JEE Advanced 2011 · Paper 2

Let P(6,3) be a point on the hyperbola x²/a² − y²/b² = 1. If the normal at the point P intersects the x-axis at (9, 0), then the eccentricity of the hyperbola is

1. A√(5/2)
2. B√(3/2)
3. C√2
4. D√3

A value of b for which the equations x² + bx − 1 = 0, x² + x + b = 0, have one root in common is

1. A−√2
2. B−i√3
3. Ci√5
4. D√2

Let ω ≠ 1 be a cube root of unity and S be the set of all non-singular matrices of the form [1 a b; ω 1 c; ω² ω 1], where each of a, b, and c is either ω or ω². Then the number of distinct matrices in the set S is

1. A2
2. B6
3. C4
4. D8

The circle passing through the point (−1, 0) and touching the y-axis at (0, 2) also passes through the point

1. A(−3/2, 0)
2. B(−5/2, 2)
3. C(−3/2, 5/2)
4. D(−4, 0)

If limₓ→₀ [1 + x ln(1 + b²)]¹/ˣ = 2b sin²θ, b > 0 and θ ∈ (−π, π], then the value of θ is

1. A±π/4
2. B±π/3
3. C±π/6
4. D±π/2

Let f : [−1, 2] → [0, ∞) be a continuous function such that f(x) = f(1 − x) for all x ∈ [−1, 2]. Let R₁ = ∫₋₁² x f(x) dx, and R₂ be the area of the region bounded by y = f(x), x = −1, x = 2, and the x-axis. Then

1. AR₁ = 2R₂
2. BR₁ = 3R₂
3. C2R₁ = R₂
4. D3R₁ = R₂

Let f(x) = x² and g(x) = sin x for all x ∈ ℝ. Then the set of all x satisfying (f ∘ g ∘ g ∘ f)(x) = (g ∘ g ∘ f)(x), where (f ∘ g)(x) = f(g(x)), is

1. A±√(nπ), n ∈ {0, 1, 2, …}
2. B±√(nπ), n ∈ {1, 2, …}
3. Cπ/2 + 2nπ, n ∈ {…, −2, −1, 0, 1, 2, …}
4. D2nπ, n ∈ {…, −2, −1, 0, 1, 2, …}

Let (x, y) be any point on the parabola y² = 4x. Let P be the point that divides the line segment from (0, 0) to (x, y) in the ratio 1 : 3. Then the locus of P is

1. Ax² = y
2. By² = 2x
3. Cy² = x
4. Dx² = 2y