JEE Advanced 2026 · Paper 2

Let ℝ denote the set of all real numbers. Consider the polynomial function f : ℝ → ℝ defined by f(x) = d¹⁰/dx¹⁰ ((x² − 1)¹⁰), for all x ∈ ℝ. Here d¹⁰/dx¹⁰ ((x² − 1)¹⁰) is the 10th order derivative of the function (x² − 1)¹⁰. Then which of the following statements is (are) TRUE ?

1. AThe coefficient of x⁸ in the polynomial f(x) is (−10)(18!/8!)
2. BThe value of f(1) + f(−1) is equal to 10! 2¹¹
3. CThe degree of the polynomial f(x) is 10
4. DThe constant term of the polynomial f(x) is −(10!/5!)

Let a, b, c be positive integers in arithmetic progression such that the equation ax² + bx + c = 0 has only integer solutions. Then which of the following statements is (are) TRUE ?

1. Ac − b is an integer multiple of a
2. BBoth the roots of the equation ax² + bx + c = 0 are odd integers
3. CIf c = 15, then ab = 8
4. DIf b = 8, then x = 3 is a root of the equation ax² + bx + c = 0

Let L be the straight line joining the points P(1, 2, −1) and Q(2, 3, 1). Let S be the foot of the perpendicular drawn from the point R(4, −1, 5) to the line L. Another line passing through R intersects L at a point T such that the point S divides the line segment PT internally in the ratio |PS| : |ST| = 1 : 2, where |PS| and |ST| are the lengths of the line segments PS and ST, respectively. Then which of the following statements is (are) TRUE ?

1. AThe orthocentre of the triangle PRT is (23/5, −4, 31/5)
2. BThe orthocentre of the triangle PRT is (4, 3, 5)
3. CThe area of the triangle PRT is 6√5
4. DThe area of the triangle PRT is 18√5

Let y = f(x) be the real valued function defined on the interval (0, ∞), satisfying y(1) = 0 and the differential equation x dy/dx = y − x³. Then which of the following statements is (are) TRUE ?

1. AThe function f has a local minimum at x = 1/√3
2. BThe function f has a local maximum at x = 1/√3
3. CThe function f is increasing in the interval (1, 2)
4. DIf g(x) = 4x³ − 5x² + (3/2)x for x > 0, then the number of elements in the set {x ∈ (0, ∞) : f(x) = g(x)} is 2

Let ℝ denote the set of all real numbers and let i = √−1. Consider the matrices S = [0 −1; 1 0] and T = [1 1; 0 1]. Let a, b, c, d be real numbers such that ST = [a b; c d]. Let H = {x + iy : x, y ∈ ℝ and y > 0}. Then which of the following statements is (are) TRUE ?

1. A(b + ia)/(d + ic) = i
2. BIf ω = (−1 + i√3)/2, then (aω + b)/(cω + d) = ω
3. CIf m is an integer greater than 2 such that (ST)² = (ST)ᵐ, then m is an integer multiple of 8
4. DIf z ∈ H, then (az + b)/(cz + d) ∈ H