JEE Main 2022 · 27 Jul 2022 · Shift 2

Q1.An expression of energy density is given by $\mathrm{u}=\frac{\alpha }{\beta }\sin \left(\frac{\alpha \mathrm{x}}{\mathrm{kt}}\right)$, where $\alpha ,\beta$ are constants, x is displacement, k is Boltzmann constant and t is the temperature. The dimensions of $\beta$ will be …

Q2.A body of mass 10 kg is projected at an angle of $45^{\circ }$ with the horizontal. The trajectory of the body is observed to pass through a point $(20,10)$. If T is the time of flight, then its momentum vector, at time $\mathrm{t}=\frac{\mathrm{T}}{\sqrt{2}}$, is ____
[ …

Q3.A block of mass M slides down on a rough inclined plane with constant velocity. The angle made by the incline plane with horizontal is $\theta$. The magnitude of the contact force will be :

Q4.A block 'A' takes 2 s to slide down a frictionless incline of $30^{\circ }$ and length ' $l$ ', kept inside a lift going up with uniform velocity ' $v$ '. If the incline is changed to $45^{\circ }$, the time taken by the block, to slide down the incline, will be approximate …

Q5.The velocity of the bullet becomes one third after it penetrates 4 cm in a wooden block. Assuming that bullet is facing a constant resistance during its motion in the block. The bullet stops completely after travelling at $(4+\mathrm{x})$ cm inside the block. The value of $x$ …

Q6.A body of mass $m$ is projected with velocity $\lambda v_e$ in vertically upward direction from the surface of the earth into space. It is given that ${\mathrm{v}}_{\mathrm{e}}$ is escape velocity and $\lambda <1$. If air resistance is considered to the negligible, then th …

Q7.A steel wire of length $3.2\mathrm{m}$ (${\mathrm{Y}}_{\mathrm{S}}=2.0\times 10^{11}{\mathrm{Nm}}^{-2}$) and a copper wire of length 4.4 M ( ${\mathrm{Y}}_{\mathrm{C}}=1.1\times 10^{11}{\mathrm{Nm}}^{-2}$ ), both of radius 1.4 mm are connected end to end. When stretched by a …

Q8.In $1^{\text{st }}$ case, Carnot engine operates between temperatures 300 K and 100 K . In $2^{\text{nd }}$ case, as shown in the figure, a combination of two engines is used. The efficiency of this combination (in $2^{\text{nd }}$ case) will be :

Q9.Which statements are correct about degrees of freedom?
A. A molecule with $n$ degrees of freedom has $n^2$ different ways of storing energy.
B. Each degree of freedom is associated with $\frac{1}{2}\mathrm{RT}$ average energy per mole.
C. A monoatomic gas molecule has 1 …

Q10.A charge of $4\mu \mathrm{C}$ is to be divided into two. The distance between the two divided charges is constant. The magnitude of the divided charges so that the force between them is maximum, will be:

Q11.A. The drift velocity of electrons decreases with the increase in the temperature of conductor.
B. The drift velocity is inversely proportional to the area of cross-section of given conductor.
C. The drift velocity does not depend on the applied potential difference to the conduc …

Q12.A compass needle of oscillation magnetometer oscillates 20 times per minute at a place P of dip $30^{\circ }$. The number of oscillations per minute become 10 at another place Q of $60^{\circ }$ dip. The ratio of the total magnetic field at the two places …

Q13.A cyclotron is used to accelerate protons. If the operating magnetic field is 1.0 T and the radius of the cyclotron 'dees' is 60 cm, the kinetic energy of the accelerated protons in MeV will be:
[use …

Q14.A series LCR circuit has $\mathrm{L}=0.01\mathrm{H},\mathrm{R}=10\Omega$ and $\mathrm{C}=1\mu \mathrm{F}$ and it is connected to ac voltage of amplitude (${\mathrm{V}}_{\mathrm{m}}$) 50 V. At frequency $60\%$ lower than resonant frequency, the amplitude of current will be appr …

