JEE (Advanced) 2010 Paper-1

If the distance between the plane $$Ax - 2y + z = d$$ and the plane containing the lines $$\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{4}$$ and $$\frac{x-2}{3} = \frac{y-3}{4} = \frac{z-4}{5}$$ is $$\sqrt{6}$$, then $$\mid d \mid$$ is

For any real numberx, let [x] denote the largest integer less than or equal to x. Let f be a real valued function defined on the interval [-10, 10] by
$$f(x) = \begin{cases}x-[x] & if & [x] & is & odd,\\1+[x]-x & if & [x] & is & even\end{cases}$$
Then the value of $$\frac{\pi^2}{10}\int_{-10}^{10}f(x) \cos \pi x dx$$ is

Let $$\omega$$ be the complex number $$\cos \frac{2 \pi}{3} + i \sin \frac{2 \pi}{3}$$. Then the number of distinct complex numbers z satisfying

Let $$S_k = 1, 2, .... 100$$, denote the sumof the infinite geometric series whose first term is $$\frac{k-1}{k!}$$ and the common ratio is $$\frac{1}{k}$$. Then the value of $$\frac{100^2}{100!}+\sum_{k=1}^{100}\mid (k^2 - 3k + 1)S_k \mid$$ is

The number of all possible values of $$\theta$$ , where $$0 < \theta < \pi$$. for which the system of equations
$$(y + z) \cos 3 \theta = (xyz) \sin 3\theta$$
$$x \sin 3 \theta = \frac{2 \cos 3\theta}{y} + \frac{2 \sin 3\theta}{z}$$
$$(xyz) \sin 3\theta = (y + 2z) \cos 3 \theta + y \sin 3 \theta$$
have a solution $$(x_0, y_0, z_0)$$ with $$y_0z_0 \leq 0$$, is

Let f be a real-valued differentiable function on R (the set of all real numbers) such that f(1) = 1. If the y-intercept of the tangent at any point P(x, y) on the curve y= f(x) is equal to the cube of the abscissa of P, then the value off(-3) is equal to

Consider a thin square sheet of side L and thickness t, made of a material of resistivity $$\rho$$. The resistance between two opposite faces, shown by the shaded areas in the figure is

A real gas behaves like an ideal gas if its

Incandescent bulbs are designed by keeping in mind that the resistance of their filament increases with the increase in temperature. If at room temperature. 100 W, 60 Wand 40 Wbulbs have filament resistances $$R_{100}, R_{60}$$ and $$R_{40}$$, respectively, the relation between these resistances is

To verify Ohm's law, a student is provided with a test resistor $$R_{T}$$, a high resistance $$R_1$$, a small resistance $$R_2$$, two identical galvanometers $$G_1$$ and $$G_2$$, and a variable voltage source V. The correct circuit to carry out the experiment is