JEE Advanced 2025 — Set 2

Let 𝑥0 be the real number such that 𝑒𝑥0 + 𝑥0 = 0 . For a given real number 𝛼, define 𝑔(𝑥) = 3𝑥𝑒𝑥+ 3𝑥−𝛼𝑒𝑥−𝛼𝑥 3(𝑒𝑥+ 1) for all real numbers 𝑥. Then which one of the following statements is TRUE?

Let ℝ denote the set of all real numbers. Then the area of the region {(𝑥, 𝑦) ∈ℝ× ℝ∶𝑥> 0, 𝑦> 1 𝑥 , 5𝑥−4𝑦−1 > 0, 4𝑥+ 4𝑦−17 < 0 } is

The total number of real solutions of the equation 𝜃= tan−1(2 tan 𝜃) −1 2 sin−1 ( 6 tan 𝜃 9 + tan2 𝜃) is (Here, the inverse trigonometric functions sin−1 𝑥 and tan−1 𝑥 assume values in [− 𝜋 2 , 𝜋 2] and (− 𝜋 2 , 𝜋 2), respectively.)