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Consider the following1. zz̅ = |z|22. z-1 = \(\rm \frac {z}{|z|^2}\), where z = complex number Which of the above statement is/are correct?1. Only 12. Only 23. Both 1 and 24. Neither 1 nor 2 |
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Answer» Correct Answer - Option 1 : Only 1 Concept: Consider a complex number, z = a + ib Conjugate of complex number = z̅ = a - ib Modulus of complex number = |z| = \(\rm \sqrt{(a^2 + b^2) }\)
Calculation: Let, z = a + ib, zz̅ = (a + ib)(a - ib) = \(\rm a^2-(ib)^2\) = \(\rm a^2-i^2(b)^2\) =\(\rm a^2-(ib)^2\) =\(\rm a^2+b^2\cdots (\because i^2=-1)\) And, |z|2 = \(\rm (\sqrt{a^2+b^2})^2\) = \(\rm {a^2+b^2}\) ∴ zz̅ = |z|2 Now, \(\rm z^{-1}=\frac1 z=\frac{1}{a+ib}\) =\(\rm \frac{1}{a+ib}\times \frac{a-ib}{a-ib}=\frac{a-ib}{a^2+b^2}\) = \(\rm \frac{\bar z}{|z|^2}\) ≠ \(\rm \frac {z}{|z|^2}\) Hence, option (1) is correct. |
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