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JEE Main 20244 AprilMorning Shift
Exam
JEE Main
Year
2024
Date
4 April
Shift
Morning Shift
Subject
Mathematics
Chapter
Complex Number

JEE Main 2024 · 4 April · Morning Shift · Mathematics · Complex Number

Let α and β be the sum and the product of all the non-zero...

Let αand\alpha and\betabethesumandtheproductofallthenon−zerosolutionsoftheequationbe the sum and the product of all the non-zero solutions of the equation(\bar{z})^2+|z|=0,,z \inC.ThenC. Then4(α2+β2)\left(\alpha^2+\beta^2\right) is equal to :

Options

  1. A6
  2. B8
  3. C2
  4. D4

Correct answer

D. 4

Step-by-step solution

z=x+iyzˉ=x−iyzˉ2=x2−y2−2ixy⇒x2−y2−2ixy+x2+y2=0⇒x=0 or y=0−y2+∣y∣=0x2+∣x∣=0∣y∣=∣y∣2⇒x=0y=0,±1⇒α=i−i=0⇒i,−iβ=i(−i)=1 are roots 4(0+1)=4\begin{array}{ll}z=x+i y & \\ \bar{z}=x-i y & \\ \bar{z}^2=x^2-y^2-2 i x y & \\ \Rightarrow x^2-y^2-2 i x y+\sqrt{x^2+y^2}=0 \\ \Rightarrow x=0 \quad \text{ or } & y=0 \\ -y^2+|y|=0 & x^2+|x|=0 \\ |y|=|y|^2 & \Rightarrow x=0 \\ y=0, \pm 1 & \Rightarrow \alpha=i-i=0 \\ \Rightarrow i,-i & \beta=i(-i)=1 \\ \text{ are roots } & 4(0+1)=4\end{array}
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