Q15.Identify the correct statements from the following descriptions of various properties of electromagnetic waves.
A. In a plane electromagnetic wave electric field and magnetic field must be perpendicular to each other and direction of propagation of wave should be along electric f …

Q16.Two coherent sources of light interfere. The intensity ratio of two sources is $1:4$. For this interference pattern if the value of $\frac{I_{\text{max}}+I_{\text{min}}}{I_{\text{max}}-I_{\text{min}}}$ is equal to $\frac{2\alpha +1}{\beta +3}$, then $\frac{\alpha }{\beta }$ will be:

Q17.With reference to the observations in photo-electric effect, identify the correct statements from below:
A. The square of maximum velocity of photoelectrons varies linearly with frequency of incident light.
B. The value of saturation current increases on moving the source of ligh …

Q18.The activity of a radioactive material is $6.4\times 10^{-4}$ curie. Its half life is 5 days. The activity will become $5\times 10^{-6}$ curie after:

Q19.For a constant collector-emitter voltage of 8 V, the collector current of a transistor reached to the value of 6 mA from 4 mA, whereas base current changed from $20\mu \mathrm{A}$ to $25\mu \mathrm{A}$ value. If transistor is in active state, small signal current gain (curren …

Q20.A square wave of the modulating signal is shown in the figure. The carrier wave is given by $C(t)=5\sin (8\pi t)$ Volt. The modulation index is:

Q21.In an experiment to determine the Young's modulus, steel wires of five different lengths ($1,2,3,4$ and 5 m) but of same cross section ($2{\mathrm{mm}}^2$) were taken and curves between extension and load were obtained. The slope (extension/load) of the curves were plotted with …

Q22.In the given figure of meter bridge experiment, the balancing length AC corresponding to null deflection of the galvanometer is 40 cm. The balancing length, if the radius of the wire AB is doubled, will be ______ cm.

Q23.A thin prism of angle $6^{\circ }$ and refractive index for yellow light (${\mathrm{n}}_{\mathrm{Y}}$) 1.5 is combined with another prism of angle $5^{\circ }$ and ${\mathrm{n}}_{\mathrm{Y}}=1.55$. The combination produces no dispersion. The net average deviation ($\delta$) produced by …

Q24.A conducting circular loop is placed in $\mathrm{X}-\mathrm{Y}$ plane in presence of magnetic field $\overrightarrow{\mathrm{B}}=\left(3{\mathrm{t}}^3\hat{\mathrm{j}}+3{\mathrm{t}}^2\hat{\mathrm{k}}\right)$ in SI unit. If the radius of the loop is 1 m, the induced emf in the …

Q25.As show in the figure, in steady state, the charge stored in the capacitor is. ______ ×10⁻⁶ C.

Q26.A parallel plate capacitor with width 4 cm, length 8 cm and separation between the plates of 4 mm is connected to a battery of 20 V. A dielectric slab of dielectric constant 5 having length 1 cm, width 4 cm and thickness 4 mm is inserted between the plates of parallel plate capac …

Q27.A wire of length 30 cm, stretched between rigid supports, has it's nᵗʰ and (n+1)ᵗʰ harmonics at 400 Hz and 450 Hz, respectively. If tension in the string is 2700 N, it's linear mass density is ______ kg/m.

Q28.A spherical soap bubble of radius 3 cm is formed inside another spherical soap bubble of radius 6 cm. If the internal pressure of the smaller bubble of radius 3 cm in the above system is equal to the internal pressure of the another single soap bubble of radius r cm. The value of …

Q29.A solid cylinder length is suspended symmetrically through two massless strings, as shown in the figure. The distance from the initial rest position, the cylinder should by unbinding the strings to achieve a speed of 4 ms⁻¹, is. ______ cm.
(take g=10 ms⁻²)

Q30.Two inclined planes are placed as shown in figure. A block is projected from the Point A of inclined plane AB along its surface with a velocity just sufficient to carry it to the top Point B at a height 10 m. After reaching the Point B the block slides down on inclined plane BC. …

