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Q12:
Find the multiplicative inverse of the complex number 
.
Find the multiplicative inverse of the complex number .
Solution :
Given Variables & Initial Setup
Let the given complex number be $z$. Based on the expression $\sqrt{5} + 3i$, we define:
$z = \sqrt{5} + 3i$
The objective is to find the multiplicative inverse, denoted as $z^{-1}$ or $\frac{1}{z}$, such that $z \cdot z^{-1} = 1$.
Step 1: Definition of Multiplicative Inverse
The multiplicative inverse of a complex number $z = a + bi$ is given by the formula:
$z^{-1} = \frac{1}{a + bi} = \frac{a - bi}{(a + bi)(a - bi)}$
[Per the property of complex conjugates where $z \cdot \bar{z} = |z|^2 = a^2 + b^2$]
Step 2: Applying the Formula
Identify the real part $a = \sqrt{5}$ and the imaginary part $b = 3$.
The conjugate of $z$ is $\bar{z} = \sqrt{5} - 3i$.
Substitute these into the multiplicative inverse expression:
$z^{-1} = \frac{\sqrt{5} - 3i}{(\sqrt{5} + 3i)(\sqrt{5} - 3i)}$
Step 3: Simplifying the Denominator
Calculate the squared modulus $|z|^2 = a^2 + b^2$:
$|z|^2 = (\sqrt{5})^2 + (3)^2$
$|z|^2 = 5 + 9$
$|z|^2 = 14$
[By the identity $(a+bi)(a-bi) = a^2 + b^2$]
Step 4: Final Algebraic Reduction
Distribute the denominator across the numerator terms:
$z^{-1} = \frac{\sqrt{5}}{14} - \frac{3}{14}i$
Visual Representation of the Complex Number
Final Solution: The multiplicative inverse of $\sqrt{5} + 3i$ is $\frac{\sqrt{5}}{14} - \frac{3}{14}i$.
More Questions from Class 11 Mathematics Complex Numbers EXERCISE 4.1
- Q1: Express the complex number in the form $a + ib$.
- Q2: Express the complex number $i^9 + i^{19}$ in the form $a + ib$.
- Q3: Express the complex number $i^{-39}$ in the form $a + ib$.
- Q4: Express the complex number $3(7 + i7) + i(7 + i7)$ in the form $a + ib$.
- Q5: Express the complex number $(1 - i) - (-1 + i6)$ in the form $a + ib$.
- Q6: Express the complex number in the form $a + ib$.
- Q7: Express the complex number in the form $a + ib$.
- Q8: Express the complex number $(1 - i)^4$ in the form $a + ib$.
- Q9: Express the complex number in the form $a + ib$.
- Q10: Express the complex number in the form $a + ib$.
- Q11: Find the multiplicative inverse of the complex number $4 - 3i$.
- Q13: Find the multiplicative inverse of the complex number $-i$.
- Q14: Express the following expression in the form of $a + ib$:
CBSE Solutions for Class 11 Mathematics Complex Numbers
Chapters in CBSE - Class 11 Mathematics
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