JEE Advanced 2023 — Set 2

Let :[1, ) f ∞→ be a differentiable function such that 1 (1) 3 f = and 3 1 3 ( ) ( ) , [1, ) 3 x x f t dt x f x x = − ∈ ∞ ∫ . Let e denote the base of the natural logarithm. Then the value of ( ) f e is

Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are same. If the probability of a random toss resulting in head is 1 3 , then the probability that the experiment stops with head is

For any y∈, let 1 cot ( ) (0, ) y π − ∈ and 1 tan ( ) , . 2 2 ( ) y ππ − ∈− Then the sum of all the solutions of the equation 2 1 1 2 6 9 2 tan cot 9 6 3 ( ) ( ) y y y y π − − − + = − for 0 | | 3, y < < is equal to