JEE Main 2022 · 29 Jul 2022 · Shift 1

Q1.Given below are two statements : One is labelled as Assertion (A) and other is labelled as Reason (R).
Assertion (A) : Time period of oscillation of a liquid drop depends on surface tension (S), if density of the liquid is $p$ and radius of the drop is $r$, then …

Q2.A ball is thrown up vertically with a certain velocity so that, it reaches a maximum height h . Find the ratio of the times in which it is at height $\frac{\mathrm{h}}{3}$ while going up and coming down respectively.

Q3.If $\mathrm{t}=\sqrt{\mathrm{x}}+4$, then ${\left(\frac{\mathrm{dx}}{\mathrm{dt}}\right)}_{\mathrm{t}=4}$ is:

Q4.A smooth circular groove has a smooth vertical wall as shown in figure. A block of mass m moves against the wall with a speed v . Which of the following curve represents the correct relation between the normal reaction on the block by the wall ( N ) and speed of the block (v) ?

Q5.A ball is projected with kinetic energy E, at an angle of $60^{\circ }$ to the horizontal. The kinetic energy of this ball at the highest point of its flight will become :

Q6.Two bodies of mass 1 kg and 3 kg have position vector $\hat{i}+2\hat{j}+\hat{k}$ and $-3\hat{i}-2\hat{j}+\hat{k}$ respectively. The magnitude of position vector of centre of mass of this system will be similar to the magnitude of vector :

Q7.Given below are two statements : One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : Clothes containing oil or grease stains cannot be cleaned by water wash.
Reason (R) : Because the angle of contact between the oil/ grease and water is obtus …

Q8.If the length of a wire is made double and radius is halved of its respective values. Then, the Young's modules of the material of the wire will :

Q9.The time period of oscillation of a simple pendulum of length L suspended from the roof of a vehicle, which moves without friction down an inclined plane of inclination $\alpha$, is given by :

Q10.A spherically symmetric charge distribution is considered with charge density varying as
$\rho (r)=\{\begin{array}{ll}{\rho }_0\left(\frac{3}{4}-\frac{r}{R}\right)& \text{ for }r\le R\\ \text{ Zero }& \text{ for }r>R\end{array}$
Where, $\mathrm{r}(\mathrm{r}<\mathrm{R})$ is th …

Q11.Given below are two statements.
Statement I: Electric potential is constant within and at the surface of each conductor.
Statement II : Electric field just outside a charged conductor is perpendicular to the surface of the conductor at every point.
In the light of the above state …

Q12.Two metallic wires of identical dimensions are connected is series. If ${\sigma }_1$ and ${\sigma }_2$ are the conductivities of the these wires respectively, the effective conductivity of the combination is:

Q13.An alternating emf $\mathrm{E}=440\sin 100\pi \mathrm{t}$ is applied to a circuit containing an inductance of $\frac{\sqrt{2}}{\pi }\mathrm{H}$. If an a.c. ammeter is connected in the circuit, its reading will be :

Q14.A coil of inductance 1 H and resistance $100\Omega$ is connected to a battery of 6 V. Determine approximately :
(a) The time elapsed before the current acquires half of its steady - state value
(b) The energy stored in the magnetic field associated with the coil at an instant …

Q15.The kinetic energy of emitted electron is E when the light incident on the metal has wavelength $\lambda$. To double the kinetic energy, the incident light must have wavelength :

Q16.Find the ratio of energies of photons produced due to transition of an election of hydrogen atom from its(i) second permitted energy level to the first level, and (ii) the highest permitted energy level to the first permitted level.

Q17.Find the modulation index of an AM wave having 8 V variation where maximum amplitude of the AM wave is 9 V .

Q18.A travelling microscope has 20 divisions per cm on the main scale while its Vernier scale has total 50 divisions and 25 Vernier scale divisions are equal to 24 main scale divisions, what is the least count of the travelling microscope ?

