JEE Advanced 2009 · Paper 1

Let P(3, 2, 6) be a point in space and Q be a point on the line r⃗ = (i⃗ − j⃗ + 2k⃗) + μ(−3i⃗ + j⃗ + 5k⃗). Then the value of μ for which the vector PQ⃗ is parallel to the plane x − 4y + 3z = 1 is

1. A1/4
2. B−1/4
3. C1/8
4. D−1/8

Tangents drawn from the point P(1, 8) to the circle x² + y² − 6x − 4y − 11 = 0 touch the circle at the points A and B. The equation of the circumcircle of the triangle PAB is

1. Ax² + y² + 4x − 6y + 19 = 0
2. Bx² + y² − 4x − 10y + 19 = 0
3. Cx² + y² − 2x + 6y − 29 = 0
4. Dx² + y² − 6x − 4y + 19 = 0

Let f be a non-negative function defined on the interval [0, 1]. If ∫₀ˣ √(1 − (f′(t))²) dt = ∫₀ˣ f(t) dt, 0 ≤ x ≤ 1, and f(0) = 0, then

1. Af(1/2) < 1/2 and f(1/3) > 1/3
2. Bf(1/2) > 1/2 and f(1/3) > 1/3
3. Cf(1/2) < 1/2 and f(1/3) < 1/3
4. Df(1/2) > 1/2 and f(1/3) < 1/3

Let z = x + iy be a complex number where x and y are integers. Then the area of the rectangle whose vertices are the roots of the equation z z̄³ + z̄ z³ = 350 is

1. A48
2. B32
3. C40
4. D80

The line passing through the extremity A of the major axis and extremity B of the minor axis of the ellipse x² + 9y² = 9 meets its auxiliary circle at the point M. Then the area of the triangle with vertices at A, M and the origin O is

1. A31/10
2. B29/10
3. C21/10
4. D27/10

If a⃗, b⃗, c⃗ and d⃗ are unit vectors such that (a⃗ × b⃗) · (c⃗ × d⃗) = 1 and a⃗ · c⃗ = 1/2, then

1. Aa⃗, b⃗, c⃗ are non-coplanar
2. Bb⃗, c⃗, d⃗ are non-coplanar
3. Cb⃗, d⃗ are non-parallel
4. Da⃗, d⃗ are parallel and b⃗, c⃗ are parallel

Let z = cos θ + i sin θ. Then the value of ∑ₘ₌₁¹⁵ Im(z²ᵐ⁻¹) at θ = 2° is

1. A1/sin 2°
2. B1/(3 sin 2°)
3. C1/(2 sin 2°)
4. D1/(4 sin 2°)

The number of seven digit integers, with sum of the digits equal to 10 and formed by using the digits 1, 2 and 3 only, is

1. A55
2. B66
3. C77
4. D88