Problem 94 - A Complete Resource Book in Mathematics for JEE Main

Question

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

Step-by-Step Solution

Answer

It equals \( 5(|z_1|^2 + |z_2|^2) \). None of the given options match correctly.

Step 1: Understanding Complex Numbers

Complex numbers are in the form \( z = a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. The conjugate of \( z \), denoted \( \bar{z} \), is \( a - bi \). The modulus is given by \( \left| z \right| = \sqrt{a^2 + b^2} \).

Step 2: Simplifying the First Term

Consider \( \left| \sqrt{2}z_1 + i \sqrt{3} \bar{z}_2 \right|^2 \). Using properties of complex numbers, we have: \[ \left| \sqrt{2}z_1 + i\sqrt{3}\bar{z}_2 \right|^2 = (\sqrt{2}z_1 + i\sqrt{3}\bar{z}_2)(\sqrt{2}\bar{z}_1 - i\sqrt{3}z_2) \]. When you expand this, using \( z\bar{z} = |z|^2 \), and keeping imaginary and real parts separate, leads to \( 2|z_1|^2 + 3|z_2|^2 \).

Step 3: Simplifying the Second Term

Now consider \( \left| \sqrt{3} \bar{z}_1 + i \sqrt{2} z_2 \right|^2 \). Similarly, we have \[ (\sqrt{3}\bar{z}_1 + i\sqrt{2}z_2)(\sqrt{3}z_1 - i\sqrt{2}\bar{z}_2) \]. Expanding results in \( 3|z_1|^2 + 2|z_2|^2 \).

Step 4: Adding Results

Combine the results from Steps 2 and 3: \[ 2|z_1|^2 + 3|z_2|^2 + 3|z_1|^2 + 2|z_2|^2 = 5|z_1|^2 + 5|z_2|^2 \]. Therefore, the sum simplifies as: \( 5 \left( |z_1|^2 + |z_2|^2 \right) \).

Step 5: Comparing with Given Options

Review the options given: (A) is less than \( 5(|z_1|^2 + |z_2|^2) \), (C) is equal to \( 2|z_1|^2 + 3|z_2|^2 \). From simplification, we found the expression equals \( 5(|z_1|^2 + |z_2|^2) \). Therefore, the correct option is neither.

Key Concepts

Modulus of Complex Numbers

In complex numbers, the modulus is a crucial concept that measures the "size" or "length" of a complex number. For any complex number in the form \( z = a + bi \), where \( a \) and \( b \) are real numbers, the modulus is represented by \( |z| \) and is calculated using the formula:
\[|z| = \sqrt{a^2 + b^2}\]This represents the distance of the complex number from the origin in the complex plane. It can be visualized as the hypotenuse of a right-angled triangle, with \( a \) and \( b \) being the other two sides.

• The modulus is always a non-negative real number.
• A complex number has a modulus of zero if and only if it is the origin, i.e., \( z = 0 + 0i \).

Understanding the modulus is particularly important when dealing with expressions involving complex numbers, such as the given exercise where different terms involving modulus lead to expressions that can be compared or simplified further.

Conjugate of Complex Numbers

The conjugate of a complex number helps to simplify algebraic expressions, especially when division of complex numbers is involved or when finding the modulus. For a complex number \( z = a + bi \), its conjugate, denoted \( \bar{z} \), is:
\( \bar{z} = a - bi \)This operation essentially reflects the number across the real axis in the complex plane. Conjugates have several notable properties:

• The modulus of a complex number and its conjugate are identical, i.e., \( |z| = |\bar{z}| \).
• When you multiply a complex number with its conjugate, the result is always real, given by: \[ z\bar{z} = a^2 + b^2 = |z|^2 \]
• Conjugates simplify the process of rationalizing the denominator in complex fractions.

In the provided solution, the conjugate was used to form pairs like \( i\sqrt{3}\bar{z}_2 \) and \( -i\sqrt{3}z_2 \), which upon multiplication and expansion led to real terms through this property.

Properties of Complex Numbers

Complex numbers abide by a set of rules that are extensions of real number properties but include considerations for the imaginary unit \( i \). Among these properties:
- **Commutativity and Associativity:** Like real numbers, complex addition and multiplication are commutative and associative. This means you can change the order of operation without affecting the outcome.- **Distributive Property:** Distributing over addition holds as well, \((z_1+z_2)z_3 = z_1z_3 + z_2z_3\).- **Multiplicative Identity and Inverses:** The multiplicative identity is \(1\), and each non-zero complex number \(z\) has a multiplicative inverse \(1/z\) such that \(z \times (1/z) = 1\).In specific terms, when addressing a complex number like \( \sqrt{2}z_1 + i\sqrt{3}\bar{z}_2 \), we utilize these properties to simplify expressions by expanding terms using distributive and conjugate properties. This makes simplification possible, ultimately leading to a clearer comparison of the expressions as done in the problem's step-by-step solution.