Q1.The correct decreasing order of energy, for the orbitals having, following set of quantum numbers:
(A) $\mathrm{n}=3,\mathrm{l}=0,\mathrm{m}=0$
(B) $n=4,1=0,m=0$
(C) $\mathrm{n}=3,\mathrm{l}=1,\mathrm{m}=0$
(D) $\mathrm{n}=3,\mathrm{l}=2,\mathrm{m}=1$

Q2.Match List-I with List-II

List-I:
(A) ${\Psi }_{\mathrm{MO}}={\Psi }_{\mathrm{A}}-{\Psi }_{\mathrm{B}}$
(B) $\mu =\mathrm{Q}\times \mathrm{r}$
(C) $\frac{{\mathrm{N}}_{\mathrm{b}}-{\mathrm{N}}_{\mathrm{a}}}{2}$
(D) ${\Psi }_{\mathrm{MO}}={\Psi }_{\mathrm{A}}+{\Psi }_{\mathrm{B}}$

List-II:
(I …

Q3.The Plot of pH-metric titration of weak base ${\mathrm{NH}}_4\mathrm{OH}$ vs strong acid HCl looks like:

Q4.Given below are two statements:

Statement I: For KI, molar conductivity increases steeply with dilution.
Statement II: For carbonic acid, molar conductivity increases slowly with dilution.
In the light of the above statements, choose the correct answer from the options given bel …

Q5.Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R)

Assertion (A): Dissolved substances can be removed from a colloidal solution by diffusion through a parchment paper.
Reason (R): Particles in a true solution cannot pass thro …

Q6.Outermost electronic configurations of four elements $\mathrm{A},\mathrm{B},\mathrm{C},\mathrm{D}$ are given below:
(A) $3{\mathrm{s}}^2$
(B) $3s^{2}3p^1$
(C) $3s^{2}3p^3$
(D) $3s^{2}3p^4$

The correct order of first ionization enthalpy for them is:

Q7.An element A of group 1 shows similarity to an element B belonging to group 2. If A has maximum hydration enthalpy in group 1 then B is:

Q8.Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R)

Assertion (A): Boron is unable to form ${\mathrm{BF}}_6^{3-}$
Reason (R): Size of B is very small.
In the light of the above statements, choose the correct answer from the …

Q9.In neutral or alkaline solution, ${\mathrm{MnO}}_4^{-}$ oxidises thiosulphate to:

Q10.Low oxidation state of metals in their complexes are common when ligands:

Q11.Given below are two statements:

Statement I: The non bio-degradable fly ash and slag from steel industry can be used by cement industry.
Statement II: The fuel obtained from plastic waste is lead free.
In the light of the above statements, choose the most appropriate answer from …

Q12.The structure of A in the given reaction is:

Q13.Major product 'B' of the following reaction sequence is:

Q14.Match List-I with List-II.

List-I
(A) (see structure)
(B) (see structure)
(C) (see structure)
(D) (see structure)

List-II
(I) Gatterman Koch reaction
(II) Etard reaction
(III) Stephen reaction
(IV) Rosenmund reaction

Choose the correct answer from the options given below:

Q15.Match List-I with List-II.

List-I (Polymer)
(A) Neoprene
(B) Teflon
(C) Acrilan
(D) Natural rubber

List-II (Monomer)
(I) Acrylonitrile
(II) Chloroprene
(III) Tetrafluoroethene
(IV) Isoprene

Choose the correct answer from the option given below:

Q16.An organic compound 'A' contains nitrogen and chlorine. It dissolves readily in water to give a solution that turns litmus red. Titration of compound 'A' with standard base indicates that the molecular weight of 'A' is $131\pm 2$. When a sample of 'A' is treated with aq. NaOH, a …

Q17.Match List-I with List-II

List-I
(A) Glucose + HI
(B) Glucose + ${\mathrm{Br}}_2$ water
(C) Glucose + acetic anhydride
(D) Glucose + ${\mathrm{HNO}}_3$

List-II
(I) Gluconic acid
(II) Glucose pentacetate
(III) Saccharic acid
(IV) Hexane

Choose the correct answer from the option …

Q18.Which of the following enhances the lathering property of soap?

