JEE Advanced 2009 · Paper 2

If the sum of first n terms of an A.P. is cn², then the sum of squares of these n terms is

1. An(4n² − 1)c²/6
2. Bn(4n² + 1)c²/3
3. Cn(4n² − 1)c²/3
4. Dn(4n² + 1)c²/6

A line with positive direction cosines passes through the point P(2, −1, 2) and makes equal angles with the coordinate axes. The line meets the plane 2x + y + z = 9 at point Q. The length of the line segment PQ equals

1. A1
2. B√2
3. C√3
4. D2

The normal at a point P on the ellipse x² + 4y² = 16 meets the x-axis at Q. If M is the mid point of the line segment PQ, then the locus of M intersects the latus rectums of the given ellipse at the points

1. A(±3√5/2, ±2/7)
2. B(±3√5/2, ±√19/4)
3. C(±2√3, ±1/7)
4. D(±2√3, ±4√3/7)

The locus of the orthocentre of the triangle formed by the lines (1 + p)x − py + p(1 + p) = 0, (1 + q)x − qy + q(1 + q) = 0, and y = 0, where p ≠ q, is

1. Aa hyperbola
2. Ba parabola
3. Can ellipse
4. Da straight line

If Iₙ = ∫₋π^π [sin nx / ((1 + πˣ) sin x)] dx, n = 0, 1, 2, …, then

1. AIₙ = Iₙ₊₂
2. B∑ₘ₌₁¹⁰ I₂ₘ₊₁ = 10π
3. C∑ₘ₌₁¹⁰ I₂ₘ = 0
4. DIₙ = Iₙ₊₁

An ellipse intersects the hyperbola 2x² − 2y² = 1 orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the ellipse are along the coordinate axes, then

1. AEquation of ellipse is x² + 2y² = 2
2. BThe foci of ellipse are (±1, 0)
3. CEquation of ellipse is x² + 2y² = 4
4. DThe foci of ellipse are (±√2, 0)

For the function f(x) = x cos(1/x), x ≥ 1,

1. Afor at least one x in the interval [1, ∞), f(x + 2) − f(x) < 2
2. Blimₓ→∞ f′(x) = 1
3. Cfor all x in the interval [1, ∞), f(x + 2) − f(x) > 2
4. Df′(x) is strictly decreasing in the interval [1, ∞)

The tangent PT and the normal PN to the parabola y² = 4ax at a point P on it meet its axis at points T and N, respectively. The locus of the centroid of the triangle PTN is a parabola whose

1. Avertex is (2a/3, 0)
2. Bdirectrix is x = 0
3. Clatus rectum is 2a/3
4. Dfocus is (a, 0)