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Complex Numbers question

2019 · 12 Apr · Shift 2 · Q32
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Complex Numbers question

2019 · 12 Apr · Shift 2 · Q32

JEE MainMathematicsComplex NumbersMCQ+4 / −1
Let z ∈\in∈ C with Im(z) = 10 and it satisfies 2z−n2z+n{{2z - n} \over {2z + n}}2z+n2z−n​ = 2i - 1 for some natural number n. Then :
  1. A
    n = 20 and Re(z) = –10
  2. B
    n = 40 and Re(z) = 10
  3. C
    n = 40 and Re(z) = –10
  4. D
    n = 20 and Re(z) = 10
View written solutionFree

Correct answer: C

  1. Let z=x+10iz = x + 10iz=x+10i where x=Re⁡(z)x = \operatorname{Re}(z)x=Re(z) since Im⁡(z)=10\operatorname{Im}(z)=10Im(z)=10.

  2. Given 2z−n2z+n=2i−1=−1+2i\frac{2z-n}{2z+n}=2i-1=-1+2i2z+n2z−n​=2i−1=−1+2i Substitute z=x+10iz=x+10iz=x+10i: 2z=2x+20i2z=2x+20i2z=2x+20i So, (2x−n)+20i(2x+n)+20i=−1+2i\frac{(2x-n)+20i}{(2x+n)+20i}=-1+2i(2x+n)+20i(2x−n)+20i​=−1+2i

  3. Cross multiply: (2x−n)+20i=(−1+2i)((2x+n)+20i)(2x-n)+20i = (-1+2i)\big((2x+n)+20i\big)(2x−n)+20i=(−1+2i)((2x+n)+20i) Now expand the right-hand side: (−1+2i)(2x+n+20i)(-1+2i)(2x+n+20i)(−1+2i)(2x+n+20i) First distribute: =−(2x+n+20i)+2i(2x+n+20i)= - (2x+n+20i) + 2i(2x+n+20i)=−(2x+n+20i)+2i(2x+n+20i) =−2x−n−20i+4xi+2ni+40i2= -2x-n-20i + 4xi + 2ni + 40i^2=−2x−n−20i+4xi+2ni+40i2 Since i2=−1i^2=-1i2=−1, =−2x−n−20i+4xi+2ni−40= -2x-n-20i + 4xi + 2ni - 40=−2x−n−20i+4xi+2ni−40 Group real and imaginary parts: =(−2x−n−40)+(4x+2n−20)i= (-2x-n-40) + (4x+2n-20)i=(−2x−n−40)+(4x+2n−20)i

  4. Equate real and imaginary parts with the left side: Left side is (2x−n)+20i(2x-n) + 20i(2x−n)+20i So,

  • Real part: 2x−n=−2x−n−402x-n = -2x-n-402x−n=−2x−n−40 4x=−404x=-404x=−40 x=−10x=-10x=−10

  • Imaginary part: 20=4x+2n−2020 = 4x+2n-2020=4x+2n−20 Substitute x=−10x=-10x=−10: 20=4(−10)+2n−2020 = 4(-10)+2n-2020=4(−10)+2n−20 20=−40+2n−2020=-40+2n-2020=−40+2n−20 20=2n−6020=2n-6020=2n−60 2n=802n=802n=80 n=40n=40n=40

  1. Therefore, Re⁡(z)=x=−10,n=40\operatorname{Re}(z)=x=-10, \qquad n=40Re(z)=x=−10,n=40

  2. Check options:

  • A: n=20n=20n=20, Re⁡(z)=−10\operatorname{Re}(z)=-10Re(z)=−10 ❌
  • B: n=40n=40n=40, Re⁡(z)=10\operatorname{Re}(z)=10Re(z)=10 ❌
  • C: n=40n=40n=40, Re⁡(z)=−10\operatorname{Re}(z)=-10Re(z)=−10 ✅
  • D: n=20n=20n=20, Re⁡(z)=10\operatorname{Re}(z)=10Re(z)=10 ❌

Hence the correct option is C.

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