Q19.Match List-I with List-II

List-I (Mixture)
(A) Chloroform & Aniline
(B) Benzoic acid & Napthalene
(C) Water & Aniline
(D) Napthalene & Sodium chloride

List-II (Purification Process)
(I) Steam distillation
(II) Sublimation
(III) Distillation
(IV) Crystallisation

Q20.${\mathrm{Fe}}^{3+}$ cation gives a prussian blue precipitate on addition of potassium ferrocyanide solution due to the formation of:

Q21.The normality of ${\mathrm{H}}_2{\mathrm{SO}}_4$ in the solution obtained on mixing 100 mL of $0.1\mathrm{M}{\mathrm{H}}_2{\mathrm{SO}}_4$ with 50 mL of 0.1 M NaOH is $\_\_\_\_$ $\times 10^{-1}\mathrm{N}$. (Nearest Integer)

Q22.For a real gas at $25^{\circ }\mathrm{C}$ temperature and high pressure (99 bar) the value of compressibility factor is 2, so the value of Vander Waal's constant 'b' should be $\_\_\_\_$ $\times 10^{-2}\mathrm{L}{\mathrm{mol}}^{-1}$ (Nearest integer)
(Given …

Q23.A gas (Molar mass $=280\mathrm{g}{\mathrm{mol}}^{-1}$) was burnt in excess ${\mathrm{O}}_2$ in a constant volume calorimeter and during combustion the temperature of calorimeter increased from 298.0 K to 298.45 K. If the heat capacity of calorimeter is …

Q24.When a certain amount of solid A is dissolved in 100 g of water at $25^{\circ }\mathrm{C}$ to make a dilute solution, the vapour pressure of the solution is reduced to one-half of that of pure water. The vapour pressure of pure water is 23.76 mmHg. The number of moles of solute A …

Q25.[A] → [B]
Reactant Product

If formation of compound [B] follows the first order of kinetics and after 70 minutes the concentration of $[\mathrm{A}]$ was found to be half of its initial concentration. Then the rate constant of the reaction is …

Q26.Among the following ores Bauxite, Siderite, Cuprite, Calamine, Haematite, Kaolinite, Malachite, Magnetite, Sphalerite, Limonite, Cryolite, the number of principal ores if (of) iron is ____.

Q27.The oxidation state of manganese in the product obtained in a reaction of potassium permanganate and hydrogen peroxide in basic medium is ____.

Q28.The number of molecule(s) or ion(s) from the following having non-planar structure is ____.
$\mathrm{NO}{}_3{}^{-}$, $\mathrm{H}{}_2\mathrm{O}{}_2$, $\mathrm{BF}{}_3$, $\mathrm{PCl}{}_3$, $\mathrm{XeF}{}_4$, $\mathrm{SF}{}_4$, $\mathrm{XeO}{}_3$, $\mathrm{PH}{}_4{}^{+}$, $\mathrm{SO}{}_3$, $[\mathrm{Al}(\mathrm{OH}){}_4]{}^{-}$

Q29.The spin only magnetic moment of the complex present in Fehling's reagent is ____ B.M. (Nearest integer).

Q30.In the above reaction, 5 g of toluene is converted into benzaldehyde with 92% yield. The amount of benzaldehyde produced is ____ $\times 10^{-2}$ g. (Nearest integer)

Q1.The domain of the function
$f(x)={\sin }^{-1}\left[2x^2-3\right]+{\log }_2\left({\log }_{\frac{1}{2}}\left(x^2-5x+5\right)\right)$,
where $[t]$ is the greatest integer function, is :

Q2.Let S be the set of all $(\alpha ,\beta ),\pi <\alpha ,\beta <2\pi$, for which the complex number $\frac{1-i\sin \alpha }{1+2i\sin \alpha }$ is purely imaginary and $\frac{1+i\cos \beta }{1-2i\cos \beta }$ is purely real. Let …

Q3.If $\alpha ,\beta$ are the roots of the equation $x^2-\left(5+3^{\sqrt{{\log }_{3}5}}-5^{\sqrt{{\log }_{5}3}}\right)x+3\left(3^{({\log }_{3}5{)}^{1/3}}-5^{({\log }_{5}3{)}^{2/3}}-1\right)=0$, then the equation whose roots are $\alpha +\frac{1}{\beta }$ and $\beta +\frac{1}{\alpha }$ is:

Q4.Let $\mathrm{A}=\left(\begin{array}{cc}4& -2\\ \alpha & \beta \end{array}\right)$

If ${\mathrm{A}}^2+\gamma \mathrm{A}+18\mathrm{I}=\mathrm{O}$, then $\det (\mathrm{A})$ is equal to
____ .

