JEE (Advanced) 2013 Paper-1

Let complex numbers $$\alpha$$ and $$\frac{1}{\alpha}$$ lie on circles $$(x - x_0)^2 + (y - y_0)^2 = r^2$$ and $$(x - x_0)^2 + (y - y_0)^2 = 4r^2$$, respectively. If $$z_0 = x_0 + iy_0$$ Satisfies the equation $$2 |z_0|^2 = r^2 + 2,$$ then $$|\alpha| = $$

Four persons independently solve a certain problem correctly with probabilities $$\frac{1}{2}, \frac{3}{4}, \frac{1}{4}, \frac{1}{8}.$$ Then the probability that the problem is solved correctly by at least one of them is

Let $$f : \left[\frac{1}{2}, 1\right] \rightarrow R$$ (the set of all real numbers) be a positive, non-constant and differentiable function such that $$f'(x) < 2 f (x)$$ and $$f\left(\frac{1}{2}\right) = 1$$ Then the value of $$\int_{\frac{1}{2}}^{1} f(x) dx$$ lies in the interval

The number of points in $$(-\infty, \infty)$$, for which $$x^2 - x \sin x - \cos x = 0,$$ is

The area enclosed by the curves $$y = \sin x + \cos x$$ and $$y = |\cos x - \sin x|$$ over the interval $$\left[0, \frac{\pi}{2}\right]$$ is

A curve passes through the point $$\left(1, \frac{\pi}{6}\right).$$ Let the slope of the curve at each point (x, y) be $$\frac{y}{x} + \sec \left(\frac{y}{x}\right), x > 0.$$ Then the equation of the curve is

The value of $$\cot \left(\sum_{n = 1}^{23}\cot^{-1} \left(1 + \sum_{k = 1}^n 2k\right)\right)$$ is

For a > b > c > 0, the distance between (1, 1) and the point of intersection of the lines ax + by + c = 0 and bx + ay + c = 0 is less than $$2 \sqrt 2$$. Then

Perpendiculars are drawn from points on the line $$\frac{x + 2}{2} = \frac{y + 1}{-1} = \frac{z}{3}$$ to the plane x + y + z = 3. The feet of perpendiculars lie on the line

Let $$\overrightarrow{PR} = 3\widehat{i} + \widehat{j} - 2\widehat{k}$$ and $$\overrightarrow{SQ} = \widehat{i} - 3 \widehat{j} - 4\widehat{k}$$ determine diagonals of a parallelogram PQRS and $$\overrightarrow{PT} = \widehat{i} + 2\widehat{j} + 3\widehat{k}$$ be another vector. Then the volume of the $$\overrightarrow{PT}, \overrightarrow{PQ}$$ and $$\overrightarrow{PS}$$ is