202560 marks16 questions
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Source: JEE (Advanced) 2025, Paper 2, Maths, published by JEE (Advanced). The official PDF
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 4x − 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 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 ___________.
A factory has a total of three manufacturing units, M₁, M₂, and M₃, which produce bulbs independent of each other. The units M₁, M₂, and M₃ produce bulbs in the proportions of 2: 2: 1, respectively. It is known that 20% of the bulbs produced in the factory are defective. It is also known that, of all the bulbs produced by M₁, 15% are defective. Suppose that, if a randomly chosen bulb produced in the factory is found to be defective, the probability that it was produced by M₂ is 2/5. If a bulb is chosen randomly from the bulbs produced by M₃, then the probability that it is defective is ___________.
Consider the vectors x⃗ = î + 2ĵ + 3k̂, y⃗ = 2î + 3ĵ + k̂, and z⃗ = 3î + ĵ + 2k̂. For two distinct positive real numbers α and β, define X⃗ = αx⃗ + βy⃗ − z⃗, Y⃗ = αy⃗ + βz⃗ − x⃗, and Z⃗ = αz⃗ + βx⃗ − y⃗. If the vectors X⃗, Y⃗, and Z⃗ lie in a plane, then the value of α + β − 3 is_____________.
Let ℝ denote the set of all real numbers. Let f: ℝ → ℝ and g: ℝ → (0, 4) be functions defined by f(x) = logₑ(x² + 2x + 4), and g(x) = 4/(1 + e⁻²ˣ). Define the composite function f ∘ g⁻¹ by (f ∘ g⁻¹)(x) = f(g⁻¹(x)), where g⁻¹ is the inverse of the function g. Then the value of the derivative of the composite function f ∘ g⁻¹ at x = 2 is ______________.