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

Let the position vectors of the points , , P Q R and S be ˆ ˆ ˆ 2 5 a i j k = + − ρ , ˆ ˆ ˆ 3 6 3 b i j k = + + ρ , 17 16 ˆ ˆ ˆ 7 5 5 c i j k = + + ρ and ˆ ˆ ˆ 2 d i j k = + + ρ , respectively. Then which of the following statements is true?

Let ( ), , {1,2,3}, ij M a i j = ∈ be the 3 3 × matrix such that 1 ij a = if 1 j + is divisible by i , otherwise 0 ij a = . Then which of the following statements is(are) true?

Let :(0,1) f → be the function defined as 2 1 1 ( ) [4 ] , 4 2 f x x x x ⎛ ⎞⎛ ⎞ = − − ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ where [ ]x denotes the greatest integer less than or equal to x . Then which of the following statements is(are) true?

Let S be the set of all twice differentiable functions f from  to  such that 2 2 ( ) 0 d f x dx > for all ( 1,1). x∈− For f S ∈ , let f X be the number of points ( 1,1) x∈− for which ( ) . f x x = Then which of the following statements is(are) true?

For x ∈, let 1 tan ( ) , 2 2 ( ) x ππ − ∈− . Then the minimum value of the function : f →   defined by 1 ( cos ) 2023 0 ( ) 1 x tan x t t e f x dt t − − = + ∫ is

For , x∈ let ( ) y x be a solution of the differential equation 2 2 2 ( 5) 2 2 ( 5) dy x x y x x dx − − =− − such that (2) 7. y = Then the maximum value of the function ( ) y x is

Let X be the set of all five digit numbers formed using 1,2,2,2,4,4,0. For example, 22240 is in X while 02244 and 44422 are not in . X Suppose that each element of X has an equal chance of being chosen. Let p be the conditional probability that an element chosen at random is a multiple of 20 given that it is a multiple of 5. Then the value of 38p is equal to