JEE Main 2022 · 27 Jul 2022 · Shift 1

Q1.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 :

Q2.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 …

Q3.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 belt moving with the same velocity will be :

Q4.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 bel …

Q5.Two cylindrical vessels of equal cross-sectional area $16{\mathrm{cm}}^2$ contain water upto heights 100 cm and 150 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 [Tak …

Q6.Two satellites A and B having masses in the ratio 4:3 are revolving in circular orbits of radii 3r and 4r respectively around the earth. The ratio of total mechanical energy of A to B is :

Q7.If ${\mathrm{K}}_1$ and ${\mathrm{K}}_2$ are the thermal conductivities, ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ are the lengths and ${\mathrm{A}}_1$ and ${\mathrm{A}}_2$ are the cross sectional areas of steel and copper rods respectively such that …

Q8.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 …

Q9.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 …

Q10.Two identical positive charges Q each are fixed at a distance of '2a' apart from each other. Another point charge ${\mathrm{q}}_0$ with mass 'm' is placed at midpoint between two fixed charges. For a small displacement along the line joining the fixed charges, the charge …

Q11.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 …

Q12.Two bar magnets oscillate in a horizontal plane in earth's magnetic field with time periods of 3 s and 4 s respectively. If their moments of inertia are in the ratio of $3:2$ then the ratio of their magnetic moments will be :

Q13.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

Q14.A direct current of 4 A and an alternating current of peak value 4 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 :

Q15.A beam of light travelling along X-axis is described by the electric field ${\mathrm{E}}_{\mathrm{y}}=900\sin \omega (\mathrm{t}-\mathrm{x}/\mathrm{c})$. The ratio of electric force to magnetic force on a charge q moving along Y-axis with a speed of …

Q16.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

Q17.An electron (mass m) with an initial velocity $\overrightarrow{\mathrm{v}}={\mathrm{v}}_0\hat{\mathrm{i}}$ (${\mathrm{v}}_0>0$) is moving in an electric field $\overrightarrow{\mathrm{E}}=-{\mathrm{E}}_0\hat{\mathrm{i}}$ (${\mathrm{E}}_0>0$) where ${\mathrm{E}}_0$ is …

Q18.What is the half-life period of a radioactive material if its activity drops to $1/16^{\text{th }}$ of its initial value of 30 years?

Q19.A logic gate circuit has two inputs A and B and output Y. The voltage waveforms of $\mathrm{A},\mathrm{B}$ and Y are shown below

The logic gate circuit is

Q20.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

Q21.In meter bridge experiment for measuring unknown resistance 'S', the null point is obtained at a distance 30 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 ______ $\Omega$.

Q22.The one division of main scale of vernier callipers reads 1 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 divi …

Q23.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 A and B respectively. The difference between the resultant intensities at the two points is xI. T …

Q24.To light, a $50\mathrm{W},100\mathrm{V}$ lamp is connected, in series with a capacitor of capacitance $\frac{50}{\pi \sqrt{\mathrm{x}}}\mu \mathrm{F}$, with $200\mathrm{V},50\mathrm{Hz}$ AC source. The value of x will be ______.

Q25.A 1 m long copper wire carries a current of 1 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 $\_\_\_\_$ $\times 10^{-23}\mathrm{N}$. …

Q26.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 $\mathrm{x}=\frac{2{\epsilon }_0}{\rho }\mathrm{m}$ from its axis is $\_\_\_\_$ ${\mathrm{Vm}}^{-1}$

Q27.A mass 0.9 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 g is placed over it and both masses move together with amplitude ${\mathrm{A}}_2$. If the ratio …

Q28.A square aluminium (shear modulus is $25\times 10^9{\mathrm{Nm}}^{-2}$) slab of side 60 cm and thickness 15 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 i …

Q29.A pulley of radius 1.5 m is rotated about its axis by a force $F=\left(12t-3t^2\right)N$ 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 …

Q30.A ball of mass m is thrown vertically upward. Another ball of mass 2m is thrown at 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 …

Q1.250 g solution of D-glucose in water contains $10.8\%$ of carbon by weight. The molality of the solution is nearest to
(Given: Atomic Weights are H, 1u ; C, 12u ; O, 16u)

Q2.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 : NaCl and $\mathrm{H}{}_2\mathrm{O}$ are weakly magnetized in opposite direction to …

Q3.Given below are two statements. One is labelled as Assertion A and the other is labelled as Reason R.

