JEE Advanced 2025 — Set 2

Let 𝑥0 be the real number such that 𝑒𝑥0 + 𝑥0 = 0 . For a given real number 𝛼, define 𝑔(𝑥) = 3𝑥𝑒𝑥+ 3𝑥−𝛼𝑒𝑥−𝛼𝑥 3(𝑒𝑥+ 1) for all real numbers 𝑥. Then which one of the following statements is TRUE?

Let ℝ denote the set of all real numbers. Then the area of the region {(𝑥, 𝑦) ∈ℝ× ℝ∶𝑥> 0, 𝑦> 1 𝑥 , 5𝑥−4𝑦−1 > 0, 4𝑥+ 4𝑦−17 < 0 } is

The total number of real solutions of the equation 𝜃= tan−1(2 tan 𝜃) −1 2 sin−1 ( 6 tan 𝜃 9 + tan2 𝜃) is (Here, the inverse trigonometric functions sin−1 𝑥 and tan−1 𝑥 assume values in [− 𝜋 2 , 𝜋 2] and (− 𝜋 2 , 𝜋 2), respectively.)

Let 𝑆 denote the locus of the point of intersection of the pair of lines 4𝑥−3𝑦= 12𝛼 , 4𝛼𝑥+ 3𝛼𝑦= 12 , where 𝛼 varies over the set of non-zero real numbers. Let 𝑇 be the tangent to 𝑆 passing through the points (𝑝, 0) and (0, 𝑞), 𝑞> 0, and parallel to the line 4 𝑥− 3 √2 𝑦= 0 . Then the value of 𝑝𝑞 is

Let 𝐼= (1 0 0 1) and 𝑃= (2 0 0 3). Let 𝑄= (𝑥 𝑦 𝑧 4) for some non-zero real numbers 𝑥, 𝑦, and 𝑧, for which there is a 2 × 2 matrix 𝑅 with all entries being non-zero real numbers, such that 𝑄𝑅= 𝑅𝑃 . Then which of the following statements is (are) TRUE?

Let 𝑆 denote the locus of the mid-points of those chords of the parabola 𝑦2 = 𝑥 , such that the area of the region enclosed between the parabola and the chord is 4 3 . Let ℛ denote the region lying in the first quadrant, enclosed by the parabola 𝑦2 = 𝑥 , the curve 𝑆, and the lines 𝑥= 1 and 𝑥= 4 . Then which of the following statements is (are) TRUE?

Let 𝑃(𝑥1, 𝑦1) and 𝑄(𝑥2, 𝑦2) be two distinct points on the ellipse 𝑥2 9 + 𝑦2 4 = 1 such that 𝑦1 > 0, and 𝑦2 > 0 . Let 𝐶 denote the circle 𝑥2 + 𝑦2 = 9 , and 𝑀 be the point (3, 0). Suppose the line 𝑥= 𝑥1 intersects 𝐶 at 𝑅, and the line 𝑥= 𝑥2 intersects C at 𝑆, such that the 𝑦-coordinates of 𝑅 and 𝑆 are positive. Let ∠𝑅𝑂𝑀= 𝜋 6 and ∠𝑆𝑂𝑀= 𝜋 3 , where 𝑂 denotes the origin (0, 0). Let |𝑋𝑌| denote the length of the line segment 𝑋𝑌 . Then which of the following statements is (are) TRUE?

Let ℝ denote the set of all real numbers. Let 𝑓: ℝ→ℝ be defined by 𝑓(𝑥) = { 6𝑥+sin𝑥 2𝑥+sin𝑥 if 𝑥≠0, 7 3 if 𝑥= 0 . Then which of the following statements is (are) TRUE?

Let 𝑦(𝑥) be the solution of the differential equation 𝑥2 𝑑𝑦 𝑑𝑥+ 𝑥𝑦= 𝑥2 + 𝑦2, 𝑥> 1 𝑒 , satisfying 𝑦(1) = 0 . Then the value of 2 (𝑦(𝑒)) 2 𝑦(𝑒2) is ___________.

Let 𝑎0, 𝑎1, … , 𝑎23 be real numbers such that (1 + 2 5 𝑥) 23 = ∑𝑎𝑖𝑥𝑖 23 𝑖=0 for every real number 𝑥. Let 𝑎𝑟 be the largest among the numbers 𝑎𝑗 for 0 ≤𝑗≤23. Then the value of 𝑟 is ___________.