JEE (Advanced) 2011 Paper-1

A boy is pushing a ring of mass 2 kg and radius 0.5 m with a stick as shown in the figure. The stick applies a force of 2 N on the ring and rolls it without slipping with an acceleration of 0.3 $$m/s^2$$. The coefficient of friction between the ground and the ring is large enough that rolling always occurs and the coefficient of friction between the stick and the ring is (P/10). The value of P is.

Four point charge, each of +q, are rigidly fixed at the four corners of a square planar soap film of side 'a'. The surface tension of the soap film is $$\gamma$$. The system of charges and planar film are in equilibrium, and $$a = k\left[\frac{q^2}{\gamma}\right]^{\frac{1}{N}}$$, where 'k' is a constant. Then N is

Four solid spheres each of diameter $$\sqrt{5}$$ cm and mass 0.5 kg are placed with their centers at the corners of a square of side 4 cm. The momentum of inertia of the system about the diagonal of the square is $$N \times 10^{-4} kg -m^2$$, then N is.

The activity of a freshly prepared radioactive sample is $$10^{10}$$ disintegrations per second, whose mean life is $$10^{9}$$ s. The mass of an atom of this radioisotope is $$10^{-25}$$ kg. The mass (in mg) of the radioactive sample is.

A long circular tube of length 10 m and radius 0.3 carries a current I along its curved surface as shown. A wire-loop of resistance 0.005 ohm and of radius 0.1 m is placed inside the tube with its axis coinciding with the axis of the tube. The current varies as $$I = I_0 \cos(300 t)$$ where $$I_0$$ is constant. If the magnetic moment of the loop is $$N \mu_0 I_0 \sin(300 t)$$. then 'N' is.

Steel wire of length 'L' at $$40^\circ C$$ is suspended from the ceiling and then a mass 'm' is hung from its free end. The wire is cooled down from $$40^\circ C$$ to $$30^\circ C$$ to region its original length 'L'. The coefficient of linear thermal expansion of the steel is $$10^{-5}/^\circ C$$, Young's modulus of steel is $$10^{11} N/m^2$$ and radius of the wire is 1 mm. Assume that L >> diameter of the wire. Then the value of 'm' in kg is nearly.

The value of $$\int_{\sqrt{\ln 2}}^{\sqrt{\ln 3}} \frac{x \sin x^2}{\sin x^2 + \sin(\ln 6 - x^2)} dx $$ is

Let the straight line x = b divide the area enclosed by $$y = (1 - x^2), y = 0$$ and x = 0 into two parts $$R_1(0 \leq x \leq b)$$ and $$R_2(b \leq x leq 1)$$ such that $$R_1 - R_2 = \frac{1}{4}$$. Then b equals

Let $$\overrightarrow{a} = \hat{i} + \hat{j} + \hat{k}, \overrightarrow{b} = \hat{i} - \hat{j} + \hat{k}$$ and $$\overrightarrow{c} = \hat{i} - \hat{j} - \hat{k}$$ be three vectors. A vector $$\overrightarrow{v}$$ in the place of $$\overrightarrow{a}$$ and $$\overrightarrow{b}$$, whose projection on $$\overrightarrow{c}$$ is $$\frac{1}{\sqrt{3}}$$, is given by

Let $$(x_0, y_0)$$ be the solution of the following equations
$$(2x)^{\ln 2} = (3y)^{\ln 3}$$
$$3^{\ln x} = 2^{\ln y}$$
Then $$x_0$$ is