Q19.In an experiment to find out the diameter of wire using screw gauge, the following observation were noted :

(a) Screw moves 0.5 mm on main scale in one complete rotation
(b) Total divisions on circular scale $=50$
(c) Main scale reading is 2.5 mm
(d) $45^{\text{th }}$ divis …

Q20.An object is projected in the air with initial velocity $u$ at an angle $\theta$. The projectile motion is such that the horizontal range R , is maximum. Another object is projected in the air with a horizontal range half of the range of first object. The initial velocity rem …

Q21.If the acceleration due to gravity experienced by a point mass at a height h above the surface of earth is same as that of the acceleration due to gravity at a depth $\alpha \mathrm{h}$ ( $\mathrm{h}\ll {\mathrm{R}}_{\mathrm{e}}$ ) from the earth surface. The value of …

Q22.The pressure $P_1$ and density $d_1$ of diatomic gas $\left(\gamma =\frac{7}{5}\right)$ changes suddenly to ${\mathrm{P}}_2\left(>{\mathrm{P}}_1\right)$ and ${\mathrm{d}}_2$ respectively during an adiabatic process. The temperature of the gas increases and becomes
_ …

Q23.One mole of a monoatomic gas is mixed with three moles of a diatomic gas. The molecular specific heat of mixture at constant volume is $\frac{{\alpha }^2}{4}\mathrm{RJ}/\mathrm{mol}\mathrm{K}$; then the value of $\alpha$ will be ____ . (Assume that the given …

Q24.The current I flowing through the given circuit will be ______ A.

Q25.A closely wounded circular coil of radius 5 cm produces a magnetic field of $37.68\times 10^{-4}\text{T}$ at its center. The current through the coil is ______ A.
[Given, number of turns in the coil is 100 and $\pi =3.14$]

Q26.Two light beams of intensities 4I and 9I interfere on a screen. The phase difference between these beams on the screen at point A is zero and at point B is $\pi$. The difference of resultant intensities, at the point A and B, will be ______ I.

Q27.A wire of length 314 cm carrying current of 14 A is bent to form a circle. The magnetic moment of the coil is ______ A$-{\text{m}}^2$. [Given $\pi =3.14$]

Q28.The X-Y plane be taken as the boundary between two transparent media ${\text{M}}_1$ and ${\text{M}}_2$. ${\text{M}}_1$ in $Z\ge 0$ has a refractive index of $\sqrt{2}$ and ${\text{M}}_2$ with $Z<0$ has a refractive index of $\sqrt{3}$. A ray of light travelling in ${\text{M}}_1$ along th …

Q29.If the potential barrier across a p-n junction is 0.6 V. Then the electric field intensity, in the depletion region having the width of $6\times 10^{-6}\text{m}$, will be ______ $\times 10^5\text{N}/\text{C}$.

Q1.Which of the following pair of molecules contain odd electron molecule and an expanded octet molecule?

Q2.${\mathrm{N}}_{2(\mathrm{g})}+3{\mathrm{H}}_{2(\mathrm{g})}\rightleftharpoons 2{\mathrm{NH}}_{3(\mathrm{g})}$

20 g 5 g

Consider the above reaction, the limiting reagent of the reaction and number of moles of ${\mathrm{NH}}_3$ formed respectively are:

Q3.100 mL of $5\%(\mathrm{w}/\mathrm{v})$ solution of NaCl in water was prepared in 250 mL beaker. Albumin from the egg was poured into NaCl solution and stirred well.

This resulted in a/ an :

Q4.The first ionization enthalpy of $\mathrm{Na},\mathrm{Mg}$ and Si , respectively, are: 496,737 and $786\mathrm{kJ}{\mathrm{mol}}^{-1}$. The first ionization enthalpy $\left(\mathrm{kJ}{\mathrm{mol}}^{-1}\right)$ of Al is:

Q5.In metallurgy the term "gangue" is used for:

Q6.The reaction of zinc with excess of aqueous alkali, evolves hydrogen gas and gives:

Q7.Lithium nitrate and sodium nitrate, when heated separately, respectively, give :

Q8.Number of lone pairs of electrons in the central atom of ${\mathrm{SCl}}_2,{\mathrm{O}}_3,{\mathrm{ClF}}_3$ and ${\mathrm{SF}}_6$, respectively, are :

Q9.In following pairs, the one in which both transition metal ions are colourless is :

Q10.In neutral or faintly alkaline medium, ${\mathrm{KMnO}}_4$ being a powerful oxidant can oxidize, thiosulphate almost quantitatively, to sulphate. In this reaction overall change in oxidation state of manganese will be :

Q11.Which among the following pairs has only herbicides?

Q12.Which among the following is the strongest Bronsted base ?

Q13.Which among the following pairs of the structures will give different products on ozonolysis? (Consider the double bonds in the structures are rigid and not delocalized.)

