Let z be a complex number with | z | = 1. Then the maximum value of | z + 1 | + | z2 – z + 1

Engineering

Mathematics

Conjugate Modulus and Argument of a Complex Number

Question

Let z be a complex number with | z | = 1. Then the maximum value of | z + 1 | + | z² – z + 1 | is

$\frac{13}{4}$

$3 \sqrt{\frac{7}{6}}$

College PredictorLive

Know your College Admission Chances Based on your Rank/Percentile, Category and Home State.

Get your JEE Main Personalised Report with Top Predicted Colleges in JoSA

| z | = 1

$∴ x \in$ [– 1, 1] (x² + y² = 1)

E = | z + 1 | + | z² – z + 1 |

$E = \sqrt{{(x + 1)}^{2} + y^{2}} + | z (z + \frac{1}{z} - 1) |$

$E = \sqrt{2 + 2 x} + | z | | z + \frac{1}{z} - 1 |$

$E = \sqrt{2} \sqrt{1 + x} + | 2 x - 1 |$

$∴ f (x) = \sqrt{2} \sqrt{1 + x} + | 2 x - 1 | in [- 1, 1]$

$f' (x) = 0 at x = \frac{- 7}{8}$

$\begin{matrix} x & - 1 & \frac{- 7}{8} & \frac{1}{2} & 1 \\ f (x) & 3 & \frac{13}{4} & \sqrt{3} & 3 \end{matrix}$

$∴ f (x) I_{\max} = \frac{13}{4}$