JEE (Advanced) 2014 Paper-1

Airplanes A and B are flying with constant velocity in the same vertical plane at angles $$30^\circ$$ and $$60^\circ$$ with respect to the horizontal respectively as shown in figure. The speed of A is $$100 \sqrt 3 ms^{−1}$$. At time t = 0 s, an observer in A finds B at a distance of 500 m. This observer sees B moving with a constant velocity perpendicular to the line of motion of A. If at $$t = t_0$$, A just escapes being hit by B, $$t_0$$ in seconds is

During Searle’s experiment, zero of the Vernier scale lies between $$3.20 \times 10^{−2} m$$ and $$3.25 \times 10^{-2} m$$ of the main scale. The $$20^{th}$$ division of the Vernier scale exactly coincides with one of the main scale divisions. When an additional load of 2 kg is applied to the wire, the zero of the Vernier scale still lies between $$3.20 \times 10^{-2} m$$ and $$3.25 \times 10^{-2} m$$ of the main scale but now the $$45^{th}$$ division of Vernier scale coincides with one of the main scale divisions. The length of the thin metallic wire is 2 m and its cross-sectional area is $$8 \times 10^{-7} m^2$$. The least count of the Vernier scale is $$1.0 \times 10^{-5} m$$. The maximum percentage error in the Young’s modulus of the wire is

A uniform circular disc of mass 1.5 kg and radius 0.5 m is initially at rest on a horizontal frictionless surface. Three forces of equal magnitude F = 0.5 N are applied simultaneously along the three sides of an equilateral triangle XYZ with its vertices on the perimeter of the disc (see figure). One second after applying the forces, the angular speed of the disc in $$rads^{-1}$$ is

Two parallel wires in the plane of the paper are distance $$X_0$$ apart. A point charge is moving with speed u between the wires in the same plane at a distance $$X_1$$ from one of the wires. When the wires carry current of magnitude I in the same direction, the radius of curvature of the path of the point charge is $$R_1$$. In contrast, if the currents I in the two wires have directions opposite to each other, the radius of curvature of the path is $$R_2$$. If $$\frac{X_0}{X_1} = 3,$$ the value of $$\frac{R_1}{R_2}$$ is

To find the distance d over which a signal can be seen clearly in foggy conditions, a railways engineer uses dimensional analysis and assumes that the distance depends on the mass density $$\rho$$ of the fog, intensity (power/area) SS of the light from the signal and its frequency f. The engineer finds that disproportional to $$S^{\frac {1}{n}}$$. The value of n is

A galvanometer gives full scale deflection with 0.006 A current. By connecting it to a $$4990 Ω$$ resistance, it can be converted into a voltmeter of range 0 - 30 V. If connected to a $$\frac{2n}{249}Ω$$ resistance, it becomes an ammeter of range 0 - 1.5 A. The value of n is

Consider an elliptically shaped rail PQ in the vertical plane with OP = 3 m and OQ = 4 m. A block of mass 1 kg is pulled along the rail from P to Q with a force of 18 N, which is always parallel to line PQ (see the figure given). Assuming no frictional losses, the kinetic energy of the block when it reaches Q is $$(n \times 10)$$ Joules. The value of n is (take acceleration due to gravity = 10 $$ms^{-2}$$)

A rocket is moving in a gravity free space with a constant acceleration of $$2 ms^{-2}$$ along +x direction (see figure). The length of a chamber inside the rocket is 4 m. A ball is thrown from the left end of the chamber in +x direction with a speed of $$0.3 ms^{-1}$$ relative to the rocket. At the same time, another ball is thrown in -x direction with a speed of $$0.2 ms^{-1}$$ from its right end relative to the rocket. The time in seconds when the two balls hit each other is

A horizontal circular platform of radius 0.5 m and mass 0.45 kg is free to rotate about its axis. Two massless spring toy-guns, each carrying a steel ball of mass 0.05 kg are attached to the platform at a distance 0.25 m from the centre on its either sides along its diameter (see figure). Each gun simultaneously fires the balls horizontally and perpendicular to the diameter in opposite directions. After leaving the platform, the balls have horizontal speed of $$9 ms^{-1}$$ with respect to the ground. The rotational speed of the platform in $$rads^{-1}$$ after the balls leave the platform is

A thermodynamic system is taken from an initial state i with internal energy $$U_i = 100 J$$ to the final state f along two different paths iaf and ibf, as schematically shown in the figure. The work done by the system along the paths af, ib and bf are $$W_{af} = 200 J, W_{ib} = 50 J$$ and $$W_{bf} = 100 J$$ respectively. The heat supplied to the system along the path iaf, ib and bf are $$Q_{iaf}, Q_{ib}$$ and $$Q_{bf}$$ respectively. If the internal energy of the system in the state b is $$U_b = 200 J$$ and $$Q_{iaf} = 500 J$$, the ratio $$\frac {Q_{bf}} {Q_{ib}}$$ is