If z=1/2-2 i , is such that z +1 =α z +β(1+ i ), i =√(-1) and α, β in R , then α+β is equal to

Mathematics · Complex Numbers

JEE Main 2024 — 29 January, Shift 1 — Question 5

If $z=\frac{1}{2}-2 i$ , is such that $|\mathrm{z}+1|=\alpha \mathrm{z}+\beta(1+\mathrm{i}), \mathrm{i}=\sqrt{-1}$ and $\alpha, \beta \in \mathrm{R} \quad$ , then $\alpha+\beta$ is equal to

Option A:
-4
Option B:
3
Correct
Option C:
2
Option D:
-1

Answer: B

Step-by-step solution

$\mathrm{z}=\frac{1}{2}-2 \mathrm{i}$

|z+1|=\alpha z+\beta(1+i)

$\left|\frac{3}{2}-2 \mathrm{i}\right|=\frac{\alpha}{2}-2 \alpha \mathrm{i}+\beta+\beta \mathrm{i}$

$\left|\frac{3}{2}-2 \mathrm{i}\right|=\left(\frac{\alpha}{2}+\beta\right)+(\beta-2 \alpha) \mathrm{i}$

$\beta=2 \alpha$

and $\frac{\alpha}{2}+\beta=\sqrt{\frac{9}{4}+4}$

$\alpha+\beta=3$

Answer key and solution verified before publishing.

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Exam: JEE Main 2024
Paper: 29 January, Shift 1
Subject: Mathematics
Chapter: Complex Numbers
Topic: Properties of Complex Numbers

Let alpha, beta be the roots of the equation x^2-x+2=0 with Im(alpha) Im(beta) . Then alpha^6+alpha^4+beta^4-5 alpha^2 is equal to

Answer: B

Step-by-step solution

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