Problem 145 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(z_{1}+z_{2}+z_{3}=A, z_{1}+z_{2} \omega+z_{3} \omega^{2}=B\) and \(z_{1}+z_{2}\) \(\omega^{2}+z_{3} \omega=C\), where \(1, \omega, \omega^{2}\) are the three cube roots of unity, then \(|A|^{2}+|B|^{2}+|C|^{2}=\) (A) \(3\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+\left|z_{3}\right|^{2}\right)\) (B) \(2\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+\left|z_{3}\right|^{2}\right)\) (C) \(\left(\left|z_{1}\right|^{2}+\left|z_{2}\right|^{2}+\left|z_{3}\right|^{2}\right)\) (D) None of these

Step-by-Step Solution

Answer

(A) 3(|z_1|^2 + |z_2|^2 + |z_3|^2)

Step 1: Understand Properties of Cube Roots of Unity

We identify the cube roots of unity: 1, \(\omega\), \(\omega^2\). These satisfy the equations \(\omega^3 = 1\) and \(1 + \omega + \omega^2 = 0\). This will be useful in simplifying expressions.

Step 2: Express Total Sums with Roots

Identify expressions for sums: \(A = z_1 + z_2 + z_3\), \(B = z_1 + z_2\omega + z_3\omega^2\), and \(C = z_1 + z_2\omega^2 + z_3\omega\). We formulate these sums by considering how the cube roots of unity affect the coefficients of \(z_2\) and \(z_3\).

Step 3: Compute the Magnitude Squares

To find \(|A|^2 + |B|^2 + |C|^2\), use the properties of roots of unity to get:1. \(|A|^2 = (z_1 + z_2 + z_3)(\overline{z_1} + \overline{z_2} + \overline{z_3})\)2. \(|B|^2 = (z_1 + z_2\omega + z_3\omega^2)(\overline{z_1} + \overline{z_2}\overline{\omega} + \overline{z_3}\overline{\omega^2})\)3. \(|C|^2 = (z_1 + z_2\omega^2 + z_3\omega)(\overline{z_1} + \overline{z_2}\overline{\omega^2} + \overline{z_3}\overline{\omega})\).

Step 4: Simplify Using Orthogonality

With complex numbers, note that the multiplication terms like \(z_2\omega\) and \(z_3\omega^2\) behave orthogonally due to properties \(\omega + \omega^2 = -1\) and \(\overline{\omega} = \omega^2\). Therefore, sums like \(z_2\omega + z_3\omega^2\) and their conjugates simplify due to cancellation of imaginary terms in products impressed by orthogonality and zero sum property.

Step 5: Evaluate Individual Magnitude Squares

Compute:- \( |B|^2 + |C|^2 = |z_1 + z_2\omega + z_3\omega^2|^2 + |z_1 + z_2\omega^2 + z_3\omega|^2 = |z_1|^2 + |z_2|^2 + |z_3|^2\) for each assignment leading to a total sum of real terms.

Step 6: Total Result and Comparison

Summing these and considering properties gives \(|A|^2 + |B|^2 + |C|^2 = 3\times(|z_1|^2 + |z_2|^2 + |z_3|^2)\). Therefore, the answer is option (A).

Key Concepts

Cube Roots of Unity

The cube roots of unity are special numbers defined as the solutions to the equation \(x^3 = 1\). These roots consist of 1, \(\omega\), and \(\omega^2\), where \(\omega = e^{2\pi i / 3}\) which is equivalent to \(-\frac{1}{2} + i\frac{\sqrt{3}}{2}\), and \(\omega^2 = e^{-2\pi i / 3}\) which equals \(-\frac{1}{2} - i\frac{\sqrt{3}}{2}\). One interesting property of these roots is that they satisfy the equation \(1 + \omega + \omega^2 = 0\).

The cube roots of unity are often used in simplifying expressions involving complex numbers.

• When you add these roots together, the result is zero, a property that can be quite handy in various calculations.
• The expression \((x-1)(x-\omega)(x-\omega^2) = x^3 - 1\) further demonstrates their significance as roots.

Understanding these roots helps in evaluating expressions and simplifying equations by leveraging their unique additive properties.

Magnitude Squares

When dealing with complex numbers, the magnitude (or modulus) is an important concept. It represents the distance of a complex number from the origin in the complex plane. The square of this magnitude for any complex number \(z = a + bi\) is given by \(|z|^2 = a^2 + b^2\). This makes use of the conjugate \(z = \overline{z}\), where \(\overline{z} = a - bi\), thus \(|z|^2 = z \cdot \overline{z}\).

In the context of cube roots of unity, calculating magnitude squares can help in understanding how complex sums behave:

• The expression \(|A|^2 = (z_1 + z_2 + z_3)(\overline{z_1} + \overline{z_2} + \overline{z_3})\) utilizes the properties of roots and their conjugates to compute magnitude squares.
• This method allows simplifying the calculations of expression magnitudes, avoiding a direct computation of vectors in complex spaces.

Through these calculations, we ensure that each part of the equation maintains consistency with the properties of magnitude in complex numbers.

Orthogonality of Complex Numbers

Orthogonality in the realm of complex numbers is tied to the concept of vectors being perpendicular in geometry. For complex numbers, orthogonality can imply that product terms cancel each other, especially when dealing with conjugate pairs over symmetric roots like the cube roots of unity.

In calculations, terms like \(z_2\omega\) and \(z_3\omega^2\) show orthogonality because the properties \(\omega + \omega^2 = -1\) help them cancel out imaginary components:

• The cube roots' property of combining to zero (\(1 + \omega + \omega^2 = 0\)) is an essential aspect where these orthogonal properties can be leveraged.
• This ensures that during calculations, imaginary components are balanced out, often leaving just the real parts.

This can simplify expressions significantly, as explored in evaluations like \(|B|^2 + |C|^2\), ensuring pure real results from otherwise complex summations.