Assertion A : Energy of 2s orbital of hydrogen atom is greater than that of 2s orbital of lithium.
Reason R: Energies of the orbitals in the same subshell decrease with increase …

Q4.Given below are two statements. One is labelled as Assertion A and the other is labelled as Reason R.

Assertion A : Activated charcoal adsorbs $\mathrm{SO}{}_2$ more efficiently than $\mathrm{CH}{}_4$.
Reason R: Gases with lower critical temperatures are readily adsorbed by activated char …

Q5.Boiling point of a $2\%$ aqueous solution of a nonvolatile 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.

Q6.The incorrect statement is

Q7.Which of the following methods are not used to refine any metal?
(A) Liquation
(B) Calcination
(C) Electrolysis
(D) Leaching
(E) Distillation

Choose the correct answer from the options given below:

Q8.Given below are two statements:

Statement I: Hydrogen peroxide can act as an oxidizing agent in both acidic and basic conditions.

Statement II: Density of hydrogen peroxide at 298 K is lower than that of $\mathrm{D}{}_2\mathrm{O}$.

In the light of the above statements. Choose the correct an …

Q9.Given below are two statements:

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

Statement II: Hydroxides of Be and Al dissolve in excess alkali to give beryllate and aluminate ions.

In the light of …

Q10.Which oxoacid of phosphorous has the highest number of oxygen atoms present in its chemical formula?

Q11.Given below are two statements:

Statement I: Iron (III) catalyst, acidified $\mathrm{K}{}_2\mathrm{Cr}{}_2\mathrm{O}{}_7$ and neutral $\mathrm{KMnO}{}_4$ have the ability to oxidise $\mathrm{I}{}^{-}$ to $\mathrm{I}{}_2$ independently.
Statement II: Manganate ion is paramagnetic in nature and involves …

Q12.The total number of $\mathrm{Mn}=\mathrm{O}$ bonds in $\mathrm{Mn}{}_2\mathrm{O}{}_7$ is

Q13.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 aromati …

Q14.In the above reaction product B is:

Q15.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

Q16.In Carius method of estimation of halogen. 0.45 g of an organic compound gave 0.36 g of AgBr . Find out the percentage of bromine in the compound.
(Molar masses : $\mathrm{AgBr}=188\mathrm{g}{\mathrm{mol}}^{-1}:\mathrm{Br}=80\mathrm{g}{\mathrm{mol}}^{-1}$ )

Q17.20 mL of $0.02\mathrm{M}{\mathrm{K}}_2{\mathrm{Cr}}_2{\mathrm{O}}_7$ solution is used for the titration of 10 mL of ${\mathrm{Fe}}^{2+}$ solution in the acidic medium.

The molarity of ${\mathrm{Fe}}^{2+}$ solution is $\_\_\_\_$ $\times 10^{-2}\mathrm{M}$.
(Nearest Integer)

Q18.$2\mathrm{NO}+2{\mathrm{H}}_2\to {\mathrm{N}}_2+2{\mathrm{H}}_2\mathrm{O}$

The above reaction has been studied at $800^{\circ }\mathrm{C}$. The related data are given in the table below

| Reaction serial number | Initial pressure of ${\mathrm{H}}_2$ / kPa | Initia …

Q19.Amongst the following the number of oxide(s) which are paramagnetic in nature is
${\mathrm{Na}}_2\mathrm{O},{\mathrm{KO}}_2,{\mathrm{NO}}_2,{\mathrm{N}}_2\mathrm{O},{\mathrm{ClO}}_2,\mathrm{NO},{\mathrm{SO}}_2,{\mathrm{Cl}}_2\mathrm{O}$

Q20.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 J upon heating it from 300 K to 500 K . The number of moles of the gas at constant volume is $\_\_\_\_$ [Nearest intege …

Q21.According to MO theory, number of species/ions from the following having identical bond order is $\_\_\_\_$ :
${\mathrm{CN}}^{-},{\mathrm{NO}}^{+},{\mathrm{O}}_2,{\mathrm{O}}_2^{+},{\mathrm{O}}_2^{2+}$

Q22.At 310 K , the solubility of ${\mathrm{CaF}}_2$ in water is
$2.34\times 10^{-3}\mathrm{g}/100\mathrm{mL}$. The solubility product of ${\mathrm{CaF}}_2$ is $\_\_\_\_$ $\times 10^{-8}(\mathrm{mol}/\mathrm{L}{)}^3$. (Given molar mass : …

