JEE 2023 — Mathematics PYQ

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Question

JEE 2023 — Mathematics PYQ

JEE | Mathematics | 2023

Let $S = {z = x + i y : \frac{2 z - 3 i}{4 z + 2 i} is a real number}$ .

Then which of the following is NOT correct?

Choose the correct answer:

Explanation

Step 1: Expression ko simplify karein

Maana $z = x + iy$ . Diya gaya complex number hai:

\frac{2(x + iy) - 3i}{4(x + iy) + 2i} = \frac{2x + i(2y - 3)}{4x + i(4y + 2)}

Step 2: Real hone ki shart (Condition for Real Number)

Koi bhi complex number $\frac{a+ib}{c+id}$ tabhi real hota hai jab uska imaginary part zero ho. Iska shortcut formula hai: $bc - ad = 0$ .

Yahan $a = 2x$ , $b = 2y - 3$ , $c = 4x$ , aur $d = 4y + 2$ hai.

(2y - 3)(4x) - (2x)(4y + 2) = 0

8xy - 12x - (8xy + 4x) = 0

8xy - 12x - 8xy - 4x = 0

-16x = 0 \implies \mathbf{x = 0}

Step 3: Denominator zero nahi hona chahiye

Condition: $4z + 2i \neq 0$

4(x + iy) + 2i \neq 0 \implies 4x + i(4y + 2) \neq 0

Chunki $x = 0$ hai, isliye:

4y + 2 \neq 0 \implies y \neq -\frac{1}{2}

Choose the correct answer:

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