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Complex Numbers question

2008 · Shift 0 · Q54
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Complex Numbers question

2008 · Shift 0 · Q54

JEE MainMathematicsComplex NumbersMCQ+4 / −1
The conjugate of a complex number is 1i−1{1 \over {i - 1}}i−11​ then that complex number is :
  1. A
    −1i−1{{ - 1} \over {i - 1}}i−1−1​
  2. B
    1i+1 {1 \over {i + 1}}\,i+11​
  3. C
    −1i+1{{ - 1} \over {i + 1}}i+1−1​
  4. D
    1i−1{1 \over {i - 1}}i−11​
View written solutionFree

Correct answer: C

  1. Let the complex number be zzz.

    We are given that its conjugate is z‾=1i−1.\overline{z} = \frac{1}{i-1}.z=i−11​.

  2. To find zzz, take conjugate on both sides: z=(1i−1)‾.z = \overline{\left(\frac{1}{i-1}\right)}.z=(i−11​)​.

  3. Use the property: (ab)‾=a‾b‾.\overline{\left(\frac{a}{b}\right)} = \frac{\overline{a}}{\overline{b}}.(ba​)​=ba​. Since 1‾=1\overline{1}=11=1 and i−1‾=−i−1\overline{i-1}=-i-1i−1​=−i−1, we get z=1−i−1.z = \frac{1}{-i-1}.z=−i−11​.

  4. Simplify: 1−i−1=−1i+1.\frac{1}{-i-1} = -\frac{1}{i+1}.−i−11​=−i+11​.

    Hence, z=−1i+1.z = -\frac{1}{i+1}.z=−i+11​.

  5. Compare with the options:

    • A: −1i−1\dfrac{-1}{i-1}i−1−1​
    • B: 1i+1\dfrac{1}{i+1}i+11​
    • C: −1i+1\dfrac{-1}{i+1}i+1−1​
    • D: 1i−1\dfrac{1}{i-1}i−11​

    This matches Option C.

Therefore, the required complex number is −1i+1.\boxed{-\frac{1}{i+1}}.−i+11​​.

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