JEE Advanced 2007 · Paper 1

Let α, β be the roots of the equation x² − px + r = 0 and α/2, 2β be the roots of the equation x² − qx + r = 0. Then the value of r is

1. A(2/9)(p − q)(2q − p)
2. B(2/9)(q − p)(2p − q)
3. C(2/9)(q − 2p)(2q − p)
4. D(2/9)(2p − q)(2q − p)

Let f(x) be differentiable on the interval (0, ∞) such that f(1) = 1, and limₜ→ₓ [t²f(x) − x²f(t)]/(t − x) = 1 for each x > 0. Then f(x) is

1. A1/(3x) + 2x²/3
2. B−1/(3x) + 4x²/3
3. C−1/x + 2/x²
4. D1/x

One Indian and four American men and their wives are to be seated randomly around a circular table. Then the conditional probability that the Indian man is seated adjacent to his wife given that each American man is seated adjacent to his wife is

1. A1/2
2. B1/3
3. C2/5
4. D1/5

The tangent to the curve y = eˣ drawn at the point (c, eᶜ) intersects the line joining the points (c − 1, eᶜ⁻¹) and (c + 1, eᶜ⁺¹)

1. Aon the left of x = c
2. Bon the right of x = c
3. Cat no point
4. Dat all points

limₓ→π/4 [∫₂^(sec²x) f(t) dt]/(x² − π²/16) equals

1. A(8/π)f(2)
2. B(2/π)f(2)
3. C(2/π)f(1/2)
4. D4f(2)

A hyperbola, having the transverse axis of length 2 sin θ, is confocal with the ellipse 3x² + 4y² = 12. Then its equation is

1. Ax² cosec² θ − y² sec² θ = 1
2. Bx² sec² θ − y² cosec² θ = 1
3. Cx² sin² θ − y² cos² θ = 1
4. Dx² cos² θ − y² sin² θ = 1

The number of distinct real values of λ, for which the vectors −λ² î + ĵ + k̂, î − λ² ĵ + k̂ and î + ĵ − λ² k̂ are coplanar, is

1. Azero
2. Bone
3. Ctwo
4. Dthree

A man walks a distance of 3 units from the origin towards the north-east (N 45° E) direction. From there, he walks a distance of 4 units towards the north-west (N 45° W) direction to reach a point P. Then the position of P in the Argand plane is

1. A3eⁱπ/⁴ + 4i
2. B(3 − 4i)eⁱπ/⁴
3. C(4 + 3i)eⁱπ/⁴
4. D(3 + 4i)eⁱπ/⁴