JEE Advanced 2025 — Set 1

Let ℝ denote the set of all real numbers. Let 𝑎𝑖, 𝑏𝑖∈ℝ for 𝑖∈{1, 2, 3}. Define the functions 𝑓: ℝ→ℝ, 𝑔: ℝ→ℝ, and ℎ: ℝ→ℝ by 𝑓(𝑥) = 𝑎1 + 10𝑥+ 𝑎2𝑥2 + 𝑎3𝑥3 + 𝑥4, 𝑔(𝑥) = 𝑏1 + 3𝑥+ 𝑏2𝑥2 + 𝑏3𝑥3 + 𝑥4, ℎ(𝑥) = 𝑓(𝑥+ 1) −𝑔(𝑥+ 2). If 𝑓(𝑥) ≠𝑔(𝑥) for every 𝑥∈ ℝ, then the coefficient of 𝑥3 in ℎ(𝑥) is

Three students 𝑆1, 𝑆2, and 𝑆3 are given a problem to solve. Consider the following events: 𝑈: At least one of 𝑆1, 𝑆2, and 𝑆3 can solve the problem, 𝑉: 𝑆1 can solve the problem, given that neither 𝑆2 nor 𝑆3 can solve the problem, 𝑊: 𝑆2 can solve the problem and 𝑆3 cannot solve the problem, 𝑇: 𝑆3 can solve the problem. For any event 𝐸, let 𝑃(𝐸) denote the probability of 𝐸. If 𝑃(𝑈) = 1 2 , 𝑃(𝑉) = 1 10 , and 𝑃(𝑊) = 1 12 , then 𝑃(𝑇) is equal to

Let ℝ denote the set of all real numbers. Define the function 𝑓: ℝ→ℝ by 𝑓(𝑥) = {2 −2𝑥2 −𝑥2 sin1 𝑥 if 𝑥≠0, 2 if 𝑥= 0 . Then which one of the following statements is TRUE?