JEE Main 2023 · 13 Apr 2023 · Shift 1

Q1.Which of the following Maxwell's equations is valid for time varying conditions but not valid for static conditions :

Q2.Different combination of 3 resistors of equal resistance $R$ are shown in the figures.
The increasing order for power dissipation is:

Q3.A vessel of depth 'd' is half filled with oil of refractive index ${\mathrm{n}}_1$ and the other half is filled with water of refractive index $n_2$. The apparent depth of this vessel when viewed from above will be-

Q4.The source of time varying magnetic field may be
(A) a permanent magnet
(B) an electric field changing linearly with time
(C) direct current
(D) a decelerating charge particle
(E) an antenna fed with a digital signal

Choose the correct answer from the options given below:

Q5.Two trains ' A ' and ' B ' of length ' $l$ ' and ' $4l$ ' are travelling into a tunnel of length ' $L$ ' in parallel tracks from opposite directions with velocities $108\mathrm{km}/\mathrm{h}$ and $72\mathrm{km}/\mathrm{h}$, respectively. If train 'A' takes 35s …

Q6.The ratio of powers of two motors is $\frac{3\sqrt{x}}{\sqrt{x}+1}$, that are capable of raising 300 kg water in 5 minutes and 50 kg water in 2 minutes respectively from a well of 100 m deep. The value of x will be

Q7.A planet having mass 9 Me and radius $4{\mathrm{R}}_{\mathrm{e}}$, where Me and Re are mass and radius of earth respectively, has escape velocity in $\mathrm{km}/\mathrm{s}$ given by: (Given escape velocity on earth
$V_e=11.2\times 10^3\mathrm{m}/\mathrm{s}$ )

Q8.The difference between threshold wavelengths for two metal surfaces A and B having work function ${\varphi }_{\mathrm{A}}=9\mathrm{eV}$ and ${\varphi }_{\mathrm{B}}=4.5\mathrm{eV}$ in nm is:
(Given, hc $=1242\mathrm{eV}\mathrm{nm}$ )

Q9.A bullet 10 g leaves the barrel of gun with a velocity of $600\mathrm{m}/\mathrm{s}$. If the barrel of gun is 50 cm long and mass of gun is 3 kg , then value of impulse supplied to the gun will be :

Q10.Two charges each of magnitude 0.01 C and separated by a distance of 0.4 mm constitute an electric dipole. If the dipole is placed in an uniform electric field ' $\overrightarrow{\mathrm{E}}$ ' of 10 dyne/C making $30^0$ angle with $\overrightarrow{\mathrm{E}}$, the magnit …

Q11.A disc is rolling without slipping on a surface. The radius of the disc is R . At $\mathrm{t}=0$, the top most point on the disc is A as shown in figure. When the disc completes half of its rotation, the displacement of point A from its initial position is

Q12.The rms speed of oxygen molecule in a vessel at particular temperature is ${\left(1+\frac{5}{x}\right)}^{\frac{1}{2}}v$, where $v$ is the average speed of the molecule. The value of $x$ will be:(Take $\pi =\frac{22}{7}$ )

Q13.A body of mass $(5\pm 0.5)$ kg is moving with a velocity of $(20\pm 0.4)$ m/s. Its kinetic energy will be

Q14.Two bodies are having kinetic energies in the ratio $16:9$. If they have same linear momentum, the ratio of their masses respectively is :

Q15.The figure shows a liquid of given density flowing steadily in horizontal tube of varying cross-section. Cross sectional areas at A is $1.5$ cm${}^2$, and B is $25$ mm${}^2$, if the speed of liquid at B is $60$ cm/s then $(P_A-P_B)$ is :
(Given $P_A$ and $P_B$ are liquid pressures …

Q16.Under isothermal condition, the pressure of a gas is given by $P=a{V}^{-3}$, where a is a constant and V is the volume of the gas. The bulk modulus at constant temperature is equal to

Q17.For the following circuit and given inputs A and B, choose the correct option for output 'Y'

Q18.Which graph represents the difference between total energy and potential energy of a particle executing SHM Vs it's distance from mean position?

