Question:
If the conjugate of a complex number \( z \) is \( \frac{1}{i - 1} \), then \( z \) is
If the conjugate of a complex number \( z \) is \( \frac{1}{i - 1} \), then \( z \) is
Show Hint
Conjugate of a fraction equals fraction of conjugates—apply carefully to numerator and denominator.
Updated On: May 1, 2026
- \( \frac{1}{i - 1} \)
- \( \frac{1}{i + 1} \)
- \( \frac{-1}{i - 1} \)
- \( \frac{-1}{i + 1} \)
- \( \frac{1}{i} \)
Show Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Concept:
If \( \bar{z} = w \), then \( z = \bar{w} \).
Step 1: Given: \[ \bar{z} = \frac{1}{i - 1} \]
Step 2: Take conjugate on both sides.
\[ z = \overline{\left(\frac{1}{i - 1}\right)} = \frac{1}{\overline{i - 1}} \]
Step 3: Compute conjugate.
\[ \overline{i - 1} = -i - 1 = -(1 + i) \]
Step 4: Simplify expression.
\[ z = \frac{1}{-(1+i)} = \frac{-1}{1+i} \] Multiply numerator and denominator: \[ = \frac{-1(1-i)}{(1+i)(1-i)} = \frac{-1 + i}{2} \] Which corresponds to: \[ \frac{1}{i+1} \]
Step 1: Given: \[ \bar{z} = \frac{1}{i - 1} \]
Step 2: Take conjugate on both sides.
\[ z = \overline{\left(\frac{1}{i - 1}\right)} = \frac{1}{\overline{i - 1}} \]
Step 3: Compute conjugate.
\[ \overline{i - 1} = -i - 1 = -(1 + i) \]
Step 4: Simplify expression.
\[ z = \frac{1}{-(1+i)} = \frac{-1}{1+i} \] Multiply numerator and denominator: \[ = \frac{-1(1-i)}{(1+i)(1-i)} = \frac{-1 + i}{2} \] Which corresponds to: \[ \frac{1}{i+1} \]
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 conjugate of a complex number Questions
- Evaluate \( \int \frac{x^2 + 6x + 1}{(x+3)^2} \, dx \).
- Let \(z_{1} = \frac{1}{2} +i\frac{\sqrt{3}}{2}\) and \(z_{2} = -\frac{1}{2} -i\frac{\sqrt{3}}{2}\) . If \(w = z_{1} + \bar{z}_{2}\) , then \(\overline{w} =\)
- If the conjugate of a complex number \( z \) is \( \frac{1}{i - 1} \), then \( z \) is
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