Problem 53 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

For any two complex numbers \(z_{1}\) and \(z_{2}\) with \(\left|z_{1}\right| \neq\left|z_{2}\right|\) \(\left|\sqrt{2} z_{1}+i \sqrt{3} \bar{z}_{2}\right|^{2}+\left|\sqrt{3} \bar{z}_{1}+i \sqrt{2} z_{2}\right|^{2}\) is (A) less than \(5\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}\) (B) greater than \(10\left|\mathrm{z}_{1} z_{2}\right|\) (C) equal to \(2\left|z_{1}\right|^{2}+3\left|z_{2}\right|^{2}\) (D) zero

Step-by-Step Solution

Answer

The expression simplifies to 5(|z_1|^2 + |z_2|^2), but this does not directly match any option, suggesting misalignment or error in provided options.

Step 1: Express Complex Numbers and Conjugates

Let the complex numbers be expressed as \( z_1 = a + bi \) and \( z_2 = c + di \). Their conjugates will be \( \bar{z}_1 = a - bi \) and \( \bar{z}_2 = c - di \).

Step 2: Simplify Modulus Expressions

We need to evaluate expressions \( \left|\sqrt{2} z_{1} + i \sqrt{3} \bar{z}_{2}\right|^{2} \) and \( \left|\sqrt{3} \bar{z}_{1} + i \sqrt{2} z_{2}\right|^{2} \). They are squares of the modulus expressions involving constants and conjugates.

Step 3: Modulus Property and Simplification

Applying \( |w|^2 = w \bar{w} \) for complex numbers, we start with: \[ |w_1|^2 = (\sqrt{2} z_1 + i \sqrt{3} \bar{z}_2)(\sqrt{2} \bar{z}_1 - i \sqrt{3} z_2) = 2|z_1|^2 + 3|z_2|^2 \]And for \( |w_2|^2 \)\[ |w_2|^2 = (\sqrt{3} \bar{z}_1 + i \sqrt{2} z_2)(\sqrt{3} z_1 - i \sqrt{2} \bar{z}_2) = 3|z_1|^2 + 2|z_2|^2 \].

Step 4: Sum of the Simplified Expressions

Add the two simplified expressions:\[ |w_1|^2 + |w_2|^2 = (2|z_1|^2 + 3|z_2|^2) + (3|z_1|^2 + 2|z_2|^2) = 5|z_1|^2 + 5|z_2|^2 \]

Step 5: Match to Given Options

Compare \( 5|z_1|^2 + 5|z_2|^2 \) with the options. It does not match any option as stated, implying the need to further check especially with simplifying 5 in both \(|z_1|^2\) and \(|z_2|^2\), leading us to a simpler common structure within the choices.

Key Concepts

Modulus of Complex Numbers

The modulus of a complex number is a measure of its size or magnitude. If you have a complex number expressed as \( z = a + bi \), where \( a \) is the real part and \( b \) is the imaginary part, the modulus is calculated using the formula:\[ |z| = \sqrt{a^2 + b^2} \] This formula comes from the Pythagorean theorem, treating the real and imaginary parts as the sides of a right triangle. The modulus represents the "length" of the vector \( z \) when plotted on the complex plane.

Understanding modulus is essential because it helps describe how large or small a complex number is, which is useful in various mathematical contexts, including equations and transformations. The modulus is always a non-negative number.

• For example, if \( z = 3 + 4i \), then \(|z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\).
• It’s important to note that the modulus of a number is simply its absolute value when dealing with real numbers.

Complex Conjugates

Complex conjugates are pairs of complex numbers that have the same real part but opposite imaginary parts. For a given complex number \( z = a + bi \), its conjugate is denoted as \( \bar{z} \), which equals \( a - bi \).

This concept is crucial in simplifying expressions and solving equations involving complex numbers. It is because often, the real component of a product of a complex number and its conjugate is significant, as it yields the square of the modulus of the number.

When you multiply a complex number by its conjugate, you eliminate the imaginary parts, which can be highly useful:

• The product is always a real number: \( z \cdot \bar{z} = (a + bi)(a - bi) = a^2 - (bi)^2 = a^2 + b^2 \).
• This product equals the modulus squared: \(|z|^2 = a^2 + b^2 \).

Using complex conjugates, we can handle division and other operations in complex arithmetic more easily.

Properties of Modulus

The properties of the modulus of complex numbers highlight its algebraic behavior, similar to those of absolute values. Some key properties include:

• For any complex numbers \( z_1 \) and \( z_2 \), the modulus function is multiplicative: \( |z_1 \times z_2| = |z_1| \times |z_2| \).
  This property illustrates that multiplying complex numbers results in a modulus that is equivalent to multiplying their individual moduli.
• The modulus of a conjugate is the same as the modulus of the original complex number: \( |\bar{z}| = |z| \).
  This reflects the idea that the magnitude remains unchanged even if the sign of the imaginary component switches.
• If \( z \) is a complex number, then the modulus is always non-negative: \(|z| \geq 0\).
  This is straightforward, as it is derived from a square root of a sum of squares.
• The triangle inequality for complex moduli states: \(|z_1 + z_2| \leq |z_1| + |z_2|\).
  This is crucial when dealing with vector addition and helps to understand the geometry of complex numbers on the plane.

These properties make calculations and proofs involving complex numbers more manageable and illustrate why the modulus is so important in complex analysis.