Q19.${}_{92}^{238}A{\to }_{90}^{234}B{+}_2^{4}D+Q$

In the given nuclear reaction, the approximate amount of energy released will be :
[Given, mass of ${}_{92}^{238}A=238.05079\times 931.5$ MeV/c${}^2$,
mass of ${}_{90}^{234}B=234.04363\times 931.5$ MeV/c${}^2$
mass of …

Q20.The elastic potential energy stored in a steel wire of length 20 m stretched through 2 cm is 80 J. The cross sectional area of the wire is ______ mm${}^2$.
(Given, $Y=2.0\times 10^{11}$ Nm${}^{-2}$)

Q21.A potential $V_0$ is applied across a uniform wire of resistance R. The power dissipation is $P_1$. The wire is then cut into two equal halves and a potential of $V_0$ is applied across the length of each half. The total power dissipation across two wires is $P_2$. The ratio …

Q22.At a given point of time the value of displacement of a simple harmonic oscillator is given as $y=A\cos (30^0)$. If amplitude is 40 cm and kinetic energy at that time is 200 J, the value of force constant is $1.0\times 10^x$ Nm${}^{-1}$. The value of $x$ is ______.

Q23.When a resistance of $5\Omega$ is shunted with a moving coil galvanometer, it shows a full scale deflection for a current of 250 mA, however when $1050\Omega$ resistance is connected with it in series, it gives full scale deflection for 25 volt. The resistance of galvanometer is …

Q24.The radius of $2^{\text{nd}}$ orbit of ${\text{He}}^{+}$ of Bohr's model is $r_1$ and that of fourth orbit of ${\text{Be}}^{3+}$ is represented as $r_2$. Now the ratio $\frac{r_2}{r_1}$ is $x:1$. The value of $x$ is ______.

Q25.A thin infinite sheet charge and an infinite line charge of respective charge densities $+\sigma$ and $+\lambda$ are placed parallel at 5 m distance from each other. Points 'P' and 'Q' are at $\frac{3}{\pi }\mathrm{m}$ and $\frac{4}{\pi }\mathrm{m}$ perpendicular distan …

Q26.In the given figure, an inductor and a resistor are connected in series with a battery of emf E volt. $\frac{E^a}{2b}\mathrm{J}/\mathrm{s}$ represents the maximum rate at which the energy is stored in the magnetic field (inductor). The numerical value of $\frac{b}{a}$ …

Q27.A fish rising vertically upward with a uniform velocity of $8{\mathrm{ms}}^{-1}$, observes that a bird is diving vertically downward towards the fish with the velocity of $12{\mathrm{ms}}^{-1}$. If the refractive index of water is $\frac{4}{3}$, then the actual velocity of …

Q28.A solid sphere is rolling on a horizontal plane without slipping. If the ratio of angular momentum about axis of rotation of the sphere to the total energy of moving sphere is $\pi :22$ then, the value of its angular speed will be ______ $\mathrm{rad}/\mathrm{s}$.

Q29.From the given transfer characteristic of a transistor in CE configuration, the value of power gain of this configuration is $10^{\mathrm{x}}$, for ${\mathrm{R}}_{\mathrm{B}}=10\mathrm{k}\Omega$, and ${\mathrm{R}}_{\mathrm{C}}=1\mathrm{k}\Omega$. The value of x is ______.

Q1.In the reaction given below
' A ' is

Q2.Given below are two statements:

Statement-I Permutit process is more efficient compared to the synthetic resin method for the softening of water.
Statement-II: Synthetic resin method results in the formation of soluble sodium salts.

In the light of the above statements, choose …

Q3.The mismatched combinations are
A. Chlorophyll - Co
B. Water hardness - EDTA
C. Photography - ${\left[\mathrm{Ag}(\mathrm{CN}{)}_2\right]}^{-}$
D. Wilkinson catalyst - $\left[{\left(P{h}_{3}P\right)}_{3}RhCl\right]$
E. Chelating ligand - D - Penicillamine

Choose the correct …

Q4.In which of the following processes, the bond order increases and paramagnetic character changes to diamagnetic one?

Q5.The incorrect statement from the following for borazine is:

Q6.Among the following compounds, the one which shows highest dipole moment is

Q7.Match the following
List-I:
(A) Nylon 6
(B) Vulcanized Rubber
(C) cis-1,4-polyisoprene
(D) Polychloroprene
List-II:
(I) Natural Rubber
(II) Cross Linked
(III) Caprolactam
(IV) Neoprene

Q8.In the above reaction. Left hand side and right hand side rings are named as ' A ' and ' B ' respectively. They undergo ring expansion. The correct statement for this process is:

Q9.The radical which mainly causes ozone depletion in the presence of UV radiations is:

Q10.In the following reaction ' X ' is ${\mathrm{CH}}_3{\left({\mathrm{CH}}_2\right)}_4{\mathrm{CH}}_3\underset{\text{ HCl, }\Delta }{\overset{\text{ Anhy. }{\mathrm{AlCl}}_3}{\to }}{}_{\text{major product}}\mathrm{X}$

Q11.2-Methyl propyl bromide reacts with ${\mathrm{C}}_2{\mathrm{H}}_5{\mathrm{O}}^{-}$and gives 'A' whereas on reaction with ${\mathrm{C}}_2{\mathrm{H}}_5\mathrm{OH}$ it gives ' $B$ '. The mechanism followed in these reactions and the products 'A' and 'B' respectively are:

Q12.D- (+)- Glyceraldehyde $\underset{\text{ (ii) }{\mathrm{H}}_2\mathrm{O}/{\mathrm{H}}^{+}}{\overset{\text{(i) }\mathrm{HCN}}{\to }}$
(iii) ${\mathrm{HNO}}_3$

The products formed in the above reaction are

Q13.Which one of the following is most likely a mismatch?

Q14.${\mathrm{ClF}}_5$ at room temperature is a:

Q15.$\mathrm{Be}(\mathrm{OH}{)}_2$ react with $\mathrm{Sr}(\mathrm{OH}{)}_2$ to yield an ionic salt. Choose the incorrect option related to this reaction from the following:

Q16.In the reaction given below

(see structure)

' B ' is:

Q17.The energy of an electron in the first Bohr orbit of hydrogen atom is $-2.18\times 10^{-18}J$. Its energy in the third Bohr orbit is ______.

Q18.What happens when a lyophilic sol is added to a lyophobic sol?

Q19.The pair of lanthanides in which both elements have high third-ionization energy is:

Q20.For the given reaction

(see structure)

The total number of possible products formed by tertiary carbocation of A is ______.

Q21.Solution of 12 g of non - electrolyte (A) prepared by dissolving it in 1000 mL of water exerts the same osmotic pressure as that of 0.05 M glucose solution at the same temperature. The empirical formula of A is ${\mathrm{CH}}_2\mathrm{O}$. The molecular mass of A is ______ g. (N …

Q22.25.0 mL of $0.050\mathrm{M}\mathrm{Ba}({\mathrm{NO}}_3{)}_2$ is mixed with 25.0 mL of 0.020 M NaF. ${\mathrm{K}}_{\mathrm{sp}}$ of ${\mathrm{BaF}}_2$ is $0.5\times 10^{-6}$ at 298 K. The ratio of $[B{a}^{2+}][F^{-}{]}^2$ and ${\mathrm{K}}_{\mathrm{sp}}$ is ______ (Nearest integer)

Q23.For the reaction ${\mathrm{A}}_2+{\mathrm{B}}_2\to 2AB$, the enthalpy change of the reaction as written is $\Delta {\mathrm{H}}^0=-200\mathrm{kJ}\mathrm{mo}{\mathrm{l}}^{-1}$. AB, ${\mathrm{A}}_2$ and ${\mathrm{B}}_2$ are diatomic molecules. If the bond enthalpies of ${\mathrm{A}}_2$, …

Q24.An organic compound gives 0.220 g of ${\mathrm{CO}}_2$ and 0.126 g of ${\mathrm{H}}_2\mathrm{O}$ on complete combustion. If the % of carbon is 24 then the % hydrogen is ______ $\times 10^{-1}$. (Nearest integer)

Q25.20 mL of calcium hydroxide was consumed when it was reacted with 10 mL of unknown solution of ${\mathrm{H}}_2{\mathrm{SO}}_4$. Also 20 mL standard solution of 0.5 M HCl containing 2 drops of phenolphthalein was titrated with calcium hydroxide the mixture showed pink colour when …

Q26.A certain quantity of real gas occupies a volume of $0.15{\mathrm{dm}}^3$ at 100 atm and 500 K when its compressibility factor is 1.07. Its volume at 300 atm and 300 K (When its compressibility factor is 1.4) is $\_\_\_\_$ $\times 10^{-4}{\mathrm{dm}}^3$ (Nearest integer)

Q27.$t_{87.5}$ is the time required for the reaction to undergo $87.5\%$ completion and $t_{50}$ is the time required for the reaction to undergo 50% completion. The relation between $t_{87.5}$ and $t_{50}$ for a first order reaction is …

Q28.${\mathrm{KMnO}}_4$ is titrated with ferrous ammonium sulphate hexahydrate in presence of dilute ${\mathrm{H}}_2{\mathrm{SO}}_4$. Number of water molecules produced for 2 molecules of ${\mathrm{KMnO}}_4$ is $\_\_\_\_$.

