Question

If \(|z_1| = 2, |z_2| = 3, |z_3| = 4\) and \(|2z_1 + 3z_2 + 4z_3| = 4\), then absolute value of \(8z_2z_3 + 27z_3z_1 + 64z_1z_2\) equals

• A. 24
• B. 48
• C. 72
• D. 96

Hint:

When you see expressions like \(k_1 z_2 z_3 + k_2 z_3 z_1 + k_3 z_1 z_2\), always try to factor out \(z_1 z_2 z_3\) and use the identity \(1/z = \bar{z}/|z|^2\). This often simplifies the problem to a known modulus given in the question.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The question asks for the absolute value (modulus) of a complex expression involving three complex numbers \(z_1, z_2, z_3\) with given moduli and a specific linear combination modulus.

Step 2: Key Formula or Approach:
For any complex number \(z\), we have \(z \cdot \bar{z} = |z|^2\).
This implies \(\bar{z} = \frac{|z|^2}{z}\) and \(z = \frac{|z|^2}{\bar{z}}\).
Also, the modulus of a conjugate is the same as the modulus of the number: \(|\bar{w}| = |w|\).

Step 3: Detailed Explanation:
Given \(|z_1| = 2 \Rightarrow |z_1|^2 = 4 \Rightarrow z_1\bar{z}_1 = 4\).
Given \(|z_2| = 3 \Rightarrow |z_2|^2 = 9 \Rightarrow z_2\bar{z}_2 = 9\).
Given \(|z_3| = 4 \Rightarrow |z_3|^2 = 16 \Rightarrow z_3\bar{z}_3 = 16\).
We need to find the modulus of \(S = 8z_2z_3 + 27z_3z_1 + 64z_1z_2\).
Let's factor out \(z_1 z_2 z_3\):
\[ S = z_1 z_2 z_3 \left( \frac{8}{z_1} + \frac{27}{z_2} + \frac{64}{z_3} \right) \] Using the conjugate relations from
Step 2:
\(\frac{1}{z_1} = \frac{\bar{z}_1}{4} \), \(\frac{1}{z_2} = \frac{\bar{z}_2}{9} \), \(\frac{1}{z_3} = \frac{\bar{z}_3}{16} \).
Substitute these into the expression for S:
\[ S = z_1 z_2 z_3 \left( 8 \cdot \frac{\bar{z}_1}{4} + 27 \cdot \frac{\bar{z}_2}{9} + 64 \cdot \frac{\bar{z}_3}{16} \right) \] \[ S = z_1 z_2 z_3 (2\bar{z}_1 + 3\bar{z}_2 + 4\bar{z}_3) \] Taking the modulus on both sides:
\[ |S| = |z_1| \cdot |z_2| \cdot |z_3| \cdot |2\bar{z}_1 + 3\bar{z}_2 + 4\bar{z}_3| \] We know \(|2\bar{z}_1 + 3\bar{z}_2 + 4\bar{z}_3| = |\overline{2z_1 + 3z_2 + 4z_3}| = |2z_1 + 3z_2 + 4z_3|\).
Substituting the given values:
\[ |S| = 2 \cdot 3 \cdot 4 \cdot 4 = 96 \]

Step 4: Final Answer:
The absolute value equals 96.