Complex NumbershardFree

Evaluate |4z2z3+9z3z1+16z1z2| with Given Moduli | JEE

JEE Maths question with a full step-by-step solution.

Question

|z_1| = 2

|z_2| = 3

|z_3| = 4

and

|z_1 + z_2 + z_3| = 5

, then

|4z_2 z_3 + 9z_3 z_1 + 16z_1 z_2|

equals

20

24

48

120

correct

Solution

Step 1: Convert the given moduli into conjugate relations. For any complex number,

|z|^2 = z\bar{z}

, so

\bar{z} = \dfrac{|z|^2}{z}

. Hence:

|z_1|^2 = 4 \implies \bar{z_1} = \frac{4}{z_1}, \quad |z_2|^2 = 9 \implies \bar{z_2} = \frac{9}{z_2}, \quad |z_3|^2 = 16 \implies \bar{z_3} = \frac{16}{z_3}

Step 2: Factor

z_1 z_2 z_3

out of the target expression:

4z_2 z_3 + 9z_3 z_1 + 16z_1 z_2 = z_1 z_2 z_3\left(\frac{4}{z_1} + \frac{9}{z_2} + \frac{16}{z_3}\right)

Step 3: Replace each fraction using Step 1:

\frac{4}{z_1} = \bar{z_1}, \quad \frac{9}{z_2} = \bar{z_2}, \quad \frac{16}{z_3} = \bar{z_3}

So:

4z_2 z_3 + 9z_3 z_1 + 16z_1 z_2 = z_1 z_2 z_3\,(\bar{z_1} + \bar{z_2} + \bar{z_3}) = z_1 z_2 z_3\,\overline{(z_1 + z_2 + z_3)}

Step 4: Take the modulus. Using

|wv| = |w||v|

and

|\bar{u}| = |u|

|4z_2 z_3 + 9z_3 z_1 + 16z_1 z_2| = |z_1|\,|z_2|\,|z_3|\,|z_1 + z_2 + z_3|

Step 5: Substitute the values:

= 2 \cdot 3 \cdot 4 \cdot 5 = 120

Correct answer: (4)

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