Q29.A metal surface of $100{\mathrm{cm}}^2$ area has to be coated with nickel layer of thickness 0.001 mm. A current of 2 A was passed through a solution of $\mathrm{Ni}{\left({\mathrm{NO}}_3\right)}_2$ for '$x$' seconds to coat the desired layer. The value of $x$ is $\_\_\_\_$ (Ne …

Q3.The set of all $\mathrm{a}\in \mathbb{R}$ for which the equation $\mathrm{x}|\mathrm{x}-1|+|\mathrm{x}+2|+\mathrm{a}=0$ has exactly one real root is :

Q4.The negation of the statement
$((\mathrm{A}\wedge (\mathrm{B}\vee \mathrm{C}))\Rightarrow (\mathrm{A}\vee \mathrm{B}))\Rightarrow \mathrm{A}$ is

Q5.The distance of the point $(-1,2,3)$ from the plane $\overrightarrow{\mathrm{r}}.(\hat{\mathrm{i}}-2\hat{\mathrm{j}}+3\hat{\mathrm{k}})=10$ parallel to the line of the shortest distance between the lines $\vec{r}=(\hat{i}-\hat{j})+\lambda (2\hat{i}+\hat{k})$ and …

Q6.A coin is biased so that the head is 3 times as likely to occur as tail. This coin is tossed until a head or three tails occur. If X denotes the number of tosses of the coin, then the mean of $X$ is

Q7.For the system of linear equations
$2\mathrm{x}+4\mathrm{y}+2\mathrm{az}=\mathrm{b}$
$x+2y+3z=4$
$2x-5y+2z=8$
which of the following is NOT correct?

Q8.For the differentiable function
$\mathrm{f}:\mathbb{R}-\{0\}\to \mathbb{R}$, let $3\mathrm{f}(\mathrm{x})+2\mathrm{f}\left(\frac{1}{\mathrm{x}}\right)=\frac{1}{\mathrm{x}}-10$, then $\left|f(3)+f^{\prime }\left(\frac{1}{4}\right)\right|$ is equal to

Q9.Let the tangent and normal at the point $(3\sqrt{3},1)$ on the ellipse $\frac{x^2}{36}+\frac{y^2}{4}=1$ meet the $y$-axis at the points A and B respectively. Let the circle C be drawn taking AB as a diameter and the line $\mathrm{x}=2\sqrt{5}$ intersect C at the po …

Q10.The area of the region enclosed by the curve $f(x)=\max \{\sin x,\cos x\},-\pi \le x\le \pi$ and the $x$-axis is

Q11.The number of symmetric matrices of order 3 , with all the entries from the set $\{0,1,2,3,4,5,6,7,8,9\}$, is :

Q12.Among :
(S1) : ${\lim }_{\mathrm{n}\to \infty }\frac{1}{{\mathrm{n}}^2}(2+4+6+\dots \dots \dots +2\mathrm{n})=1$
(S2) : …

Q13.Let PQ be a focal chord of the parabola $y^2=36x$ of length 100, making an acute angle with the positive x-axis. Let the ordinate of P be positive and M be the point on the line segment PQ such that $\mathrm{PM}:\mathrm{MQ}=3:1$. Then which of the following points does NO …

Q14.For $x\in \mathbb{R}$, two real valued functions $f(x)$ and $g(x)$ are such that $g(x)=\sqrt{x}+1$ and $f\circ g(x)=x+3-\sqrt{x}$. Then $f(0)$ is equal to

Q15.Fractional part of the number $\frac{4^{2022}}{15}$ is equal to

Q16.Let $\vec{a}=\hat{i}+4\hat{j}+2\hat{k}$, $\vec{b}=3\hat{i}-2\hat{j}+7\hat{k}$ and $\vec{c}=2\hat{i}-\hat{j}+4\hat{k}$. If a vector $\vec{d}$ satisfies $\vec{d}\times \vec{b}=\vec{c}\times \vec{b}$ and $\vec{d}\cdot \vec{a}=24$, then $|\vec{d}{|}^2$ is equal to

