JEE (Advanced) 2009 Paper-2

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

The normal at a point P on the ellipse $$x^2 + 4y^2 = 16$$ 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

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 $$p \neq q$$, is

If $$I_n = \int_{-\pi}^{\pi}\frac{\sin nx}{(1 + \pi^x) \sin x}dx$$, n = 0, 1, 2, 3, ....,
then

An ellipse intersects the hyperbola $$2x^2 - 2y^2 = 1$$ 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

For the function
$$f(x) = x \cos \frac{1}{x}, x \geq 1$$,

The tangent PT and the normal PN to the parabola $$y^2 = 4ax$$ 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

For $$0 < \theta < \frac{\pi}{2}$$, the solution(s) of
$$ \sum_{m=1}^{6}\cosec\left(\theta + \frac{(m - 1)\pi}{4}\right)\cosec\left(\theta + \frac{m\pi}{4}\right) = 4\sqrt{2}$$
is(are)

Match the statements/expressions given in Column I with the values given in Column II.

Match the statements/expressions given in Column I with the values given in Column II.