Q14.Considering the above reactions, the compound 'A' and compound 'B' respectively are:

Q15.Consider the above reaction sequence, the Product 'C' is:

Q16.'A' $(\mathrm{C}{}_8\mathrm{H}{}_6\mathrm{Cl}{}_2\mathrm{O})\stackrel{\mathrm{NH}{}_3}{\to }\mathrm{C}{}_8\mathrm{H}{}_8\mathrm{ClNO}\underset{\mathrm{NaOH}}{\overset{\mathrm{Br}{}_2}{\to }}$

Consider the above reaction, the compound 'A' is:

Q17.Which among the following represent reagent 'A'?

Q18.Consider the following reaction sequence:

The product 'B' is:

Q19.Which of the following compounds is an example of hypnotic drug?

Q20.A compound 'X' is acidic and it is soluble in NaOH solution, but insoluble in $\mathrm{NaHCO}{}_3$ solution. Compound 'X' also gives violet colour with neutral $\mathrm{FeCl}{}_3$ solution. The compound 'X' is:

Q21.Resistance of a conductivity cell (cell constant $129{\mathrm{m}}^{-1}$) filled with 74.5 ppm solution of KCl is $100\Omega$ (labelled as solution 1). When the same cell is filled with KCl solution of 149 ppm, the resistance is $50\Omega$ (labelled as solution 2). The ratio of …

Q22.Ionic radii of cation ${\mathrm{A}}^{+}$ and anion ${\mathrm{B}}^{-}$ are 102 and 181 pm respectively. These ions are allowed to crystallize into an ionic solid. This crystal has cubic close packing for ${\mathrm{B}}^{-}$. ${\mathrm{A}}^{+}$ is present in all octahedral voids. The edge l …

Q23.The minimum uncertainty in the speed of an electron in an one dimensional region of length $2a_O$ (Where $a_0=$ Bohr radius 52.9 pm) is ______ $\mathrm{km}{\mathrm{s}}^{-1}$.
(Given: Mass of electron $=9.1\times 10^{-31}\mathrm{kg}$, Planck's constant …

Q24.When 600 mL of $0.2\mathrm{M}{\mathrm{HNO}}_3$ is mixed with 400 mL of 0.1 M NaOH solution in a flask, the rise in temperature of the flask is ______ $\times 10^{-2}$ ${}^0\mathrm{C}$.
(Enthalpy of neutralisation $=57\mathrm{kJ}{mo1}^{-1}$ and Specific heat of water …

Q25.If ${\mathrm{O}}_2$ gas is bubbled through water at 303 K, the number of millimoles of ${\mathrm{O}}_2$ gas that dissolve in 1 litre of water is ______. (Nearest Integer)
(Given: Henry's Law constant for ${\mathrm{O}}_2$ at 303 K is 46.82 k bar and partial pressure of …

Q26.If the solubility product of PbS is $8\times 10^{-28}$, then the solubility of PbS in pure water at 298 K is $x\times 10^{-16}\mathrm{mol}{\mathrm{L}}^{-1}$. The value of $x$ is ______.
(Nearest Integer)
[Given $\sqrt{2}=1.41$]

Q27.The reaction between X and Y is first order with respect to X and zero order with respect to Y.
| Experiment | [X] mol L${}^{-1}$ | [Y] mol L${}^{-1}$ | Initial rate mol L${}^{-1}$ min${}^{-1}$ ||------------|------------------|------------------|--------------------------------------- …

Q28.In a linear tetrapeptide (Constituted with different amino acids), (number of amino acids) - (number of peptide bonds) is ____.

Q29.In bromination of Propyne, with Bromine 1, 1, 2, 2-tetrabromopropane is obtained in 27% yield. The amount of 1,1,2,2 tetrabromopropane obtained from 1 g of Bromine in this reaction is ____ × 10⁻¹ g. (Nearest integer)
(Molar Mass : Bromine = 80 g/mol)

Q30.[Fe(CN)₆]³⁻ should be an inner orbital complex. Ignoring the pairing energy, the value of crystal field stabilization energy for this complex is (-) ____ Δₒ. (Nearest integer)

