Let integers a, b ∈[-3,3] be such that a+b ≠ 0 . Then the number of all possible ordered pairs (a, b), for which | z-a z+b |=1 and | array ccc z+1 & ω & ω^2 ω & z+ω^2 & 1 ω^2 & 1 & z+ω array |=1, z ∈ C , where ω and ω^2

Correct answer: 0 aligned & a, b ∈ I,-3 ≤ a, b ≤ 3, a+b ≠ 0 & |z-a|=|z+b| & | array ccc z+1 & ω & ω^2 ω & z+ω^2 & 1 ω^2 & 1 & z+ω array |=1 & | array ccc z & z & z ω & z+ω^2 & 1 ω^2 & 1 & z+ω array |=1 aligned aligned & z| array ccc 1 & 1 & 1 ω & z+ω^2 & 1 ω^2 & 1 & z+ω array |=1 & z| array ccc 1 & 0 & 0 ω & z+ω^2-ω & 1-ω ω^2 & 1-ω^2 & z+ω-ω^2 array |=1 & z^3=1 & z=ω, ω^2, 1 aligned Now aligned & |1-a|=|^ 1 +b| & 10 pairs aligned

JEE Main 202529 JanuaryEvening Shift

Exam: JEE Main
Year: 2025
Date: 29 January
Shift: Evening Shift
Subject: Mathematics
Chapter: Complex Number

JEE Main 2025 · 29 January · Evening Shift · Mathematics · Complex Number

Let integers a, b ∈[-3,3] be such that a+b ≠ 0 . Then the number of...

Let integers

\mathrm{a}, \mathrm{b} \in[-3,3]

be such that

\mathrm{a}+\mathrm{b} \neq 0

. Then the number of all possible ordered pairs (a, b), for which

\left|\frac{z-\mathrm{a}}{z+\mathrm{b}}\right|=1 and \left|

\begin{array}{ccc}z+1 & \omega & \omega^2 \\ \omega & z+\omega^2 & 1 \\ \omega^2 & 1 & z+\omega\end{array}

\right|=1, z \in

\mathrm{C}

, where

\omega

and

\omega^2

are the roots of

x^2+x+1=0$, is equal to ________.

Correct answer

Step-by-step solution

\begin{aligned} & a, b \in I,-3 \leq a, b \leq 3, a+b \neq 0 \\ & |z-a|=|z+b| \\ & \left|\begin{array}{ccc}z+1 & \omega & \omega^2 \\ \omega & z+\omega^2 & 1 \\ \omega^2 & 1 & z+\omega\end{array}\right|=1 \\ & \Rightarrow\left|\begin{array}{ccc}z & z & z \\ \omega & z+\omega^2 & 1 \\ \omega^2 & 1 & z+\omega\end{array}\right|=1\end{aligned}

\begin{aligned} & \Rightarrow z\left|\begin{array}{ccc}1 & 1 & 1 \\ \omega & z+\omega^2 & 1 \\ \omega^2 & 1 & z+\omega\end{array}\right|=1 \\ & \Rightarrow z\left|\begin{array}{ccc}1 & 0 & 0 \\ \omega & z+\omega^2-\omega & 1-\omega \\ \omega^2 & 1-\omega^2 & z+\omega-\omega^2\end{array}\right|=1 \\ & \Rightarrow z^3=1 \\ & \Rightarrow z=\omega, \omega^2, 1\end{aligned}

Now

\begin{aligned} & |1-\mathrm{a}|=|^{1}+\mathrm{b}| \\ & \Rightarrow 10 \text{ pairs } \end{aligned}

Practice Complex Number on Quantrex Academy →

More from Complex Number