Q23.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 ____

Q24.In the titration of ${\mathrm{KMnO}}_4$ and oxalic acid in acidic medium, the change in oxidation number of carbon at the end point is ____

Q25.Optical activity of an enantiomeric mixture is $+12.6^{\circ }$ and the specific rotation of ( + ) isomer is $+30^{\circ }$. The optical purity is ____ %

Q26.In the following reaction

The % yield for reaction I is 60% and that of reaction II is $50\%$. The overall yield of the complete reaction is __ % [nearest integer]

Q1.Let $R_1$ and $R_2$ be two relations defined on $\mathbb{R}$ by $a{\mathrm{R}}_{1}b\iff ab\ge 0$ and $a{R}_{2}b\iff a\ge b$, then

Q2.Let $f,g:\mathbb{N}-\{1\}\to \mathbb{N}$ be functions defined by $f(\mathrm{a})=\alpha$, where $\alpha$ is the maximum of the powers of those primes p such that $p^{\alpha }$ divides $a$, and $g(a)=a+1$, for all $a\in \mathbb{N}-\{1\}$. Then, the function …

Q3.Let the minimum value $v_0$ of $v=|z{|}^2+|z-3{|}^2+|z-6i{|}^2$, $z\in \mathbb{C}$ is attained at $z=z_0$. Then ${\left|2z_0^2-{\bar{z}}_0^3+3\right|}^2+v_0^2$ is equal to

Q4.Let $A=\left(\begin{array}{cc}1& 2\\ -2& -5\end{array}\right)$. Let $\alpha ,\beta \in \mathbb{R}$ be such that $\alpha A^2+\beta A=2I$. Then $\alpha +\beta$ is equal to -

Q5.The remainder when $(2021{)}^{2022}+(2022{)}^{2021}$ is divided by 7 is

Q6.Suppose $a_1,a_2,\dots ,a_{\mathrm{n}},\dots$ be an arithmetic progression of natural numbers. If the ratio of the sum of the first five terms of the sum of first nine terms of the progression is $5:17$ and $110<a_{15}<120$, then the sum of the first ten terms of …

Q7.Let $f:\mathbb{R}\to \mathbb{R}$ be a function defined as $f(x)=a\sin \left(\frac{\pi [x]}{2}\right)+[2-x],a\in \mathbb{R}$, where $[\mathrm{t}]$ is the greatest integer less than or equal to $t$. If ${\lim }_{x\to -1}f(x)$ exists, then the value of …

Q8.$I={\int }_{\pi /4}^{\pi /3}\left(\frac{8\sin x-\sin 2x}{x}\right)dx$. Then

Q9.The area of the smaller region enclosed by the curves $y^2=8x+4$ and $x^2+y^2+4\sqrt{3}x-4=0$ is equal to

Q10.Let $\mathrm{y}={\mathrm{y}}_1(\mathrm{x})$ and $\mathrm{y}={\mathrm{y}}_2(\mathrm{x})$ be two distinct solutions of the differential equation $\frac{dy}{dx}=x+y$, with ${\mathrm{y}}_1(0)=0$ and ${\mathrm{y}}_2(0)=1$ respectively. Then, the number of points of interse …

Q11.Let $\mathrm{P}(a,\mathrm{b})$ be a point on the parabola ${\mathrm{y}}^2=8\mathrm{x}$ such that the tangent at P passes through the centre of the circle $x^2+y^2-10x-14y+65=0$. Let $A$ be the product of all possible values of $a$ and $B$ be the product of al …

Q12.Let $\vec{a}=\alpha \hat{i}+\hat{j}+\beta \hat{k}$ and $\vec{b}=3\hat{i}-5\hat{j}+4\hat{k}$ be two vectors, such that $\vec{a}\times \vec{b}=-\hat{i}+9\hat{j}+12\hat{k}$. Then the projection of $\vec{b}-2\vec{a}$ on $\vec{b}+\vec{a}$ is equal to

Q13.Let $\vec{a}=2\hat{i}-\hat{j}+5\hat{k}$ and $\vec{b}=\alpha \hat{i}+\beta \hat{j}+2\hat{k}$. If $((\vec{a}\times \vec{b})\times \hat{i})\cdot \hat{k}=\frac{23}{2}$, then $|\vec{b}\times 2\hat{j}|$ is equal to

Q14.Let S be the sample space of all five digit numbers. If $p$ is the probability that a randomly selected number from S, is a multiple of 7 but not divisible by 5, then $9p$ is equal to