Q1.Let R be a relation from the set $\{1,2,3\dots \dots \dots ,60\}$ to itself such that $\mathrm{R}=\{(\mathrm{a},\mathrm{b}):\mathrm{b}=\mathrm{pq}$, where $\mathrm{p},\mathrm{q}\ge 3$ are prime numbers $\}$. Then, the number of elements in R is :

Q2.If $\mathrm{z}=2+3\mathrm{i}$, then ${\mathrm{z}}^5+(\overline{\mathrm{z}}{)}^5$ is equal to :

Q3.Let A and B be two $3\times 3$ non-zero real matrices such that $AB$ is a zero matrix. Then

Q4.If $\frac{1}{(20-a)(40-a)}+\frac{1}{(40-a)(60-a)}+\dots \dots +\frac{1}{(180-a)(200-a)}=\frac{1}{256}$, then the maximum value of $a$ is :

Q5.If ${\lim }_{x\to 0}\frac{\alpha e^x+\beta e^{-x}+\gamma \sin x}{x{\sin }^{2}x}=\frac{2}{3}$,
where $\alpha ,\beta ,\gamma \in \mathrm{R}$, then which of the following is NOT correct ?

Q6.The integral ${\int }_0^{\frac{\pi }{2}}\frac{1}{3+2\sin x+\cos x}dx$ is equal to:

Q7.Let the solution curve $\mathrm{y}=\mathrm{y}(\mathrm{x})$ of the differential equation $\left(1+e^{2x}\right)\left(\frac{dy}{dx}+y\right)=1$ pass through the point $\left(0,\frac{\pi }{2}\right)$. Then, ${\lim }_{x\to \infty }{e}^{x}y(x)$ is equal to :

Q8.Let a line L pass through the point of intersection of the lines $bx+10y-8=0$ and $2x-3y=0$, $\mathrm{b}\in \mathrm{R}-\left\{\frac{4}{3}\right\}$. If the line L also passes through the point $(1,1)$ and touches the circle $17\left(x^2+y^2\right)=16$, then the ecce …

Q9.If the foot of the perpendicular from the point $\mathrm{A}(-1,4,3)$ on the plane $\mathrm{P}:2\mathrm{x}+\mathrm{my}+\mathrm{nz}=4$, is $\left(-2,\frac{7}{2},\frac{3}{2}\right)$, then the distance of the point A from the plane P, measured parallel to a line with directi …

Q10.Let $\vec{a}=3\hat{i}+\hat{j}$ and $\vec{b}=\hat{i}+2\hat{j}+\hat{k}$. Let $\vec{c}$ be a vector satisfying $\vec{a}\times (\vec{b}\times \vec{c})=\vec{b}+\lambda \vec{c}$. If $\vec{b}$ and $\vec{c}$ are non-parallel, then the value of $\lambda$ is :

Q11.The angle of elevation of the top of a tower from a point A due north of it is $\alpha$ and from a point B at a distance of 9 units due west of $A$ is ${\cos }^{-1}\left(\frac{3}{\sqrt{13}}\right)$. If the distance of the point $B$ from the tower is 15 units, then …

Q12.The statement $(\mathrm{p}\wedge \mathrm{q})\Rightarrow (\mathrm{p}\wedge \mathrm{r})$ is equivalent to :

Q13.Let the circumcentre of a triangle with vertices $\mathrm{A}(\mathrm{a},3),\mathrm{B}(\mathrm{b},5)$ and $\mathrm{C}(\mathrm{a},\mathrm{b}),\mathrm{ab}>0$ be $\mathrm{P}(1,1)$. If the line AP intersects the line BC at the point …

Q14.Let $\hat{a}$ and $\hat{b}$ be two unit vectors such that the angle between them is $\frac{\pi }{4}$. If $\theta$ is the angle between the vectors $(\hat{a}+\hat{b})$ and $(\hat{a}+2\hat{b}+2(\hat{a}\times \hat{b}))$, then the value of $164{\cos }^2\theta$ is equ …

Q15.If $\mathrm{f}(\alpha )={\int }_1^{\alpha }\frac{{\log }_{10}t}{1+t}dt,\alpha >0$, then $\mathrm{f}\left({\mathrm{e}}^3\right)+\mathrm{f}\left({\mathrm{e}}^{-3}\right)$ is equal to :

Q16.The area of the region $\left\{(x,y):|x-1|\le y\le \sqrt{5-x^2}\right\}$ is equal to :