Q17.Let $B=\left[\begin{array}{ccc}1& 3& \alpha \\ 1& 2& 3\\ \alpha & \alpha & 4\end{array}\right]$, $\alpha >2$ be the adjoint of a matrix A and $|A|=2$, then $\left[\begin{array}{ccc}\alpha & -2\alpha & \alpha \end{array}\right]B\left[\begin{array}{c}\alpha \\ -2\alpha \\ \alpha \end{array}\right]$ is equal to

Q18.Let $s_1,s_2,s_3,\dots ,s_{10}$ respectively be the sum to 12 terms of 10 A.P.s whose first terms are $1,2,3,\dots ,10$ and the common differences are $1,3,5,\dots ,19$ respectively. Then ${\sum }_{i=1}^{10}{s}_i$ is equal to

Q19.Let $y=y_1(x)$ and $y=y_2(x)$ be the solution curves of the differential equation $\frac{dy}{dx}=y+7$ with initial conditions $y_1(0)=0$, $y_2(0)=1$ respectively. Then the curves $y=y_1(x)$ and $y=y_2(x)$ intersect at

Q20.Let the equation of plane passing through the line of intersection of the planes $x+2y+az=2$ and $x-y+z=3$ be $5x-11y+bz=6a-1$. For $c\in \mathbb{Z}$, if the distance of this plane from the point $(a,-c,c)$ is $\frac{2}{\sqrt{a}}$, then $\frac{a+b}{c}$ is equal to

Q21.Let $\alpha$ be the constant term in the binomial expansion of ${\left(\sqrt{x}-\frac{6}{x^{3/2}}\right)}^n$, $n\le 15$. If the sum of the coefficients of the remaining terms in the expansion is 649 and the coefficient of $x^{-n}$ is $\lambda \alpha$, then $\lambda$ …

Q22.If $S=\left\{x\in \mathbb{R}:{\sin }^{-1}\left(\frac{x+1}{\sqrt{x^2+2x+2}}\right)-{\sin }^{-1}\left(\frac{x}{\sqrt{x^2+1}}\right)=\frac{\pi }{4}\right\}$, then ${\sum }_{x\in S}\left(\sin \left((x^2+x+5)\frac{\pi }{2}\right)-\cos \left((x^2+x+5)\pi \right)\right)$ is equal to __ …

Q23.Let $\omega =z\overline{z}+k_{1}z+k_{2}iz+\lambda (1+i)$, $k_1,k_2\in \mathbb{R}$. Let $\mathrm{Re}(\omega )=0$ be the circle C of radius 1 in the first quadrant touching the line $y=1$ and the y-axis. If the curve $\mathrm{Im}(\omega )=0$ intersects C at …

Q24.Let for $x\in \mathbb{R}$, $S_0(x)=x$, $S_k(x)=C_{k}x+k{\int }_0^x{S}_{k-1}(t)dt$, where $C_0=1$, $C_k=1-{\int }_0^1{S}_{k-1}(x)dx$, $k=1,2,3\dots$. Then $S_2(3)+6C_3$ is equal to ______.

Q25.The sum to 20 terms of the series
$2.2^2-3^2+2.4^2-5^2+2.6^2-$ _____ is equal to _____.

Q26.The number of seven digit positive integers formed using the digits 1, 2, 3 and 4 only and sum of the digits equal to 12 is _____.

Q27.Let $m_1$ and $m_2$ be the slopes of the tangents drawn from the point $P(4,1)$ to the hyperbola $H:\frac{y^2}{25}-\frac{x^2}{16}=1$. If $Q$ is the point from which the tangents drawn to $H$ have slopes $|m_1|$ and $|m_2|$ and they make positive intercepts $\alpha$ and …

Q28.Let the image of the point $\left(\frac{5}{3},\frac{5}{3},\frac{8}{3}\right)$ in the plane $x-2y+z-2=0$ be $P$. If the distance of the point $Q(6,-2,\alpha )$, $\alpha >0$, from $P$ is 13, then $\alpha$ is equal to _____.

Q29.Let $\vec{a}=3\hat{i}+\hat{j}-\hat{k}$ and $\vec{c}=2\hat{i}-3\hat{j}+3\hat{k}$. If $\vec{b}$ is a vector such that $\vec{a}=\vec{b}\times \vec{c}$ and $|\vec{b}{|}^2=50$, then $\left|72-|\vec{b}+\vec{c}{|}^2\right|$ is equal to _____.

Q30.Let the mean of the data

be 5. If $m$ and ${\sigma }^2$ are respectively the mean deviation about the mean and the variance of the data, then $\frac{3\alpha }{m+{\sigma }^2}$ is equal …