JEE Advanced 2025 · Paper 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

1. A8
2. B2
3. C−4
4. D−6

Let ℝ denote the set of all real numbers. Define the function f: ℝ → ℝ by f(x) = { 2 − 2x² − x² sin(1/x) if x ≠ 0, 2 if x = 0. Then which one of the following statements is TRUE?

1. AThe function f is NOT differentiable at x = 0
2. BThere is a positive real number δ, such that f is a decreasing function on the interval (0, δ)
3. CFor any positive real number δ, the function f is NOT an increasing function on the interval (−δ, 0)
4. Dx = 0 is a point of local minima of f

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

1. A32
2. B8
3. C16
4. D24

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?

1. AThe length of the line segment PQ is 9√3
2. BThe length of the line segment QR is 15
3. CThe area of ΔPQR is 3/2 √234
4. DThe acute angle between the line segments PQ and PR is cos⁻¹(1/(2√3))

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?

1. Ag ∘ f is NOT one-one and g ∘ f is NOT onto
2. Bf ∘ g is NOT one-one but f ∘ g is onto
3. Cg is one-one and g is onto
4. Df is NOT one-one but f is onto