JEE (Advanced) 2013 Paper-1

A horizontal stretched string, fixed at two ends, is vibrating in its fifth harmonic according to the equation, $$y(x, t) = (0.01 m) \sin [(62.8 m^{-1}) x] \cos [(628 s^{-1})t].$$ Assuming $$\pi = 3.14$$, the correct statement(s) is (are)

A solid sphere of radius R and density $$\rho$$ is attached to one end of a mass-less spring of force constant k. The other end of the spring is connected to another solid sphere of radius R and density $$3 \rho$$. The complete arrangement is placed in a liquid of density $$2 \rho$$ and is allowed to reach equilibrium. The correct statement(s) is (are)

A particle of mass M and positive charge Q, moving with a constant velocity $$\overrightarrow{u}_1 = 4\widehat{i} ms^{-1}$$, enters a region of uniform static magnetic field normal to the x-y plane. The region of the magnetic field extends from x = 0 to x = L for all values of y. After passing through this region, the particle emerges on the other side after 10 milliseconds with a velocity $$\overrightarrow{u}_2 = 2\left(\sqrt 3 \widehat{i} + \widehat{j}\right) ms^{-1}$$. The correct statement(s) is (are)

Two non-conducting solid spheres of radii R and 2R, having uniform volume charge densities $$\rho_1$$ and $$\rho_2$$ respectively, touch each other. The net electric field at a distance 2R from the centre of the smaller sphere, along the line joining the centres of the spheres, is zero. The ratio $$\frac{\rho_1}{\rho_2}$$ can be

In the circuit shown in the figure, there are two parallel plate capacitors each of capacitance C. The switch $$S_1$$ is pressedfirst to fully charge the capacitor $$C_1$$ and then released. The switch $$S_2$$ is then pressed to charge the capacitor $$C_2$$. After some time, $$S_2$$ is released and then $$S_3$$ is pressed. After sometime,

The work functions of Silver and Sodium are 4.6 and 2.3 eV, respectively. The ratio of the slope of the stopping potential versus frequencyplot for Silver to that of Sodium is

A freshly prepared sample of a radioisotope of half-life 1386 s has activity $$10^3$$ disintegrations per second. Given that ln 2 = 0.693, the fraction of the initial number of nuclei (expressed in nearest integer percentage) that will decay in the first 80 s after preparation of the sample is

A particle of mass 0.2 kg is moving in one dimension under a force that delivers a constant power 0.5 W to the particle. If the initial speed (in $$ms^{-1})$$ of the particle is zero, the speed (in $$ms^{-1})$$ after 5 s is

A uniform circular disc of mass 50 kg and radius 0.4 m is rotating with an angular velocity of 10 rad $$s^{-1}$$ aboutits own axis, which is vertical. Two uniform circular rings, each of mass 6.25 kg and radius 0.2 m, are gently placed symmetrically on the disc in such a manner that they are touching each other along the axis of the disc and are horizontal. Assume that the friction is large enough such that the rings are at rest relative to the disc and the system rotates about the original axis. The new angular velocity (in rad $$s^{-1}$$) of the system is

A bob of mass m, suspended by string of length $$l_1$$, is given a minimum velocity required to completea full circle in the vertical plane. At the highestpoint,it collides elastically with another bob of mass m suspendedbya string of length $$l_2$$, which is initially at rest. Both the strings are mass-less and inextensible. If the second bob, after collision acquires the minimum speed required to complete a full circle in the vertical plane, the ratio $$\frac{l_1}{l_2}$$ is