Q17.Let the focal chord of the parabola $P:y^2=4x$ along the line $\mathrm{L}:\mathrm{y}=\mathrm{mx}+\mathrm{c},\mathrm{m}>0$ meet the parabola at the points M and N . Let the line L be a tangent to the hyperbola $\mathrm{H}:{\mathrm{x}}^2-{\mathrm{y}}^2=4$. If O is the …

Q18.The number of points, where the function $\mathrm{f}:\mathbf{R}\to \mathbf{R},\mathrm{f}(\mathrm{x})=|\mathrm{x}-1|\cos |\mathrm{x}-2|\sin |\mathrm{x}-1|+(\mathrm{x}-3)\left|{\mathrm{x}}^2-5\mathrm{x}+4\right|$, is NOT differentiable, is :

Q19.Let $\mathrm{S}=\{1,2,3,\dots ,2022\}$. Then the probability, that a randomly chosen number n from the set S such that $\mathrm{HCF}(\mathrm{n},2022)=1$, is :

Q20.Let $f(x)=3^{{\left(x^2-2\right)}^3+4},x\in \mathbf{R}$. Then which of the following statements are true ?
$\mathrm{P}:\mathrm{x}=0$ is a point of local minima of f
$\mathrm{Q}:\mathrm{x}=\sqrt{2}$ is a point of inflection of f
R : ${\mathrm{f}}^{\prime }$ is increasin …

Q21.Let $\mathrm{S}=\{\theta \in (0,2\pi ):7{\cos }^2\theta -3{\sin }^2\theta -2{\cos }^{2}2\theta =2\}$. Then, the sum of roots of all the equations $x^2-2\left({\tan }^2\theta +{\cot }^2\theta \right)x+6{\sin }^2\theta =0\theta \in \mathrm{S}$, is …

Q22.Let the mean and the variance of 20 observations $x_1,x_2,\dots x_{20}$ be 15 and 9 , respectively. For $\alpha \in R$, if the mean of ${\left(x_1+\alpha \right)}^2,{\left(x_2+\alpha \right)}^2,\dots ,{\left(x_{20}+\alpha \right)}^2$ is 178, then the square of the …

Q23.Let a line with direction ratios a, $-4\mathrm{a},-7$ be perpendicular to the lines with direction ratios 3, $-1,2b$ and $b,a,-2$. If the point of intersection of the line $\frac{x+1}{a^2+b^2}=\frac{y-2}{a^2-b^2}=\frac{z}{1}$ and the plane …

Q24.Let $a_1,a_2,a_3,\dots$ be an A.P. If ${\sum }_{r=1}^{\infty }\frac{a_r}{2^r}=4$, then $4a_2$ is equal to ____ .

Q25.Let the ratio of the fifth term from the beginning to the fifth term from the end in the binomial expansion of ${\left(\sqrt[4]{2}+\frac{1}{\sqrt[4]{3}}\right)}^n$, in the increasing powers of $\frac{1}{\sqrt[4]{3}}$ be $\sqrt[4]{6}:1$. If the sixth term from the beginning …

Q26.The number of matrices of order $3\times 3$, whose entries are either 0 or 1 and the sum of all the entries is a prime number, is ______.

Q27.Let $p$ and $p+2$ be prime numbers and let $\Delta =\left|\begin{array}{ccc}p!& (p+1)!& (p+2)!\\ (p+1)!& (p+2)!& (p+3)!\\ (p+2)!& (p+3)!& (p+4)!\end{array}\right|$

Then the sum of the maximum values of $\alpha$ and $\beta$, such that ${\mathrm{p}}^{\alpha }$ and …

Q28.If $\frac{1}{2\times 3\times 4}+\frac{1}{3\times 4\times 5}+\frac{1}{4\times 5\times 6}+\dots +\frac{1}{100\times 101\times 102}=\frac{\mathrm{k}}{101}$, then 34 k is equal to ______.

Q29.Let $\mathrm{S}=\{4,6,9\}$ and $\mathrm{T}=\{9,10,11,\dots ,1000\}$. If …

Q30.Let the mirror image of a circle $c_1:x^2+y^2-2x-6y+\alpha =0$ in line $y=x+1$ be $c_2:5x^2+5y^2+10gx+10\mathrm{fy}+38=0$. If r is the radius of circle ${\mathrm{c}}_2$, then $\alpha +6{\mathrm{r}}^2$ is equal to ______.