JAMIA 2023 — Mathematics PYQ

Unimodular complex number ka matlab hai $|z|=1$ (Unit Circle: $x^2+y^2=1$).

Semicircle aur Unit Circle ke intersection points nikalne ke liye:

1. Semicircle ki equation (diameter endpoints $(1,0)$ aur $(0,-1)$ ke saath):

   $$(x-1)x+(y-0)(y+1)=0$$
   $$x^2-x+y^2+y=0$$

2. Ismein $|z|=1$ yaani $x^2+y^2=1$ ki value rakho:

   $$1-x+y=0⟹y=x-1$$

3. Isse $x^2+y^2=1$ mein wapas rakho:

   $$x^2+(x-1{)}^2=1⟹2x^2-2x=0$$
   $$x=0\text{ ya }x=1$$

• Agar $x=1$ hai, toh $y=0⟹z=1$

• Agar $x=0$ hai, toh $y=-1⟹z=-i$

Step 2: Final Check

Argument function $\arg \left(\frac{z-z_1}{z-z_2}\right)=\theta$ mein endpoints ($z_1$ aur $z_2$) hamesha exclude kiye jaate hain kyunki wahan argument undefined hota hai.

• $z=1$ par numerator $0$ ho jayega.

• $z=-i$ par denominator $0$ ho jayega.

Dono points exclude hone ki wajah se, aisa koi bhi point nahi bacha jo condition satisfy kare.

Final Answer:

Number of such complex numbers = 0