JEE Advanced 2012 · Paper 1

The total number of ways in which 5 balls of different colours can be distributed among 3 persons so that each person gets at least one ball is

1. A75
2. B150
3. C210
4. D243

Let f(x) = { x² |cos(π/x)|, x ≠ 0; 0, x = 0 }, x ∈ R, then f is

1. Adifferentiable both at x = 0 and at x = 2
2. Bdifferentiable at x = 0 but not differentiable at x = 2
3. Cnot differentiable at x = 0 but differentiable at x = 2
4. Ddifferentiable neither at x = 0 nor at x = 2

The function f : [0, 3] → [1, 29], defined by f(x) = 2x³ − 15x² + 36x + 1, is

1. Aone-one and onto.
2. Bonto but not one-one.
3. Cone-one but not onto.
4. Dneither one-one nor onto.

If limₓ→∞ ((x² + x + 1)/(x + 1) − ax − b) = 4, then

1. Aa = 1, b = 4
2. Ba = 1, b = −4
3. Ca = 2, b = −3
4. Da = 2, b = 3

Let z be a complex number such that the imaginary part of z is nonzero and a = z² + z + 1 is real. Then a cannot take the value

1. A−1
2. B1/3
3. C1/2
4. D3/4

The ellipse E₁ : x²/9 + y²/4 = 1 is inscribed in a rectangle R whose sides are parallel to the coordinate axes. Another ellipse E₂ passing through the point (0, 4) circumscribes the rectangle R. The eccentricity of the ellipse E₂ is

1. A√2/2
2. B√3/2
3. C1/2
4. D3/4

Let P = [aᵢⱼ] be a 3×3 matrix and let Q = [bᵢⱼ], where bᵢⱼ = 2ⁱ⁺ʲaᵢⱼ for 1 ≤ i, j ≤ 3. If the determinant of P is 2, then the determinant of the matrix Q is

1. A2¹⁰
2. B2¹¹
3. C2¹²
4. D2¹³

The integral ∫ sec²x/(sec x + tan x)⁹⁄² dx equals (for some arbitrary constant K)

1. A−1/(sec x + tan x)¹¹⁄² {1/11 − 1/7 (sec x + tan x)²} + K
2. B1/(sec x + tan x)¹¹⁄² {1/11 − 1/7 (sec x + tan x)²} + K
3. C−1/(sec x + tan x)¹¹⁄² {1/11 + 1/7 (sec x + tan x)²} + K
4. D1/(sec x + tan x)¹¹⁄² {1/11 + 1/7 (sec x + tan x)²} + K