JEE Advanced 2026 · Paper 1

Consider the function f: (0, ∞) → (−∞, ∞) given by f(x) = √x logₑ(x) − x + 1. Then which one of the following statements is TRUE?

1. AThe derivative of the function f is decreasing in the interval (0, 1)
2. BThe function f has a local maximum at some point a ∈ (0, ∞)
3. CThe function f has a local minimum at some point b ∈ (0, ∞)
4. DThe function f has NEITHER a point of local maximum NOR a point of local minimum in the interval (0, ∞)

Let P be the point on the parabola y = x² such that the slope of the tangent to the parabola at the point P is 4. Let Q be the point in the first quadrant lying on the circle x² + y² = 2 such that the slope of the tangent to the circle at the point Q is −1. Let R be the point in the first quadrant lying on the ellipse x² + 4y² = 8 such that the slope of the tangent to the ellipse at the point R is −1/2. Then the radius of the circle passing through the points P, Q and R is

1. A√10
2. B√5
3. C√(5/2)
4. D2√5

Which one of the following matrices can be obtained by performing elementary row transformations on the 3 × 3 identity matrix?

1. A[1 1 1; 1 1 1; 1 1 1]
2. B[1 1 1; 2 3 4; 1 2 1]
3. C[1 1 1; 2 3 4; 2 5 8]
4. D[1 1 1; −1 1 2; 0 2 3]

Considering only the principal values of the inverse trigonometric functions, the value of cot⁻¹(cot(−11)) + 10 sin (2 cos⁻¹ (1/√2)) + 10 sin(2 tan⁻¹(2)) is

1. A3π + 7
2. B7
3. C4π + 7
4. D3π − 5

Suppose that Box I contains 6 red balls and 9 green balls, and Box II contains 8 red balls and 12 green balls. All the balls of Box I and Box II are mixed together and a ball is chosen at random from them. Let E₁ be the event that the ball chosen belonged to Box I and let E₂ be the event that the ball chosen belonged to Box II. Let F₁ be the event that the ball chosen is red and let F₂ be the event that the ball chosen is green. Then which of the following statements is (are) TRUE?

1. AThe events E₁ and F₁ are independent
2. BThe events E₂ and F₂ are dependent
3. CThe conditional probability P(F₁ | E₁) is equal to the conditional probability P(F₁ | E₂)
4. DThe conditional probability P(F₁ | E₁) is greater than the conditional probability P(F₂ | E₂)

Let P be the plane such that it contains the straight line (x−1)/2 = (y−3)/3 = (z+2)/1 and is perpendicular to the plane x + 2y + 3z = 4. Let P₁ be the plane which passes through the point (4, 2, 2) and is parallel to P. Then which of the following statements is (are) TRUE?

1. AThe equation of the plane P is 7x − 5y + z = −10
2. BThe distance between the planes P and P₁ is 30
3. CThe distance of the plane P from the origin is 2√3
4. DThe acute angle between the plane P and the plane 2x + 2y + z = 3 is cos⁻¹(1/(3√3))

Let ℝ denote the set of all real numbers. Let f: ℝ → ℝ be an arbitrary function and let g: ℝ → ℝ be the function defined by g(x) = xf(x), for all x ∈ ℝ. Then which of the following statements is (are) TRUE?

1. AThe function g is always continuous at x = 0
2. BIf f is continuous at x = 0, then g is differentiable at x = 0
3. CIf g is differentiable at x = 0, then f is continuous at x = 0
4. DIf g is differentiable at x = 0, then limₓ→₀ f(x) exists