JEE Advanced 2024 — Set 2

Considering only the principal values of the inverse trigonometric functions, the value of 1 1 3 2 tan sin 2cos 5 5 − − ⎛ ⎞ ⎛ ⎞ ⎛ ⎞− ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ is

Let   2 2 ( , ) : 0, 0, 4 , 12 2 and 3 8 5 8 . S x y x y y x y x y x = ∈ × ≥ ≥ ≤ ≤ − + ≤ If the area of the region S is 2, α then αis equal to

Let . k ∈ If ( ) 2 6 0 lim , sin(sin ) cos x x e kx x x →+ = + + then the value of k is

Let : f → be a function defined by 2 2 sin , if 0, ( ) 0, if 0. x x f x x x π ⎧ ⎛ ⎞ ≠  ⎜ ⎟ = ⎝ ⎠ ⎨  = ⎩ Then which of the following statements is TRUE?

Let S be the set of all ( , ) αβ∈ × such that 2 2 1 sin( )(log ) sin lim 0. (log (1 )) e x e x x x x x α αβ β →∞ ⎛ ⎞ ⎜ ⎟ ⎝ ⎠= + Then which of the following is (are) correct?

A straight line drawn from the point (1,3,2), P parallel to the line 2 4 6 , 1 2 1 x y z − − − = = intersects the plane 1 : 3 6 L x y z − + = at the point . Q Another straight line which passes through Q and is perpendicular to the plane 1L intersects the plane 2 : 2 4 L x y z − + = − at the point . R Then which of the following statements is (are) TRUE?

Let 1 1 1 , , A B C be three points in the xy -plane. Suppose that the lines 1 1 AC and 1 1 B C are tangents to the curve 2 8 y x = at 1A and 1, B respectively. If (0,0) O = and 1 ( 4,0), C = − then which of the following statements is (are) TRUE?

Let : f → be a function such that ( ) ( ) ( ) for all , , f x y f x f y x y + = + ∈ and : (0, ) g → ∞ be a function such that ( ) ( ) ( ) for all , . g x y g x g y x y + = ∈ If 3 1 12 and 2, 5 3 f g − − ⎛ ⎞ ⎛ ⎞ = = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ then the value of ( ) 1 + 2 8 (0) 4 f g g ⎛ ⎞ ⎛ ⎞ − − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ is ________.

A bag contains N balls out of which 3balls are white, 6 balls are green, and the remaining balls are blue. Assume that the balls are identical otherwise. Three balls are drawn randomly one after the other without replacement. For 1,2,3, i = let , , and i i i W G B denote the events that the ball drawn in the thi draw is a white ball, green ball, and blue ball, respectively. If the probability 1 2 3 2 ( ) 5 P W G B N = and the conditional probability 3 1 2 2 ( ) , 9 P B W G = then N equals ______.

Let the function : f → be defined by ( ) ( ) ( ) ( ) 2023 2023 2 2 2024 2025 2024 2025 sin 2 ( ) 3 3 x x x x x x x f x e e x x x x π π + + + + = + − + − + . Then the number of solutions of ( ) 0 f x = in is ______.