JEE 2023 — Mathematics PYQ

Humein equation di gayi hai:

Step 1: Formula ka upyog

Humein pata hai ki $2{\tan }^{-1}x$ ko ${\sin }^{-1}$ ki terms mein aise likha ja sakta hai:

Kyunki interval $x\in (-1,1]$ diya gaya hai, hum ye formula apply kar sakte hain.

Step 2: Equation ko compare karna

Ab equation banti hai:

Dono sides se ${\sin }^{-1}$ hatane par:

Step 3: $x$ ki values nikalna

Equation ko simplify karte hain:

Isse humein teen possible values milti hain:

1. $x=0$

2. $x^2-1=0\Rightarrow x=1$ ya $x=-1$

Step 4: Interval Check

Humein condition di gayi hai $x\in (-1,1]$. Iska matlab hai $x$, $-1$ se bada hona chahiye aur $1$ ke barabar ya usse chota ho sakta hai.

• $x=0$: Is interval mein hai.

• $x=1$: Is interval mein hai (bracket $1]$ hai).

• $x=-1$: Is interval mein nahi hai (kyunki open bracket $(-1$ hai).

Final Answer:

Solutions ki kul sankhya (number of solutions) 2 hai ($x=0$ aur $x=1$).