JEE Main 2024 Jan 31 Shift 1 — Physics

A force is represented by $F = ax^2 + bt^{\frac{1}{2}}$

where $x$ = distance and $t$ = time. The dimensions of $b^2/a$ are:

The relation between time $'t'$ and distance $'x'$ is $t = \alpha x^2 + \beta x$, where $\alpha$ and $\beta$ are constants. The relation between acceleration $(a)$ and velocity $(v)$ is :

A coin is placed on a disc. The coefficient of friction between the coin and the disc is $\mu$. If the distance of the coin from the center of the disc is $r$, the maximum angular velocity which can be given to the disc, so that the coin does not slip away, is :

In the given arrangement of a doubly inclined plane two blocks of masses $M$ and $m$ are placed. The blocks are connected by a light string passing over an ideal pulley as shown. The coefficient of friction between the surface of the plane and the blocks is 0.25. The value of $m$, for which $M = 10$ kg will move down with an acceleration of $2\ \mathrm{m/s^2}$, is: (take $g = 10\ \mathrm{m/s^2}$ and $\tan 37° = 3/4$)

double inclined plane with block M on the 53° incline and block m on the 37° incline, connected by a string over a pulley at the apex

An artillery piece of mass $M_1$ fires a shell of mass $M_2$ horizontally. Instantaneously after the firing, the ratio of kinetic energy of the artillery and that of the shell is:

Four identical particles of mass $m$ are kept at the four corners of a square. If the gravitational force exerted on one of the masses by the other masses is $\left(\frac{2\sqrt{2}+1}{32}\right)\frac{Gm^2}{L^2}$, the length of the sides of the square is

A small steel ball is dropped into a long cylinder containing glycerine. Which one of the following is the correct representation of the velocity time graph for the transit of the ball?

The given figure represents two isobaric processes for the same mass of an ideal gas, then

V–T graph with two straight lines from origin O, line $P_2$ steeper (above) line $P_1$

The parameter that remains the same for molecules of all gases at a given temperature is :

Two charges $q$ and $3q$ are separated by a distance $'r'$ in air. At a distance $x$ from charge $q$, the resultant electric field is zero. The value of $x$ is :

Two conductors have the same resistances at $0°\mathrm{C}$ but their temperature coefficients of resistance are $\alpha_1$ and $\alpha_2$. The respective temperature coefficients for their series and parallel combinations are :

A rigid wire consists of a semicircular portion of radius $R$ and two straight sections. The wire is partially immerged in a perpendicular magnetic field $B = B_0\,\hat{j}$ as shown in figure. The magnetic force on the wire if it has a current $i$ is:

U-shaped wire with semicircular top of radius R inside a region of magnetic field pointing out of the page (dots), with x–y axes shown; the two straight sections extend downward out of the field region

A coil is places perpendicular to a magnetic field of 5000 T. When the field is changed to 3000 T in 2 s, an induced emf of 22 V is produced in the coil. If the diameter of the coil is 0.02 m, then the number of turns in the coil is:

In a plane EM wave, the electric field oscillates sinusoidally at a frequency of $5 \times 10^{10}$ Hz and an amplitude of $50\ \mathrm{Vm^{-1}}$. The total average energy density of the electromagnetic field of the wave is : [Use $\varepsilon_0 = 8.85 \times 10^{-12}\ \mathrm{C^2/Nm^2}$]

The refractive index of a prism with apex angle $A$ is $\cot A/2$. The angle of minimum deviation is :

When a metal surface is illuminated by light of wavelength $\lambda$, the stopping potential is 8V. When the same surface is illuminated by light of wavelength $3\lambda$, stopping potential is 2V. The threshold wavelength for this surface is:

If the wavelength of the first member of Lyman series of hydrogen is $\lambda$. The wavelength of the second member will be

Identify the logic operation performed by the given circuit.

logic circuit — inputs A and B each pass through a NOT gate (inverter), whose outputs feed a NAND gate giving output Y

The fundamental frequency of a closed organ pipe is equal to the first overtone frequency of an open organ pipe. If length of the open pipe is 60 cm, the length of the closed pipe will be:

If the percentage errors in measuring the length and the diameter of a wire are 0.1% each. The percentage error in measuring its resistance will be:

A body starts falling freely from height $H$ hits an inclined plane in its path at height $h$. As a result of this perfectly elastic impact, the direction of the velocity of the body becomes horizontal. The value of $\frac{H}{h}$ for which the body will take the maximum time to reach the ground is ________.

A solid circular disc of mass 50 kg rolls along a horizontal floor so that its center of mass has a speed of 0.4 m/s. The absolute value of work done on the disc to stop it is __________ J.

The depth below the surface of sea to which a rubber ball be taken so as to decrease its volume by 0.02% is _______ $m$.

(Take density of sea water $= 10^3\ \mathrm{kgm^{-3}}$, Bulk modulus of rubber $= 9 \times 10^8\ \mathrm{Nm^{-2}}$, and $g = 10\ \mathrm{ms^{-2}}$)

A particle performs simple harmonic motion with amplitude $A$. Its speed is increased to three times at an instant when its displacement is $\frac{2A}{3}$. The new amplitude of motion is $\frac{nA}{3}$. The value of $n$ is ________.

A parallel plate capacitor with plate separation 5 mm is charged up by a battery. It is found that on introducing a dielectric sheet of thickness 2 mm , while keeping the battery connections intact, the capacitor draws 25% more charge from the battery than before. The dielectric constant of the sheet is ________.

Equivalent resistance of the following network is ________ $\Omega$.

resistor network between terminals A and B — three parallel branches ($R_1$ 6 $\Omega$ shown with $R_2$, $R_3$ 2 $\Omega$, and $R_5$ 3 $\Omega$ with $R_6$ 3 $\Omega$) linked by series resistors of 2 $\Omega$ ($R_2$ top-left) and 2 $\Omega$ ($R_4$ top-right)

An electron moves through a uniform magnetic field $\vec{B} = B_0\,\hat{i} + 2B_0\,\hat{j}\ T$. At a particular instant of time, the velocity of electron is $\vec{u} = 3\hat{i} + 5\hat{j}\ \mathrm{m/s}$. If the magnetic force acting on electron is $\vec{F} = 5e\,\hat{k}\ N$, where $e$ is the charge of electron, then the value of $B_0$ is ________ $T$.

A small square loop of wire of side $l$ is placed inside a large square loop of wire of side $L$ $(L = l^2)$. The loops are coplanar and their centers coincide. The value of the mutual inductance of the system is $\sqrt{x} \times 10^{-7}\ H$, where $x =$ ______.

Two waves of intensity ratio 1:9 cross each other at a point. The resultant intensities at that point, when (a) Waves are incoherent is $I_1$ (b) Waves are coherent is $I_2$ and differ in phase by $60°$. If $\frac{I_1}{I_2} = \frac{10}{x}$ then $x =$ ______.

The mass defect in a particular reaction is 0.4g. The amount of energy liberated is $n \times 10^7$ kWh, where $n =$ ______.

(speed of light $= 3 \times 10^8$ m/s)