JEE Advanced 2021 · Paper 2

Let S₁ = {(i, j, k) : i, j, k ∈ {1,2, … ,10}}, S₂ = {(i, j) : 1 ≤ i < j + 2 ≤ 10, i, j ∈ {1,2, … ,10}}, S₃ = {(i, j, k, l) : 1 ≤ i < j < k < l, i, j, k, l ∈ {1,2, … ,10}} and S₄ = {(i, j, k, l) : i, j, k and l are distinct elements in {1,2, … ,10}}. If the total number of elements in the set Sᵣ is nᵣ, r = 1,2,3,4, then which of the following statements is (are) TRUE ?

1. An₁ = 1000
2. Bn₂ = 44
3. Cn₃ = 220
4. Dn₄/12 = 420

Consider a triangle PQR having sides of lengths p, q and r opposite to the angles P, Q and R, respectively. Then which of the following statements is (are) TRUE ?

1. Acos P ≥ 1 − p²/(2qr)
2. Bcos R ≥ ((q − r)/(p + q)) cos P + ((p − r)/(p + q)) cos Q
3. C(q + r)/p < 2√(sin Q sin R)/sin P
4. DIf p < q and p < r, then cos Q > p/r and cos R > p/q

For any real numbers α and β, let y₍α,β₎(x), x ∈ ℝ, be the solution of the differential equation dy/dx + αy = xe^(βx), y(1) = 1. Let S = {y₍α,β₎(x) : α, β ∈ ℝ}. Then which of the following functions belong(s) to the set S ?

1. Af(x) = (x²/2)e⁻ˣ + (e⁻¹/2)e⁻ˣ
2. Bf(x) = −(x²/2)e⁻ˣ + (e + 1/2)e⁻ˣ
3. Cf(x) = (eˣ/2)(x − 1/2) + (e − e²/4)e⁻ˣ
4. Df(x) = (eˣ/2)(1/2 − x) + (e + e²/4)e⁻ˣ

Let E denote the parabola y² = 8x. Let P = (−2, 4), and let Q and Q′ be two distinct points on E such that the lines PQ and PQ′ are tangents to E. Let F be the focus of E. Then which of the following statements is (are) TRUE ?

1. AThe triangle PFQ is a right-angled triangle
2. BThe triangle QPQ′ is a right-angled triangle
3. CThe distance between P and F is 5√2
4. DF lies on the line joining Q and Q′