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Q14:
Express the following expression in the form of $a + ib$: 
Express the following expression in the form of $a + ib$:
Solution :
Given Expression
The expression to be simplified is:
$$Z = \frac{(3 + i\sqrt{5})(3 - i\sqrt{5})}{(\sqrt{3} + \sqrt{2}i) - (\sqrt{3} - i\sqrt{2})}$$
Step 1: Simplifying the Numerator
The numerator is of the form $(x + iy)(x - iy)$, which follows the identity $(x + y)(x - y) = x^2 - y^2$ [Algebraic Difference of Squares].
- Let $x = 3$ and $y = i\sqrt{5}$.
- Numerator $= (3)^2 - (i\sqrt{5})^2$
- $= 9 - (i^2 \cdot 5)$
- Since $i^2 = -1$ [Definition of Imaginary Unit]:
- $= 9 - (-1 \cdot 5) = 9 + 5 = 14$
Step 2: Simplifying the Denominator
The denominator involves the subtraction of two complex numbers:
- Denominator $= (\sqrt{3} + \sqrt{2}i) - (\sqrt{3} - i\sqrt{2})$
- Distribute the negative sign: $\sqrt{3} + \sqrt{2}i - \sqrt{3} + i\sqrt{2}$
- Combine real and imaginary parts [Commutative and Associative Properties of Addition]:
- $= (\sqrt{3} - \sqrt{3}) + (\sqrt{2}i + \sqrt{2}i)$
- $= 0 + 2\sqrt{2}i = 2\sqrt{2}i$
Step 3: Division and Rationalization
Now, substitute the simplified numerator and denominator back into $Z$:
$$Z = \frac{14}{2\sqrt{2}i} = \frac{7}{\sqrt{2}i}$$
To express this in $a + ib$ form, multiply the numerator and denominator by $i$:
- $Z = \frac{7 \cdot i}{\sqrt{2}i \cdot i}$
- $Z = \frac{7i}{\sqrt{2}i^2}$
- Since $i^2 = -1$:
- $Z = \frac{7i}{-\sqrt{2}} = -\frac{7}{\sqrt{2}}i$
Step 4: Final Standardization
Rationalize the denominator by multiplying by $\frac{\sqrt{2}}{\sqrt{2}}$ [Property of Real Numbers]:
- $Z = -\frac{7\sqrt{2}}{2}i$
- In standard $a + ib$ form, where $a=0$ and $b=-\frac{7\sqrt{2}}{2}$:
- $Z = 0 - \frac{7\sqrt{2}}{2}i$
Final Solution: $0 - \frac{7\sqrt{2}}{2}i$
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 $4 - 3i$.
- Q12: Find the multiplicative inverse of the complex number .
- Q13: Find the multiplicative inverse of the complex number $-i$.
CBSE Solutions for Class 11 Mathematics Complex Numbers
Chapters in CBSE - Class 11 Mathematics
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