JEE Main 2019 January 09, Shift 2 Question Paper with Solutions

The position co-ordinates of a particle moving in a $3D$ coordinate system is given by
$x=a\cos\omega t$
$y=a\sin\omega t$
and $z=a\omega t$
The speed of the particle is:

Expression for time in terms of $G$ (universal gravitational constant), $h$ (Planck constant) and $c$ (speed of light) is proportional to:

In a car race on straight road, car A takes a time t less than car B at the finish and passes finishing point with a speed v more than that of car B. Both the cars start from rest and travel with constant acceleration $a_1$ and $a_2$ respectively. Then v is equal to:

A mass of $10$ kg is suspended vertically by a rope from the roof. When a horizontal force is applied on the rope at some point, the rope deviated at an angle of $45^{\circ}$ at the roof point. If the suspended mass is at equilibrium, the magnitude of the force applied is $(g=10\ ms^{-1})$

A force acts on a $2\ kg$ object so that its position is given as a function of time as $x=3t^2+5$. What is the work done by this force in first $5$ seconds?

A rod of length $50\ cm$ is pivoted at one end. It is raised such that it makes an angle of $30^{\circ}$ from the horizontal as shown and released from rest. Its angular speed when it passes through the horizontal (in $\text{rad}\ s^{-1}$) will be $(g=10\ ms^{-2})$

The energy required to take a satellite to a height h above the Earth surface (radius of Earth $=6.4\times10^3$ km) is $E_1$, and the kinetic energy required for the satellite to be in a circular orbit at this height is $E_2$. The value of h for which $E_1$ and $E_2$ are equal, is

The top of a water tank is open to air and its water level is maintained. It is giving out $0.74\ m^3$ water per minute through a circular opening of $2\ cm$ radius in its wall. The depth of the centre of the opening from the level of water in the tank is close to:

Two Carnot engines A and B are operated in series. The first one, A, receives heat at $T_1(=600K)$ and rejects to a reservoir at temperature $T_2$. The second engine B receives heat rejected by the first engine and, in turn, rejects to a heat reservoir at $T_3(=400K)$. Calculate the temperature $T_2$ if the work outputs of the two engines are equal:

A $15\ g$ mass of nitrogen gas is enclosed in a vessel at a temperature, $27^{\circ}C$. The amount of heat transferred to the gas, so that R.M.S. velocity of molecules is doubled, is about:
$\left[R=8.3\ J\ (K\ \text{mole})^{-1}\right]$

A particle is executing simple harmonic motion ($SHM$) of amplitude $A$, along the $x$-axis, about $x=0$. When its potential energy ($PE$) equals kinetic energy ($KE$), the position of the particle will be:

A rod of mass M and length $2L$ is suspended at its middle by a wire. It exhibits torsional oscillations; If two masses, each of mass m, are attached at distance $L/2$ from its centre on both sides, it reduces the oscillation frequency by $20\%$. The value of ratio $m/M$ is close to:

A musician using an open flute of length $50\ cm$ produces second harmonic sound waves. A person runs towards the musician from another end of a hall at a speed of $10\ km\ h^{-1}$. If the wave speed is $330\ m\ s^{-1}$, the frequency heard by the running person shall be close to:

Charge is distributed within a sphere of radius R with a volume charge density $\rho(r)=\dfrac{A}{r^2}e^{-\frac{2r}{a}}$, where A and a are constants. If Q is the total charge of this charge distribution, the radius R is:

Two point charges $q_1\left(\sqrt{10}\ \mu C\right)$ and $q_2(-25\ \mu C)$ are placed on the x-axis at $x=1\ m$ and $x=4\ m$ respectively. The electric field (in V/m) at a point $y=3\ m$ on y-axis is,
$\left[\text{Take}\ \dfrac{1}{4\pi\varepsilon_0}=9\times10^9\ N\ m^2C^{-2}\right]$

A parallel plate capacitor with square plates is filled with four dielectrics of dielectric constants $K_1, K_2, K_3, K_4$ arranged as shown in the figure. The effective dielectric constant K will be:

A carbon resistance has a following colour code. What is the value of the resistance?

In the given circuit the internal resistance of the $18V$ cell is negligible. If $R_1 = 400\ \Omega$, $R_3 = 100\ \Omega$ and $R_4 = 500\ \Omega$ and the reading of an ideal voltmeter across $R_4$ is $5\ V$, then the value of $R_2$ will be:

A particle having the same charge as of electron moves in a circular path of radius $0.5\ cm$ under the influence of a magnetic field of $0.5\ T$. If an electric field of $100\ V/m$ makes it to move in a straight path, then the mass of the particle is (Given charge of electron $= 1.6 \times 10^{-19}\ {}^{0}C$)

One of the two identical conducting wires of length L is bent in the form of a circular loop and the other into a circular coil of N identical turns. If the same current is passed in both, the ratio of the magnetic field at the centre of the loop $(B_L)$ to that at the centre of the coil $(B_C)$, i.e. $\frac{B_L}{B_C}$ will be:

A power transmission line feeds input power at $2300\ V$ to a step down transformer with its primary windings having $4000$ turns. The output power is delivered at $230\ V$ by the transformer. If the current in the primary of the transformer is $5\ A$ and its efficiency is $90\ \%$, the output current would be:

A series AC circuit containing an inductor $(20\ \text{mH})$, a capacitor $(120\ \mu F)$ and a resistor $(60\ \Omega)$ is driven by an AC source of $24\ V/50\ Hz$. The energy dissipated in the circuit in $60\ s$ is:

The energy associated with electric field is $(U_E)$ and with magnetic field is $(U_B)$ for an electromagnetic wave in free space. Then:

Two plane mirrors are inclined to each other such that a ray of light incident on the first mirror $(M_1)$ and parallel to the second mirror $(M_2)$ is finally reflected from the second mirror $(M_2)$ parallel to the first mirror $(M_1)$. The angle between the two mirrors will be: