JEE Advanced 2023 — Set 1

Let (0,1) (1,2) (3,4) S = ∪ ∪ and {0,1,2,3} T = . Then which of the following statements is(are) true?

Let 1T and 2T be two distinct common tangents to the ellipse 2 2 : 1 6 3 x y E + = and the parabola 2 : 12 P y x = . Suppose that the tangent 1T touches P and E at the points 1A and 2 A , respectively and the tangent 2T touches P and E at the points 4 A and 3A , respectively. Then which of the following statements is(are) true?

Let :[0,1] [0,1] f → be the function defined by 3 2 5 17 ( ) 3 9 36 x f x x x = − + + . Consider the square region [0,1] [0,1] S = × . Let {( , ) : ( )} G x y S y f x = ∈ > be called the green region and {( , ) : ( )} R x y S y f x = ∈ < be called the red region. Let {( , ) : [0,1]} h L x h S x = ∈ ∈ be the horizontal line drawn at a height [0,1] h∈ . Then which of the following statements is(are) true?

Let :(0,1) f → be the function defined as ( ) f x n = if 1 1 , 1 x n n ⎡ ⎞ ∈ ⎟ ⎢ + ⎣ ⎠ where n∈∞. Let :(0,1) g → be a function such that 2 1 ( ) 2 x x t dt g x x t − < < ∫ for all (0,1) x∈ . Then 0 lim ( ) ( ) x f x g x →

Let Q be the cube with the set of vertices   3 1 2 3 1 2 3 ( , , ) : , , { 0,1} x x x x x x ∈ ∈  . Let F be the set of all twelve lines containing the diagonals of the six faces of the cube Q. Let S be the set of all four lines containing the main diagonals of the cube Q; for instance, the line passing through the vertices (0,0,0) and (1,1,1) is in S . For lines 1λ and 2λ, let 1 2 ( , ) d λλ denote the shortest distance between them. Then the maximum value of 1 2 ( , ) d λλ , as 1λ varies over F and 2λ varies over S , is

Let 2 2 2 ( , ) : 1 and 5 8 20 x y X x y y x ⎧ ⎫ = ∈ × + < < ⎨ ⎬ ⎩ ⎭  . Three distinct points P, Q and R are randomly chosen from X . Then the probability that P, Q and R form a triangle whose area is a positive integer, is

Let P be a point on the parabola 2 4 y ax = , where 0 a > . The normal to the parabola at P meets the x -axis at a point Q. The area of the triangle PFQ , where F is the focus of the parabola, is 120. If the slope m of the normal and a are both positive integers, then the pair ( , ) a m is

Let 1 tan ( ) , , 2 2 x ππ − ⎛ ⎞ ∈− ⎜ ⎟ ⎝ ⎠ for x ∈. Then the number of real solutions of the equation 1 1 cos(2 ) 2 tan (tan ) x x − + = in the set 3 3 , , , 2 2 2 2 2 2 π π ππ π π ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ − − ∪− ∪ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ is equal to

Let 2 n ≥ be a natural number and :[0,1] f → be the function defined by ( ) 1 (1 2 ) if 0 2 1 3 2 (2 1) if 2 4 ( ) 3 1 4 (1 ) if 4 1 1 if 1 1 n nx x n n nx x n n f x n nx x n n n nx x n n ⎧ − ≤ ≤    − ≤ ≤  =⎨  − ≤ ≤    − ≤ ≤ − ⎩ If n is such that the area of the region bounded by the curves 0 x = , 1 x = , 0 y = and ( ) y f x = is 4 , then the maximum value of the function f is

Let  75 57 r Λ denote the ( 2) r + digit number where the first and the last digits are 7 and the remaining r digits are 5. Consider the sum  98 77 757 7557 75 57. S = + + + + Λ Λ If  99 75 57 , m S n + = Λ where m and n are natural numbers less than 3000, then the value of m n + is SECTION 3 (Maximum Marks: 24) • This section contains SIX (06) questions. • The answer to each question is a NON-NEGATIVE INTEGER. • For each question, enter the correct integer corresponding to the answer using the mouse and the on- screen virtual numeric keypad in the place designated to enter the answer. • Answer to each question will be evaluated according to the following marking scheme: Full Marks : ൅4 If ONLY the correct integer is entered; Zero Marks : 0 In all other cases.