JEE Advanced 2007 · Paper 2

Let O(0, 0), P(3, 4), Q(6, 0) be the vertices of the triangle OPQ. The point R inside the triangle OPQ is such that the triangles OPR, PQR, OQR are of equal area. The coordinates of R are

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

If |z| = 1 and z ≠ ±1, then all the values of z/(1 − z²) lie on

1. Aa line not passing through the origin
2. B|z| = √2
3. Cthe x-axis
4. Dthe y-axis

Let Eᶜ denote the complement of an event E. Let E, F, G be pairwise independent events with P(G) > 0 and P(E ∩ F ∩ G) = 0. Then P(Eᶜ ∩ Fᶜ | G) equals

1. AP(Eᶜ) + P(Fᶜ)
2. BP(Eᶜ) − P(Fᶜ)
3. CP(Eᶜ) − P(F)
4. DP(E) − P(Fᶜ)

d²x/dy² equals

1. A(d²y/dx²)⁻¹
2. B−(d²y/dx²)⁻¹(dy/dx)⁻³
3. C(d²y/dx²)(dy/dx)⁻²
4. D−(d²y/dx²)(dy/dx)⁻³

The differential equation dy/dx = √(1 − y²)/y determines a family of circles with

1. Avariable radii and a fixed centre at (0, 1)
2. Bvariable radii and a fixed centre at (0, −1)
3. Cfixed radius 1 and variable centres along the x-axis
4. Dfixed radius 1 and variable centres along the y-axis

Let a⃗, b⃗, c⃗ be unit vectors such that a⃗ + b⃗ + c⃗ = 0. Which one of the following is correct?

1. Aa⃗ × b⃗ = b⃗ × c⃗ = c⃗ × a⃗ = 0⃗
2. Ba⃗ × b⃗ = b⃗ × c⃗ = c⃗ × a⃗ ≠ 0⃗
3. Ca⃗ × b⃗ = b⃗ × c⃗ = a⃗ × c⃗ ≠ 0⃗
4. Da⃗ × b⃗, b⃗ × c⃗, c⃗ × a⃗ are mutually perpendicular

Let ABCD be a quadrilateral with area 18, with side AB parallel to the side CD and AB = 2 CD. Let AD be perpendicular to AB and CD. If a circle is drawn inside the quadrilateral ABCD touching all the sides, then its radius is

1. A3
2. B2
3. C3/2
4. D1

Let f(x) = x/(1 + xⁿ)¹/ⁿ for n ≥ 2 and g(x) = (f ∘ f ∘ … ∘ f)(x), f occurs n times. Then ∫xⁿ⁻²g(x) dx equals

1. A1/[n(n − 1)](1 + nxⁿ)¹⁻¹/ⁿ + K
2. B1/(n − 1)(1 + nxⁿ)¹⁻¹/ⁿ + K
3. C1/[n(n + 1)](1 + nxⁿ)¹⁺¹/ⁿ + K
4. D1/(n + 1)(1 + nxⁿ)¹⁺¹/ⁿ + K