Question:
The imaginary part of \(\frac{1 - i\sqrt{3}}{1 + i\sqrt{3}}\) is
The imaginary part of \(\frac{1 - i\sqrt{3}}{1 + i\sqrt{3}}\) is
Show Hint
To find imaginary part, first convert expression into $a+ib$ form.
Updated On: Apr 30, 2026
- $-\frac{1}{2}$
- $\frac{1}{2}$
- $\frac{\sqrt{3}}{2}$
- $-\frac{\sqrt{3}}{2}$
- $\frac{\sqrt{3}}{4}$
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Concept:
Convert to standard form using conjugate.
Step 1: Multiply by conjugate
\[ \frac{1 - i\sqrt{3}}{1 + i\sqrt{3}} \cdot \frac{1 - i\sqrt{3}}{1 - i\sqrt{3}} \]
Step 2: Simplify denominator
\[ 1 + 3 = 4 \]
Step 3: Simplify numerator
\[ (1 - i\sqrt{3})^2 = 1 - 2i\sqrt{3} - 3 = -2 - 2i\sqrt{3} \] \[ = \frac{-2 - 2i\sqrt{3}}{4} = -\frac{1}{2} - \frac{\sqrt{3}}{2}i \]
Step 4: Imaginary part
\[ -\frac{\sqrt{3}}{2} \] Final Conclusion:
Option (D)
Step 1: Multiply by conjugate
\[ \frac{1 - i\sqrt{3}}{1 + i\sqrt{3}} \cdot \frac{1 - i\sqrt{3}}{1 - i\sqrt{3}} \]
Step 2: Simplify denominator
\[ 1 + 3 = 4 \]
Step 3: Simplify numerator
\[ (1 - i\sqrt{3})^2 = 1 - 2i\sqrt{3} - 3 = -2 - 2i\sqrt{3} \] \[ = \frac{-2 - 2i\sqrt{3}}{4} = -\frac{1}{2} - \frac{\sqrt{3}}{2}i \]
Step 4: Imaginary part
\[ -\frac{\sqrt{3}}{2} \] Final Conclusion:
Option (D)
Was this answer helpful?
0
0
Top KEAM Mathematics Questions
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
- The value of $ \cos [{{\tan }^{-1}}\{\sin ({{\cot }^{-1}}x)\}] $ is
- The solutions set of inequation $\cos^{-1}x < \,\sin^{-1}x$ is
- Let $\Delta= \begin{vmatrix}1&1&1\\ 1&-1-w^{2}&w^{2}\\ 1&w&w^{4}\end{vmatrix}$, where $w \neq 1$ is a complex number such that $w^3 = 1$. Then $\Delta$ equals
- Let $p : 57$ is an odd prime number, $\quad \, q : 4$ is a divisor of $12$ $\quad$ $r : 15$ is the $LCM$ of $3$ and $5$ Be three simple logical statements. Which one of the following is true?
View More Questions
Top KEAM Complex numbers Questions
- The principal argument of the complex numb $Z=\frac{1+\sin \frac{\pi}{3}+i \cos\frac{\pi}{3} }{1+\sin \frac{\pi}{3} - i \cos\frac{\pi}{3} }$ is
- If $ \sqrt{x+iy}=\pm (a+ib), $ then $ \sqrt{-x-iy} $ is equal to
- If $a = e^{i \theta}$ , then $\frac{1 + a}{1-a}$ is equal to
- If the conjugate of a complex number $z$ is $\frac{1}{i - 1}$ , then $z$ is
- The sum of odd integers from $1$ to $2001$ is
View More Questions
Top KEAM Questions
- i.$\quad$ They help in respiration ii.$\quad$ They help in cell wall formation iii.$\quad$ They help in DNA replication iv. $\quad$They increase surface area of plasma membrane Which of the following prokaryotic structures has all the above roles?
- A body oscillates with SHM according to the equation (in SI units), $x = 5 cos \left(2\pi t +\frac{\pi}{4}\right) .$ Its instantaneous displacement at $t = 1$ second is
- The pH of a solution obtained by mixing 60 mL of 0.1 M BaOH solution at 40m of 0.15m HCI solution is
Kepler's second law (law of areas) of planetary motion leads to law of conservation of
- If $\int e^{2x}f' \left(x\right)dx =g \left(x\right)$, then $ \int\left(e^{2x}f\left(x\right) + e^{2x} f' \left(x\right)\right)dx =$
View More Questions