Q15.Let a vertical tower AB of height $2h$ stands on a horizontal ground. Let from a point P on the ground a man can see upto height $h$ of the tower with an angle of elevation $2\alpha$. When from $P$, he moves a distance d in the direction of $\overrightarrow{AP}$, he can see the t …

Q16.$(p^{\wedge }r)\iff (p^{\wedge }(\sim q))$ is equivalent to $(\sim p)$ when $r$ is

Q17.If the plane P passes through the intersection of two mutually perpendicular planes $2x+ky-5z=1$ and $3kx-ky+z=5$, $k<3$ and intercepts a unit length on positive $x$-axis, then the intercept made by the plane P on the y-axis is

Q18.Let $A(1,1)$, $B(-4,3)$, $C(-2,-5)$ be vertices of a triangle $ABC$, P be a point on side BC, and ${\Delta }_1$ and ${\Delta }_2$ be the areas of triangle APB and ABC respectively.

If ${\Delta }_1:{\Delta }_2=4:7$, then the area enclosed by the lines AP, AC and the x-axis is

Q19.If the circle $x^2+y^2-2gx+6y-19c=0$, $g,c\in \mathbb{R}$ passes through the point $(6,1)$ and its centre lies on the line $x-2cy=8$, then the length of intercept made by the circle on x-axis is

Q20.Let a function $f:\mathbb{R}\to \mathbb{R}$ be defined as:
$f(x)=\{\begin{array}{ll}{\int }_0^x(5-|t-3|)dt,& x>4\\ x^2+bx,& x\le 4\end{array}$
where $b\in \mathbb{R}$. If $f$ is continuous at $x=4$, then which of the following statements is NOT true?

Q21.For $k\in \mathbb{R}$, let the solutions of the equation $\cos \left({\sin }^{-1}\left(x\cot \left({\tan }^{-1}\left(\cos \left({\sin }^{-1}x\right)\right)\right)\right)\right)=k$, $0<|x|<\frac{1}{\sqrt{2}}$ be $\alpha$ and $\beta$, where the inverse trigonometric functions take only p …

Q22.The mean and variance of 10 observations were calculated as 15 and 15 respectively by a student who took by mistake 25 instead of 15 for one observation. Then, the correct standard deviation is ____.

Q23.Let the line $\frac{x-3}{7}=\frac{y-2}{-1}=\frac{z-3}{-4}$ intersect the plane containing the lines $\frac{x-4}{1}=\frac{y+1}{-2}=\frac{z}{1}$ and $4ax-y+5z-7a=0=2x-5y-z-3$, $a\in \mathbb{R}$ at the point $P(\alpha ,\beta ,\gamma )$. Then the value of $\alpha +\beta +\gamma$ equals …

Q24.An ellipse $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ passes through the vertices of the hyperbola $H:\frac{x^2}{49}-\frac{y^2}{64}=-1$. Let the major and minor axes of the ellipse E coincide with the transverse and conjugate axes of the hyperbola H. Let the product of t …

Q25.Let $y=y(x)$ be the solution curve of the differential equation
$\sin \left(2x^2\right){\log }_e\left(\tan x^2\right)dy+\left(4xy-4\sqrt{2}x\sin \left(x^2-\frac{\pi }{4}\right)\right)dx=0$,
$0<x<\sqrt{\frac{\pi }{2}}$, which passes through the point …

Q26.Let $M$ and $N$ be the number of points on the curve $y^5-9xy+2x=0$, where the tangents to the curve are parallel to x-axis and y-axis, respectively. Then the value of $M+N$ equals ______.

Q27.Let $f(x)=2x^2-x-1$ and $S=\{n\in \mathbb{Z}:|f(n)|\le 800\}$. Then, the value of ${\sum }_{n\in S}f(n)$ is equal to ______.

Q28.Let S be the set containing all $3\times 3$ matrices with entries from $\{-1,0,1\}$. The total number of matrices $A\in S$ such that the sum of all the diagonal elements of $A^{T}A$ is 6 is ______.

Q29.If the length of the latus rectum of the ellipse $x^2+4y^2+2x+8y-\lambda =0$ is 4, and $l$ is the length of its major axis, then $\lambda +l$ is equal to ______.

Q30.Let $S=\left\{z\in \mathbb{C}:z^2+\bar{z}=0\right\}$. Then ${\sum }_{z\in S}(\mathrm{Re}(z)+\mathrm{Im}(z))$ is equal to ______.