JEE Main 2022 27th July Morning Shift Question Paper with Solutions

A torque meter is calibrated to reference standards of mass, length and time each with $5 \%$ accuracy. After calibration, the measured torque with this torque meter will have net accuracy of :

Sand is being dropped from a stationary dropper at a rate of $0.5 \,\mathrm{kgs}^{-1}$ on a conveyor belt moving with a velocity of $5 \mathrm{~ms}^{-1}$. The power needed to keep the belt moving with the same velocity will be :

A square aluminum (shear modulus is $25 \times 10^{9}\, \mathrm{Nm}^{-2}$) slab of side $60 \mathrm{~cm}$ and thickness $15 \mathrm{~cm}$ is subjected to a shearing force (on its narrow face) of $18.0 \times 10^{4}$ $\mathrm{N}$. The lower edge is riveted to the floor. The displacement of the upper edge is ____________ $\mu$m.

A bullet is shot vertically downwards with an initial velocity of $100 \mathrm{~m} / \mathrm{s}$ from a certain height. Within 10 s, the bullet reaches the ground and instantaneously comes to rest due to the perfectly inelastic collision. The velocity-time curve for total time $\mathrm{t}=20 \mathrm{~s}$ will be:

(Take g = 10 m/s²).

The one division of main scale of Vernier callipers reads $1 \mathrm{~mm}$ and 10 divisions of Vernier scale is equal to the 9 divisions on main scale. When the two jaws of the instrument touch each other, the zero of the Vernier lies to the right of zero of the main scale and its fourth division coincides with a main scale division. When a spherical bob is tightly placed between the two jaws, the zero of the Vernier scale lies in between $4.1 \mathrm{~cm}$ and $4.2 \mathrm{~cm}$ and $6^{\text {th }}$ Vernier division coincides scale division. The diameter of the bob will be ____________ $\times$ 10^$-$2 cm.

Two cylindrical vessels of equal cross-sectional area $16 \mathrm{~cm}^{2}$ contain water upto heights $100 \mathrm{~cm}$ and $150 \mathrm{~cm}$ respectively. The vessels are interconnected so that the water levels in them become equal. The work done by the force of gravity during the process, is [Take, density of water $=10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ and $\mathrm{g}=10 \mathrm{~ms}^{-2}$ ] :

Read the following statements :

A. When small temperature difference between a liquid and its surrounding is doubled, the rate of loss of heat of the liquid becomes twice.

B. Two bodies $P$ and $Q$ having equal surface areas are maintained at temperature $10^{\circ} \mathrm{C}$ and $20^{\circ} \mathrm{C}$. The thermal radiation emitted in a given time by $\mathrm{P}$ and $\mathrm{Q}$ are in the ratio $1: 1.15$.

C. A Carnot Engine working between $100 \mathrm{~K}$ and $400 \mathrm{~K}$ has an efficiency of $75 \%$.

D. When small temperature difference between a liquid and its surrounding is quadrupled, the rate of loss of heat of the liquid becomes twice.

Choose the correct answer from the options given below :

To light, a $50 \mathrm{~W}, 100 \mathrm{~V}$ lamp is connected, in series with a capacitor of capacitance $\frac{50}{\pi \sqrt{x}} \mu F$, with $200 \mathrm{~V}, 50 \mathrm{~Hz} \,\mathrm{AC}$ source. The value of $x$ will be ___________.

At a particular station, the TV transmission tower has a height of 100 m. To triple its coverage range, height of the tower should be increased to

In a meter bridge experiment, for measuring unknown resistance 'S', the null point is obtained at a distance $30 \mathrm{~cm}$ from the left side as shown at point D. If R is $5.6$ $\mathrm{k} \Omega$, then the value of unknown resistance 'S' will be __________ $\Omega$.

A direct current of $4 \mathrm{~A}$ and an alternating current of peak value $4 \mathrm{~A}$ flow through resistance of $3\, \Omega$ and $2\,\Omega$ respectively. The ratio of heat produced in the two resistances in same interval of time will be :

A logic gate circuit has two inputs A and B and output Y. The voltage waveforms of A, B and Y are shown below.

