JEE (Advanced) 2012 Paper-2

Which of the following is correct ?

The value of $$b_6$$ is

A common tangent of the two circles is

A possible equation of L is

Let X and Y be two events such that $$P(X \mid Y) = \frac{1}{2}, P(Y \mid X) = \frac{1}{3}$$ and $$P(X \cap Y) = \frac{1}{6}$$. Which of the following is (are) correct?

If $$f(x) = \int_{0}^{x} e^{t^2}(t - 2)(t - 3) dt$$ for all $$x \in (0, \infty)$$, then

For every integer n, let $$a_n$$ and $$b_n$$ be real numbers. Let function $$f : IR \rightarrow IR$$ be given by
$$ f(x) = \begin{cases}a_n + \sin \pi x & for x \in [2n , 2n + 1]\\b_n + \cos \pi x & for x \in [2n - 1 , 2n]\end{cases}$$, for all integers n.
If f is continuous, then which of the following hold(s) for all n?

If the straight lines $$\frac{x - 1}{2} = \frac{y + 1}{k} = \frac{z}{2}$$ and $$\frac{x + 1}{5} = \frac{y + 1}{2} = \frac{z}{k}$$ are coplanar, then the plane(s) containing these two lines is(are)

If the adjoint of a $$3 \times 3$$ matrix P is $$\begin{bmatrix}1 & 4 & 4 \\2 & 1 & 7\\1 & 1 & 3 \end{bmatrix}$$, then the possible value(s) of the determinant of P is (are)

Let $$f : (-1, 1) \rightarrow IR$$ be such that $$f(\cos 4 \theta) = \frac{2}{2 - \sec^2 \theta}$$ for $$\theta \in \left(0, \frac{\pi}{4}\right)\cup \left(\frac{\pi}{4}, \frac{\pi}{2}\right)$$ Then the value(s) of $$f\left(\frac{1}{3}\right)$$ is (are)