JEE (Advanced) 2008 Paper-2

A particle P starts from the point $$z_0 = 1 + 2i$$, where $$i - \sqrt{-1}$$. It moves first horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point $$z_1$$. From $$z_1$$ the particle moves $$\sqrt{2}$$ units in the direction of the vector $$\hat{i} + \hat{j}$$ and then it moves through an angle $$\frac{\pi}{2}$$ in anticlockwise direction on a circle with center at origin, to reach a point $$z_2$$. The point $$z_2$$ is given by

Let the function $$g : (-\infty, \infty)\rightarrow \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$$ be given by $$g(u) = 2 \tan^{-1}(e^u) - \frac{\pi}{2}$$. Then, g is

Consider a branch of the hyperbola
$$x^2 - 2y^2 - 2\sqrt{2}x - 4\sqrt{2}y - 6 = 0$$
with vertex at the point A. Let B be oneof the endpoints of its latus rectum. If C is the focus of the hyperbola nearest to the point A, then the area of the triangle ABC is

The area of the region between the curves $$y = \sqrt{\frac{1 + \sin x}{\cos x}}$$ and $$y = \sqrt{\frac{1 - \sin x}{\cos x}}$$ bounded by the lines x = 0 and $$x = \frac{\pi}{4}$$ is

Consider three points $$P = (-\sin (\beta - \alpha), -\cos \beta), Q = (\cos(\beta - \alpha), \sin \beta)$$ and $$R = (\cos(\beta - \alpha + \theta), \sin(\beta - \theta))$$, where $$0 < \alpha, \beta, \theta < \frac{\pi}{4}$$. Then,

An experiment has 10 equally likely outcomes. Let A and B be two non-empty events of the experiment. If A consists of 4 outcomes, the number of outcomes that B must have so that A and B are independent, is

Let two non-collinear unit vectors $$\hat{a}$$ and $$\hat{b}$$ form an acute angle. A point P moves so that at any time t the position vector $$\overrightarrow{OP}$$(where O is the origin) is given by $$\hat{a} \cos t + \hat{b} \sin t$$. When is farthest from origin O, let M be the length of $$\overrightarrow{OP}$$ and $$\hat{u}$$ be the unit vector along $$\overrightarrow{OP}$$ . Then

Let
$$I = \int\frac{e^x}{e^{4x}+e^{2x}+1}dx, J = \int\frac{e^{-x}}{e^{-4x}+e^{-2x}+1}dx$$.
Then, for an arbitrary constant C, the value of J - I equals

Let $$g(x) = \log f(x)$$ where $$f(x)$$ is a twice differentiable positive function on $$(0, \infty)$$ such that $$f(x + 1) = x f(x)$$. Then, for $$N = 1, 2, 3,...,$$
$$g''\left(N + \frac{1}{2}\right) - g''\left(\frac{1}{2}\right) =$$

Suppose four distinct positive numbers $$a_1, a_2, a_3, a_4$$ are in G.P. Let $$b_1 = a_1, b_2 = b_1 + a_2, b_3 = b_2 + a_3$$ and $$b_4 = b_3 + a_4$$.
STATEMENT-1: The numbers $$b_1, b_2, b_3, b_4$$ are neither in A.P. nor in G.P.
and
STATEMENT-2: The numbers $$b_1, b_2, b_3, b_4$$ are in H.P.