JEE (Advanced) 2010 Paper-2

Let S = {1, 2, 3, 4}. The total number of unordered pairs of disjoint subsets of S is equal to

Let f be a real-valued function defined on the interval (-1, 1) such that $$e^{-x}f(x) = 2 + \int_{0}^{x}\sqrt{t^4 + 1} dt$$, for all $$x \in (-1, 1)$$, and let $$f^{-1}$$ be the inverse function of f. Then $$(f^{-1})'(2)$$ is equal to

If the distance of the point P(1, -2, 1) from the plane $$x + 2y - 2z = \alpha$$, where $$\alpha > 0$$, is 5, then the foot of the perpendicular from P to the plane is

Two adjacent sides of a parallelogram ABCD are given by
$$\overrightarrow{AB} = 2\hat{i} + 10\hat{j} + 11\hat{k}$$ and $$\overrightarrow{AD} = -\hat{i} + 2\hat{j} + 2\hat{k}$$
The side AD is rotated by an acute angle $$\alpha$$ in the plane of the parallelogram so that AD becomes AD'. If AD' makes a right angle with the side AB, then the cosine of the angle $$\alpha$$ is given by

A signal which can be green or red with probability $$\frac{4}{5}$$ and $$\frac{1}{5}$$ respectively, is received by station A and then transmitted to station B. The probability of each station receiving the signal correctly is $$\frac{3}{4}$$. If the signal received at station B is green, then the probability that the original signal was green is

Two parallel chords of a circle of radius 2 are at a distance $$\sqrt{3} + 1$$ apart. If the chords subtend at the center, angles of $$\frac{\pi}{k}$$ and $$\frac{2 \pi}{k}$$, where k > 0, then the value of [k] is
[Note : [k] denotes the largest integer less then or equal to k]

Consider a triangle ABC and let a, b and c denote the lengths of the sides opposite to vertices A, B and C respectively. Suppose a = 6, b = 10 and the area of the triangle is $$15\sqrt{3}$$. If $$\angle ACB$$ is obtuse andif r denotes the radius of the incircle of the triangle, then $$r^2$$ is equal to

Let f be a function defined on R (the set of all real numbers) such that
$$f'(x) = 2010(x-2009)(x-2010)^2(x-2011)^3(x-2012)^4$$, for all $$x \in R$$.
If g is a function defined on R with values in the interval $$(0, \infty)$$ such that $$f(x) = \ln (g(x))$$, for all $$x \in R$$,
then the number of points in R at which g has a local maximum is

Let $$a_1, a_2, a_3, ...., a_{11}$$ be real numbers satisfying
$$a_1 = 15, 27 - 2a_2 > 0$$ and $$a_k = 2a_{k-1} - a_{k-2}$$, for k = 3, 4, ...., 11.
If $$\frac{a_{1}^{2} + a_{2}^{2} + ..... + a_{11}^{2}}{11} = 90$$, then the value of $$\frac{a_1 + a_2 + .... + a_{11}}{11}$$ is equal to

Let k be a positive real number and let
$$A = \begin{bmatrix}2k-1 & 2\sqrt{k} & 2\sqrt{k} \\2\sqrt{k} & 1 & -2k \\-2\sqrt{k} & 2k & -1 \end{bmatrix}$$ and $$B = \begin{bmatrix}0 & 2k-1 & \sqrt{k} \\1-2k & 0 & 2\sqrt{k} \\-\sqrt{k} & -2\sqrt{k} & 0 \end{bmatrix}$$.If $$det(adj A) + det(adj B) = 10^6$$, then [k] is equal to
[Note : adj M denotes the adjoint of a square matrix M and [k] denotes the largest integer less than or equal to k].