SATHEE: Complex Numbers 2 Question 42

Complex Numbers 2 Question 42

43. If $i z^{3} + z^{2} - z + i = 0$ , then show that $| z | = 1$ .

(1995, 5M)

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Solution:

Given, $i z^{3} + z^{2} - z + i = 0$

$\begin{aligned} \Rightarrow i z^{3} - i^{2} z^{2} - z + i = 0 [∵ i^{2} = - 1] \\ \Rightarrow i z^{2} (z - i) - 1 (z - i) = 0 \\ \Rightarrow (i z^{2} - 1) (z - i) = 0 \\ \Rightarrow z - i = 0 or i z^{2} - 1 = 0 \\ \Rightarrow z = i or z^{2} = \frac{1}{i} = - i \\ If z = i, then | z | = | i | = 1 \\ If z^{2} = - i, then | z^{2} | = | - i | = 1 \\ \Rightarrow | z |^{2} = 1 \Rightarrow | z | = 1 \end{aligned}$

Complex Numbers 2 Question 42

43. If $i z^{3} + z^{2} - z + i = 0$ , then show that $| z | = 1$ .

Solution:

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Complex Numbers 2 Question 42

43. If iz3+z2−z+i=0, then show that |z|=1.

Solution:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

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NCERT Exercise Video Solution

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43. If $i z^{3} + z^{2} - z + i = 0$ , then show that $| z | = 1$ .