Q5.If for $\mathrm{p}\ne \mathrm{q}\ne 0$, then function
$f(x)=\frac{\sqrt[7]{p(729+x)}-3}{\sqrt[3]{729+qx}-9}$ is continuous at $x=0$, then:

Q6.Let $\mathrm{f}(\mathrm{x})=2+|\mathrm{x}|-|\mathrm{x}-1|+|\mathrm{x}+1|,\mathrm{x}\in \mathbf{R}$. Consider
(Sl): …

Q7.Let the sum of an infinite G.P., whose first term is a and the common ratio is r , be 5 . Let the sum of its first five terms be $\frac{98}{25}$. Then the sum of the first 21 terms of an AP, whose first term is $10\mathrm{ar},{\mathrm{n}}^{\text{th }}$ term is …

Q8.The area of the region enclosed by
$\mathrm{y}\le 4{\mathrm{x}}^2,{\mathrm{x}}^2\le 9\mathrm{y}$ and $\mathrm{y}\le 4$, is equal to :

Q9.${\int }_0^2\left(\left|2x^2-3x\right|+\left[x-\frac{1}{2}\right]\right)dx$,
where $[t]$ is the greatest integer function, is equal to:

Q10.Consider a curve $y=y(x)$ in the first quadrant as shown in the figure. Let the area ${\mathrm{A}}_1$ is twice the area ${\mathrm{A}}_2$. Then the normal to the curve perpendicular to the line $2\mathrm{x}-12\mathrm{y}=15$ does NOT pass through the point.

Q11.The equations of the sides $\mathrm{AB},\mathrm{BC}$ and CA of a triangle ABC are $2\mathrm{x}+\mathrm{y}=0,x+\mathrm{py}=39$ and $\mathrm{x}-\mathrm{y}=3$ respectively and $\mathrm{P}(2,3)$ is its circumcentre. Then which of the following is NOT true :

Q12.A circle ${\mathrm{C}}_1$ passes through the origin O and has diameter 4 on the positive x -axis. The line $\mathrm{y}=2\mathrm{x}$ gives a chord OA of a circle ${\mathrm{C}}_1$. Let ${\mathrm{C}}_2$ be the circle with OA as a diameter. If the tangent to …

Q13.If the length of the latus rectum of a parabola, whose focus is (a, a) and the tangent at its vertex is $x+y=a$, is 16, then $|a|$ is equal to :

Q14.If the length of the perpendicular drawn from the point $P(a,4,2)$, $a>0$ on the line $\frac{x+1}{2}=\frac{y-3}{3}=\frac{z-1}{-1}$ is $2\sqrt{6}$ units and $Q({\alpha }_1,{\alpha }_2,{\alpha }_3)$ is the image of the point P in this line, then $a+{\sum }_{i=1}^3{\alpha }_i$ is eq …

Q15.If the line of intersection of the planes $ax+by=3$ and $ax+by+cz=0$, $a>0$ makes an angle $30^{\circ }$ with the plane $y-z+2=0$, then the direction cosines of the line are :

Q16.Let X have a binomial distribution $B(n,p)$ such that the sum and the product of the mean and variance of X are 24 and 128 respectively. If $P(X>n-3)=\frac{k}{2^n}$, then k is equal to

Q17.A six faced die is biased such that $3\times P(\text{a prime number})=6\times P(\text{a composite number})=2\times P(1)$. Let X be a random variable that counts the number of times one gets a perfect square on some throws of this die. If the die is thrown twice, then the m …