The logic gate circuit is :

A magnet hung at $45^{\circ}$ with magnetic meridian makes an angle of $60^{\circ}$ with the horizontal. The actual value of the angle of dip is -

An electron (mass $\mathrm{m}$) with an initial velocity $\vec{v}=v_{0} \hat{i}\left(v_{0}>0\right)$ is moving in an electric field $\vec{E}=-E_{0} \hat{i}\left(E_{0}>0\right)$ where $E_{0}$ is constant. If at $\mathrm{t}=0$ de Broglie wavelength is $\lambda_{0}=\frac{h}{m v_{0}}$, then its de Broglie wavelength after time t is given by

Same gas is filled in two vessels of the same volume at the same temperature. If the ratio of the number of molecules is $1: 4$, then

A. The r.m.s. velocity of gas molecules in two vessels will be the same.

B. The ratio of pressure in these vessels will be $1: 4$.

C. The ratio of pressure will be $1: 1$.

D. The r.m.s. velocity of gas molecules in two vessels will be in the ratio of $1: 4$.

Choose the correct answer from the options given below :

Two bar magnets oscillate in a horizontal plane in earth's magnetic field with time periods of $3 \mathrm{~s}$ and $4 \mathrm{~s}$ respectively. If their moments of inertia are in the ratio of $3: 2$, then the ratio of their magnetic moments will be:

A pulley of radius $1.5 \mathrm{~m}$ is rotated about its axis by a force $F=\left(12 \mathrm{t}-3 \mathrm{t}^{2}\right) N$ applied tangentially (while t is measured in seconds). If moment of inertia of the pulley about its axis of rotation is $4.5 \mathrm{~kg} \mathrm{~m}^{2}$, the number of rotations made by the pulley before its direction of motion is reversed, will be $\frac{K}{\pi}$. The value of K is ___________.

A ball of mass m is thrown vertically upward. Another ball of mass $2 \mathrm{~m}$ is thrown at an angle $\theta$ with the vertical. Both the balls stay in air for the same period of time. The ratio of the heights attained by the two balls respectively is $\frac{1}{x}$. The value of x is _____________.

Two beams of light having intensities I and 4I interfere to produce a fringe pattern on a screen. The phase difference between the two beams are $\pi / 2$ and $\pi / 3$ at points $\mathrm{A}$ and $\mathrm{B}$ respectively. The difference between the resultant intensities at the two points is $x I$. The value of $x$ will be ________.

What is the half-life period of a radioactive material if its activity drops to $1 / 16^{\text {th }}$ of its initial value in 30 years?

A $1 \mathrm{~m}$ long copper wire carries a current of $1 \mathrm{~A}$. If the cross section of the wire is $2.0 \mathrm{~mm}^{2}$ and the resistivity of copper is $1.7 \times 10^{-8}\, \Omega \mathrm{m}$, the force experienced by moving electron in the wire is ____________ $\times 10^{-23} \mathrm{~N}$.

(charge on electorn $=1.6 \times 10^{-19} \,\mathrm{C}$)

A bag is gently dropped on a conveyor belt moving at a speed of $2 \mathrm{~m} / \mathrm{s}$. The coefficient of friction between the conveyor belt and bag is $0.4$. Initially, the bag slips on the belt before it stops due to friction. The distance travelled by the bag on the belt during slipping motion, is : [Take $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{-2}$ ]

A microscope was initially placed in air (refractive index 1). It is then immersed in oil (refractive index 2). For a light whose wavelength in air is $\lambda$, calculate the change of microscope's resolving power due to oil and choose the correct option.

Two satellites $\mathrm{A}$ and $\mathrm{B}$, having masses in the ratio $4: 3$, are revolving in circular orbits of radii $3 \mathrm{r}$ and $4 \mathrm{r}$ respectively around the earth. The ratio of total mechanical energy of $\mathrm{A}$ to $\mathrm{B}$ is :

A mass $0.9 \mathrm{~kg}$, attached to a horizontal spring, executes SHM with an amplitude $\mathrm{A}_{1}$. When this mass passes through its mean position, then a smaller mass of $124 \mathrm{~g}$ is placed over it and both masses move together with amplitude $A_{2}$. If the ratio $\frac{A_{1}}{A_{2}}$ is $\frac{\alpha}{\alpha-1}$, then the value of $\alpha$ will be ___________.

