 2^n\). By equating exponents, we deduce that \(n/2 = n\), leading to \(n = 0\) as the solution, since the magnitudes only match for \(n = 0\).

Exponentiation Equality

Exponentiation equality involves solving equations where two expressions raised to a power are set equal. In this exercise, we deal with \((1-i)^n = 2^n\). Exponentiation equality ensures that both the base and exponent operations result in the same value on both sides of an equation.
The strategy involves aligning magnitude and arguments when dealing with complex numbers or directly comparing expressions when they are real numbers.

• For complex numbers, one checks both the magnitude and argument, making sure both the length and angle point to the same result.
• Specifically in our task, the magnitudes yield \(2^{n/2} = 2^n\) by simplifying the magnitudes which gives \(n/2 = n\).

This results in \(n = 0\), as any non-zero \(n\) would not maintain the equality. Exponential equality becomes straightforward when the expressions on both sides of the equation have common bases, but can get complicated if they involve complex transformations or changes.