Q18.The angle of elevation of the top P of a vertical tower PQ of height 10 from a point A on the horizontal ground is $45^{\circ }$. Let R be a point on AQ and from a point B, vertically above R, the angle of elevation of P is $60^{\circ }$. If $\angle BAQ=30^{\circ }$, $AB=d$ and the …

Q19.Let $S=\left\{\theta \in \left(0,\frac{\pi }{2}\right):{\sum }_{m=1}^9\sec \left(\theta +(m-1)\frac{\pi }{6}\right)\sec \left(\theta +\frac{m\pi }{6}\right)=-\frac{8}{\sqrt{3}}\right\}$

Then

Q20.If the truth value of the statement $(P\wedge (\sim R))\to ((\sim R)\wedge Q)$ is F, then the truth value of which of the following is F ?

Q21.Consider a matrix $A=\left[\begin{array}{ccc}\alpha & \beta & \gamma \\ {\alpha }^2& {\beta }^2& {\gamma }^2\\ \beta +\gamma & \gamma +\alpha & \alpha +\beta \end{array}\right]$, where $\alpha ,\beta ,\gamma$ are three distinct natural numbers. If …

Q22.The number of functions f, from the set $A=\left\{x\in N:x^2-10x+9\le 0\right\}$ to the set $B=\left\{n^2:n\in N\right\}$ such that $f(x)\le (x-3{)}^2+1$, for every $x\in A$, is ______.

Q23.Let for the $9^{\text{th}}$ term in the binomial expansion of $(3+6x{)}^n$, in the increasing powers of $6x$, to be the greatest for $x=\frac{3}{2}$, the least value of n is $n_0$. If k is the ratio of the coefficient of $x^6$ to the coefficient of $x^3$, then $k+n_0$ is …

Q24.$\frac{2^3-1^3}{1\times 7}+\frac{4^3-3^3+2^3-1^3}{2\times 11}+\frac{6^3-5^3+4^3-3^3+2^3-1^3}{3\times 15}+\dots +\frac{30^3-29^3+28^3-27^3+\dots +2^3-1^3}{15\times 63}$ is equal to

Q25.A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semivertical angle is ${\tan }^{-1}\frac{3}{4}$. Water is poured in it at a constant rate of 6 cubic meter per hour. The rate (in square meter per hour), at which the wet curved surface …

Q26.For the curve $C:(x^2+y^2-3)+(x^2-y^2-1{)}^5=0$, the value of $3y^{\prime }-y^3{y}^{\prime \prime }$, at the point $(\alpha ,\alpha )$, $\alpha >0$, on $C$, is equal to ____.

Q27.Let $f(x)=\min \{[x-1],[x-2],\dots ,[x-10]\}$ where $[t]$ denotes the greatest integer $\le t$. Then ${\int }_0^{10}f(x)dx+{\int }_0^{10}(f(x){)}^{2}dx+{\int }_0^{10}|f(x)|dx$ is equal to ____.

Q28.Let $f$ be a differentiable function satisfying $f(x)=\frac{2}{\sqrt{3}}{\int }_0^{\sqrt{3}}f\left(\frac{{\lambda }^{2}x}{3}\right)d\lambda$, $x>0$ and $f(1)=\sqrt{3}$. If $y=f(x)$ passes through the point $(\alpha ,6)$, then $\alpha$ is equal to ____.

Q29.A common tangent $T$ to the curves $C_1:\frac{x^2}{4}+\frac{y^2}{9}=1$ and $C_2:\frac{x^2}{42}-\frac{y^2}{143}=1$ does not pass through the fourth quadrant. If $T$ touches $C_1$ at $(x_1,y_1)$ and $C_2$ at $(x_2,y_2)$, then $|2x_1+x_2|$ is equal to ____.

Q30.Let $\vec{a},\vec{b},\vec{c}$ be three non-coplanar vectors such that $\vec{a}\times \vec{b}=4\vec{c}$, $\vec{b}\times \vec{c}=9\vec{a}$ and $\vec{c}\times \vec{a}=\alpha \vec{b}$, $\alpha >0$. If $|\vec{a}|+|\vec{b}|+|\vec{c}|=36$, then $\alpha$ is equal to ____.