If z is any complex number and Given: iz3+z2−z+i=0, where i=−1,$$\text{Given: }$$ iz^3 + z^2 - z + i = 0, $$\text{ where }$$ i = $$\sqrt{-1}$$, \quad Given: iz3+z2−z+i=0, where i=−1,then what is the value of Find: (∣z∣+1)$$2\text{Find: }$$ (|z| + 1)^2Find: (∣z∣+1)2 ?

Given: iz3+z2−z+i=0, where i=−1,$$\\text{Given: }$$ iz^3 + z^2 - z + i = 0, $$\\text{ where }$$ i = $$\\sqrt{-1}$$, \\quad Given: iz3+z2−z+i=0, where i=−1,Solution:iz3+z2−z+i=0iz2(z−i)−1(z−i)=0(iz2−1)(z−i)=0z2=1i,(z=i)∣z∣=1(∣z∣+1)2=(1+1)2=$$4\\begin{aligned}$$&iz^3 + z^2 - z + i = 0 \\\\&iz^2(z - i) - 1(z - i) = 0 \\\\&(iz^2 - 1)(z - i) = 0 \\\\&z^2 = $$\\frac{1}{i}$$, \\quad (z = i) \\\\&|z| = 1 \\\\&(|z| + 1)^2 = (1 + 1)^2 = $$4\\end{aligned}$$iz3+z2−z+i=0iz2(z−i)−1(z−i)=0(iz2−1)(z−i)=0z2=i1,(z=i)∣z∣=1(∣z∣+1)2=(1+1)2=4

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If z is any complex number and Given: iz3+z2−z+i=0, where i=−1,\text{Given: } iz^3 + z^2 - z + i = 0, \text{ where } i = \sqrt{-1}

Question

If z is any complex number and $\text{Given: } iz^3 + z^2 - z + i = 0, \text{ where } i = \sqrt{-1}, \quad$ then what is the value of $\text{Find: } (|z| + 1)^2$ ?

1

4

81

121

Solution

Correct option is B

$\text{Given: } iz^3 + z^2 - z + i = 0, \text{ where } i = \sqrt{-1}, \quad$

Solution:

$\begin{aligned}&iz^3 + z^2 - z + i = 0 \\&iz^2(z - i) - 1(z - i) = 0 \\&(iz^2 - 1)(z - i) = 0 \\&z^2 = \frac{1}{i}, \quad (z = i) \\&|z| = 1 \\&(|z| + 1)^2 = (1 + 1)^2 = 4\end{aligned}$