JEE (Advanced) 2012 Paper-1

Let $$\theta, \phi \in [0, 2 \pi]$$ be such that
$$2 \cos \theta (1 - \sin \phi) = \sin^2 \theta \left(\tan \frac{\theta}{2} + \cot \frac{\theta}{2}\right) \cos \phi - 1, \tan(2 \pi - \theta) > 0$$ and $$-1 < \sin \theta < -\frac{\sqrt{3}}{2}$$. Then $$\phi$$ cannot satisfy

Let S be the area of the region enclosed by $$y = e^{-x^2}, y = 0, x = 0$$ and x = 1. Then

A ship is fitted with three engines $$E_1, E_2$$ and $$E_3$$. The engines function independently of each other with respective probabilities $$\frac{1}{2}, \frac{1}{4}$$ and $$\frac{1}{4}$$. For the ship to be operational at least two of its engines must function. LetX denote the eventthat the ship is operational and let $$X_1, X_2$$, and $$X_3$$ denote respectively the events that the engines $$E_1, E_2$$, and $$E_3$$ are functioning. Which of the following is (are) true ?

Tangents are drawnto the hyperbola $$\frac{x^2}{9} - \frac{y^2}{4} = 1$$, parallel to the straight line 2x - y = 1. The points of contact of the tangents on the hyperbola are

If y (x) satisfies the differential equation y'- ytan x = 2x sec x and y(0) = 0, then

Let $$f : IR \rightarrow IR$$ be defined as $$f(x) = \mid x \mid + \mid x^2 - 1 \mid$$. The total numberof points at which fattains either a local maximum or a local minimum is

The value of $$6 + \log_{\frac{3}{2}}\left(\frac{1}{3\sqrt{2}}\sqrt{4 - \frac{1}{3\sqrt{2}}\sqrt{4 - \frac{1}{3\sqrt{2}}\sqrt{4 - \frac{1}{3\sqrt{2}}...}}}\right)$$ is

Let p(x) be a real polynomial of least degree which has a local maximum at x = 1 and local minimum at x = 3. If p(1) = 6 and p(3) = 2, then p'(0) is

If $$\overrightarrow{a}, \overrightarrow{b}$$ and $$\overrightarrow{c}$$ are unit vectors satisfying $$\mid \overrightarrow{a} - \overrightarrow{b} \mid^2 + \mid \overrightarrow{b} - \overrightarrow{c} \mid^2 + \mid \overrightarrow{c} - \overrightarrow{a} \mid^2 = 9$$, then $$\mid 2 \overrightarrow{a} + 5 \overrightarrow{b} + 5 \overrightarrow{c}\mid$$ is

Let S be the focus of the parabola $$y^2 = 8x$$ and let PQ be the common chord of the circle $$x^2 + y^2 - 2x - 4y = 0$$ and the given parabola. The area ofthe triangle PQS is