A beam of light travelling along $X$-axis is described by the electric field $E_{y}=900 \sin \omega(\mathrm{t}-x / c)$. The ratio of electric force to magnetic force on a charge $\mathrm{q}$ moving along $Y$-axis with a speed of $3 \times 10^{7} \mathrm{~ms}^{-1}$ will be :

(Given speed of light $=3 \times 10^{8} \mathrm{~ms}^{-1}$)

Two sources of equal emfs are connected in series. This combination is connected to an external resistance R. The internal resistances of the two sources are $r_{1}$ and $r_{2}$ $\left(r_{1}>r_{2}\right)$. If the potential difference across the source of internal resistance $r_{1}$ is zero, then the value of R will be :

If $K_{1}$ and $K_{2}$ are the thermal conductivities, $L_{1}$ and $L_{2}$ are the lengths and $A_{1}$ and $A_{2}$ are the cross sectional areas of steel and copper rods respectively such that $\frac{K_{2}}{K_{1}}=9, \frac{A_{1}}{A_{2}}=2, \frac{L_{1}}{L_{2}}=2$. Then, for the arrangement as shown in the figure, the value of temperature $\mathrm{T}$ of the steel - copper junction in the steady state will be:

Two identical positive charges $Q$ each are fixed at a distance of '2a' apart from each other. Another point charge $q_{0}$ with mass 'm' is placed at midpoint between two fixed charges. For a small displacement along the line joining the fixed charges, the charge $\mathrm{q}_{0}$ executes $\mathrm{SHM}$. The time period of oscillation of charge $\mathrm{q}_{0}$ will be :

A long cylindrical volume contains a uniformly distributed charge of density $\rho \,\mathrm{Cm}^{-3}$. The electric field inside the cylindrical volume at a distance $x=\frac{2 \varepsilon_{0}}{\rho} \mathrm{m}$ from its axis is ________ $\mathrm{Vm}^{-1}$.

A sugar 'X' dehydrates very slowly under acidic condition to give furfural which on further reaction with resorcinol gives the coloured product after sometime. Sugar 'X' is

Boiling point of a $2 \%$ aqueous solution of a non-volatile solute A is equal to the boiling point of $8 \%$ aqueous solution of a non-volatile solute B. The relation between molecular weights of A and B is

Amongst the following, the number of oxide(s) which are paramagnetic in nature is

$\mathrm{Na}_{2} \mathrm{O}, \mathrm{KO}_{2}, \mathrm{NO}_{2}, \mathrm{~N}_{2} \mathrm{O}, \mathrm{ClO}_{2}, \mathrm{NO}, \mathrm{SO}_{2}, \mathrm{Cl}_{2} \mathrm{O}$

The molar heat capacity for an ideal gas at constant pressure is $20.785 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$. The change in internal energy is $5000 \mathrm{~J}$ upon heating it from $300 \mathrm{~K}$ to $500 \mathrm{~K}$. The number of moles of the gas at constant volume is ____________. [Nearest integer] (Given: $\mathrm{R}=8.314 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$)

Given below are two statements: one is labelled as Assertion A and, the other is labelled as Reason R.

Assertion A: [6] Annulene, [8] Annulene and cis-[10] Annulene, are respectively aromatic, not-aromatic and aromatic.

Reason R: Planarity is one of the requirements of aromatic systems.

In the light of the above statements, choose the most appropriate answer from the options given below.

The conductivity of a solution of complex with formula $\mathrm{CoCl}_{3}\left(\mathrm{NH}_{3}\right)_{4}$ corresponds to 1 : 1 electrolyte, then the primary valency of central metal ion is __________.

Given below are two statements.

Statement I: $\mathrm{O}_{2}, \mathrm{Cu}^{2+}$, and $\mathrm{Fe}^{3+}$ are weakly attracted by magnetic field and are magnetized in the same direction as magnetic field.

Statement II: $\mathrm{NaCl}$ and $\mathrm{H}_{2} \mathrm{O}$ are weakly magnetized in opposite direction to magnetic field.

In the light of the above statements, choose the most appropriate answer from the options given below.

$250 \mathrm{~g}$ solution of $\mathrm{D}$-glucose in water contains $10.8 \%$ of carbon by weight. The molality of the solution is nearest to

(Given: Atomic Weights are, $\mathrm{H}, 1 \,\mathrm{u} ; \mathrm{C}, 12 \,\mathrm{u} ; \mathrm{O}, 16 \,\mathrm{u}$)

In the above reaction product B is :

Given below are two statements.

Statement I: The chlorides of $\mathrm{Be}$ and $\mathrm{Al}$ have Cl-bridged structure. Both are soluble in organic solvents and act as Lewis bases.

Statement II: Hydroxides of $\mathrm{Be}$ and $\mathrm{Al}$ dissolve in excess alkali to give beryllate and aluminate ions.

In the light of the above statements, choose the correct answer from the options given below.

Match List I with List II.