JEE Advanced 2011 · Paper 1

Let (x₀, y₀) be the solution of the following equations (2x)ˡⁿ² = (3y)ˡⁿ³ 3ˡⁿˣ = 2ˡⁿʸ. Then x₀ is

1. A1/6
2. B1/3
3. C1/2
4. D6

The value of ∫ from √(ln 2) to √(ln 3) [x sin x²/(sin x² + sin(ln 6 − x²))] dx is

1. A(1/4) ln(3/2)
2. B(1/2) ln(3/2)
3. Cln(3/2)
4. D(1/6) ln(3/2)

Let a⃗ = î + ĵ + k̂, b⃗ = î − ĵ + k̂ and c⃗ = î − ĵ − k̂ be three vectors. A vector v⃗ in the plane of a⃗ and b⃗, whose projection on c⃗ is 1/√3, is given by

1. Aî − 3ĵ + 3k̂
2. B−3î − 3ĵ − k̂
3. C3î − ĵ + 3k̂
4. Dî + 3ĵ − 3k̂

Let P = {θ : sin θ − cos θ = √2 cos θ} and Q = {θ : sin θ + cos θ = √2 sin θ} be two sets. Then

1. AP ⊂ Q and Q − P ≠ ∅
2. BQ ⊄ P
3. CP ⊄ Q
4. DP = Q

Let the straight line x = b divide the area enclosed by y = (1 − x)², y = 0, and x = 0 into two parts R₁ (0 ≤ x ≤ b) and R₂ (b ≤ x ≤ 1) such that R₁ − R₂ = 1/4. Then b equals

1. A3/4
2. B1/2
3. C1/3
4. D1/4

Let α and β be the roots of x² − 6x − 2 = 0, with α > β. If aₙ = αⁿ − βⁿ for n ≥ 1, then the value of (a₁₀ − 2a₈)/(2a₉) is

1. A1
2. B2
3. C3
4. D4

A straight line L through the point (3, −2) is inclined at an angle 60° to the line √3x + y = 1. If L also intersects the x-axis, then the equation of L is

1. Ay + √3x + 2 − 3√3 = 0
2. By − √3x + 2 + 3√3 = 0
3. C√3y − x + 3 + 2√3 = 0
4. D√3y + x − 3 + 2√3 = 0

The vector(s) which is/are coplanar with vectors î + ĵ + 2k̂ and î + 2ĵ + k̂, and perpendicular to the vector î + ĵ + k̂ is/are

1. Aĵ − k̂
2. B−î + ĵ
3. Cî − ĵ
4. D−ĵ + k̂