JEE Advanced 2025 — Set 1

Let ℝ denote the set of all real numbers. Let 𝑎𝑖, 𝑏𝑖∈ℝ for 𝑖∈{1, 2, 3}. Define the functions 𝑓: ℝ→ℝ, 𝑔: ℝ→ℝ, and ℎ: ℝ→ℝ by 𝑓(𝑥) = 𝑎1 + 10𝑥+ 𝑎2𝑥2 + 𝑎3𝑥3 + 𝑥4, 𝑔(𝑥) = 𝑏1 + 3𝑥+ 𝑏2𝑥2 + 𝑏3𝑥3 + 𝑥4, ℎ(𝑥) = 𝑓(𝑥+ 1) −𝑔(𝑥+ 2). If 𝑓(𝑥) ≠𝑔(𝑥) for every 𝑥∈ ℝ, then the coefficient of 𝑥3 in ℎ(𝑥) is

Three students 𝑆1, 𝑆2, and 𝑆3 are given a problem to solve. Consider the following events: 𝑈: At least one of 𝑆1, 𝑆2, and 𝑆3 can solve the problem, 𝑉: 𝑆1 can solve the problem, given that neither 𝑆2 nor 𝑆3 can solve the problem, 𝑊: 𝑆2 can solve the problem and 𝑆3 cannot solve the problem, 𝑇: 𝑆3 can solve the problem. For any event 𝐸, let 𝑃(𝐸) denote the probability of 𝐸. If 𝑃(𝑈) = 1 2 , 𝑃(𝑉) = 1 10 , and 𝑃(𝑊) = 1 12 , then 𝑃(𝑇) is equal to

Let ℝ denote the set of all real numbers. Define the function 𝑓: ℝ→ℝ by 𝑓(𝑥) = {2 −2𝑥2 −𝑥2 sin1 𝑥 if 𝑥≠0, 2 if 𝑥= 0 . Then which one of the following statements is TRUE?

Consider the matrix 𝑃= ( 2 0 0 0 2 0 0 0 3 ) . Let the transpose of a matrix 𝑋 be denoted by 𝑋𝑇. Then the number of 3 × 3 invertible matrices 𝑄 with integer entries, such that 𝑄−1 = 𝑄𝑇 and 𝑃𝑄= 𝑄𝑃 , is

Let 𝐿1 be the line of intersection of the planes given by the equations 2𝑥+ 3𝑦+ 𝑧= 4 and 𝑥+ 2𝑦+ 𝑧= 5 . Let 𝐿2 be the line passing through the point 𝑃(2, −1, 3) and parallel to 𝐿1. Let 𝑀 denote the plane given by the equation 2𝑥+ 𝑦−2𝑧= 6 . Suppose that the line 𝐿2 meets the plane 𝑀 at the point 𝑄. Let 𝑅 be the foot of the perpendicular drawn from 𝑃 to the plane 𝑀 . Then which of the following statements is (are) TRUE?

Let ℕ denote the set of all natural numbers, and ℤ denote the set of all integers. Consider the functions 𝑓: ℕ→ℤ and 𝑔: ℤ→ℕ defined by 𝑓(𝑛) = { (𝑛+ 1)/2 if 𝑛 is odd , (4 −𝑛)/2 if 𝑛 is even , and 𝑔(𝑛) = { 3 + 2𝑛 if 𝑛 ≥ 0 , −2𝑛 if 𝑛 < 0 . Define (𝑔∘𝑓)(𝑛) = 𝑔(𝑓(𝑛)) for all 𝑛∈ℕ, and (𝑓∘𝑔)(𝑛) = 𝑓(𝑔(𝑛)) for all 𝑛∈ℤ. Then which of the following statements is (are) TRUE?

Let ℝ denote the set of all real numbers. Let 𝑧1 = 1 + 2𝑖 and 𝑧2 = 3𝑖 be two complex numbers, where 𝑖= √−1 . Let 𝑆= {(𝑥, 𝑦) ∈ℝ× ℝ∶ |𝑥+ 𝑖𝑦−𝑧1| = 2|𝑥+ 𝑖𝑦−𝑧2| }. Then which of the following statements is (are) TRUE?

Let the set of all relations 𝑅 on the set {𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓}, such that 𝑅 is reflexive and symmetric, and 𝑅 contains exactly 10 elements, be denoted by 𝒮. Then the number of elements in 𝒮 is _________________.

For any two points 𝑀 and 𝑁 in the 𝑋𝑌-plane, let 𝑀𝑁 ⃗⃗⃗⃗⃗⃗⃗ denote the vector from 𝑀 to 𝑁, and 0⃗ denote the zero vector. Let 𝑃, 𝑄 and 𝑅 be three distinct points in the 𝑋𝑌-plane. Let 𝑆 be a point inside the triangle Δ𝑃𝑄𝑅 such that 𝑆𝑃 ⃗⃗⃗⃗ + 5 𝑆𝑄 ⃗⃗⃗⃗⃗ + 6 𝑆𝑅 ⃗⃗⃗⃗⃗ = 0⃗ . Let 𝐸 and 𝐹 be the mid-points of the sides 𝑃𝑅 and 𝑄𝑅, respectively. Then the value of length of the line segment 𝐸𝐹 length of the line segment 𝐸𝑆 is _______________.

Let 𝑆 be the set of all seven-digit numbers that can be formed using the digits 0, 1 and 2. For example, 2210222 is in 𝑆, but 0210222 is NOT in 𝑆. Then the number of elements 𝑥 in 𝑆 such that at least one of the digits 0 and 1 appears exactly twice in 𝑥, is equal to ____________. SECTION 3 (Maximum Marks: 24) • This section contains SIX (06) questions. • The answer to each question is a NUMERICAL VALUE. • For each question, enter the correct numerical value of the answer using the mouse and the on- screen virtual numeric keypad in the place designated to enter the answer. • If the numerical value has more than two decimal places, truncate/round-off the value to TWO decimal places. • Answer to each question will be evaluated according to the following marking scheme: Full Marks : +4 If ONLY the correct numerical value is entered in the designated place; Zero Marks : 0 In all other cases. Answer: 105 Answer: [1.15 to 1.25] Answer: 762