JEE Main 2023 · 24 Jan 2023 · Shift 1

Q1.Two long straight wires P and Q carrying equal current 10A each were kept parallel to each other at 5 cm distance. Magnitude of magnetic force experienced by 10 cm length of wire P is ${\mathrm{F}}_1$. If distance between wires is halved and currents on them are doubled, force …

Q2.Given below are two statements :

Statement-I : An elevator can go up or down with uniform speed when its weight is balanced with the tension of its cable.
Statement-II : Force exerted by the floor of an elevator on the foot of a person standing on it is more than his/her weight …

Q3.From the photoelectric effect experiment, following observations are made. Identify which of these are correct
A. The stopping potential depends only on the work function of the metal.
B. The saturation current increases as the intensity of incident light increases.
C. The maximu …

Q4.The weight of a body at the surface of earth is 18 N . The weight of the body at an altitude of 3200 km above the earth's surface is (given, radius of earth ${\mathrm{R}}_{\mathrm{e}}=6400\mathrm{km}$)

Q5.A 100 m long wire having cross-sectional area 6.25 $\times 10^{-4}{\mathrm{m}}^2$ and Young's modulus is $10^{10}{\mathrm{Nm}}^{-2}$ is subjected to a load of 250 N , then the elongation in the wire will be :

Q6.1 g of a liquid is converted to vapour at $3\times 10^5\mathrm{Pa}$ pressure. If $10\%$ of the heat supplied is used for increasing the volume by $1600{\mathrm{cm}}^3$ during this phase change, then the increase in internal energy in the process will be :

Q7.A modulating signal is a square wave, as shown in the figure.

If the carrier wave is given as $\mathrm{c}(\mathrm{t})=2\sin (8\pi \mathrm{t})$ volts, the modulation index is :

Q8.As per given figure, a weightless pulley P is attached on a double inclined frictionless surface. The tension in the string (massless) will be (if $\mathrm{g}=10\mathrm{m}/{\mathrm{s}}^2$)

Q9.Given below are two statements : one is labelled as Assertion A and the other is labelled as Reason R
Assertion A : Photodiodes are preferably operated in reverse bias condition for light intensity measurement.
Reason R : The current in the forward bias is more than the current i …

Q10.If $\overrightarrow{\mathrm{E}}$ and $\overrightarrow{\mathrm{K}}$ represent electric field and propagation vectors of the EM waves in vacuum, then magnetic field vector is given by : ( $\omega$ angular frequency) :

Q11.A circular loop of radius $r$ is carrying current I A. The ratio of magnetic field at the centre of circular loop and at a distance r from the center of the loop on its axis is :

Q12.A travelling wave is described by the equation
$y(x,t)=[0.05\sin (8x-4t]m$
The velocity of the wave is : [all the quantities are in SI unit]

Q13.As shown in the figure, a network of resistors is connected to a battery of 24 V with an internal resistance of $3\Omega$. The currents through the resistors ${\mathrm{R}}_4$ and ${\mathrm{R}}_5$ are ${\mathrm{I}}_4$ and ${\mathrm{I}}_5$ respectively. The values of …

Q14.Given below are two statements :

Statement I: If the Brewster's angle for the light propagating from air to glass is ${\theta }_{\mathrm{B}}$, then Brewster's angle for the light propagating from glass to air is $\frac{\pi }{2}-{\theta }_{\mathrm{B}}$.
Statement II : The Brewster's …

Q15.If two charges ${\mathrm{q}}_1$ and ${\mathrm{q}}_2$ are separated with distance ' $d$ ' and placed in a medium of dielectric constant K . What will be the equivalent distance between charges in air for the same electrostatic force ?

Q16.Consider the following radioactive decay process
${}_{84}^{218}A\stackrel{\alpha }{\to }{A}_1\stackrel{\beta }{\to }{A}_2\stackrel{\gamma }{\to }{A}_3\stackrel{\alpha }{\to }{A}_4\stackrel{{\beta }^{+}}{\to }{A}_5\stackrel{\gamma }{\to }{A}_6$
The mass number and the atomic number …

Q17.Given below are two statements :

Statements I : The temperature of a gas is $-73^{\circ }\mathrm{C}$. When the gas is heated to $527^{\circ }\mathrm{C}$, the root mean square speed of the molecules is doubled.
Statement II : The product of pressure and volume of an ideal gas …

Q18.The maximum vertical height to which a man can throw a ball is 136 m . The maximum horizontal distance upto which he can throw the same ball is

Q19.A conducting loop of radius $\frac{10}{\sqrt{\pi }}\mathrm{cm}$ is placed perpendicular to a uniform magnetic field of 0.5 T . The magnetic field is decreased to zero in 0.5 s at a steady rate. The induced emf in the circular loop at 0.25 s is:

Q20.A spherical body of mass 2 kg starting from rest acquires a kinetic energy of 10000 J at the end of $5^{\text{th }}$ second. The force acted on the body is ____ N.

Q21.A block of mass 2 kg is attached with two identical springs of spring constant $20\mathrm{N}/\mathrm{m}$ each. The block is placed on a frictionless surface and the ends of the springs are attached to rigid supports (see figure). When the mass is displaced from its equilibr …

Q22.A hole is drilled in a metal sheet. At $27^{\circ }\mathrm{C}$, the diameter of hole is 5 cm . When the sheet is heated to $177^{\circ }\mathrm{C}$, the change in the diameter of hole is $\mathrm{d}\times 10^{-3}\mathrm{cm}$. The value of d will be ____ if coefficien …

Q23.In the circuit shown in the figure, the ratio of the quality factor and the band width is ____ s.

Q24.A hollow cylindrical conductor has length of 3.14 m, while its inner and outer diameters are 4 mm and 8 mm respectively. The resistance of the conductor is $n\times 10^{-3}\Omega$.
If the resistivity of the material is $2.4\times 10^{-8}\Omega \mathrm{m}$. The value of $n$ i …

Q25.As shown in the figure, a combination of a thin plano concave lens and a thin plano convex lens is used to image an object placed at infinity. The radius of curvature of both the lenses is 30 cm and refraction index of the material for both the lenses is 1.75. Both the lenses are …

Q26.Solid sphere A is rotating about an axis PQ. If the radius of the sphere is 5 cm then its radius of gyration about PQ will be $\sqrt{x}$ cm. The value of x is ______.

Q27.Vectors $a\hat{i}+b\hat{j}+\hat{k}$ and $2\hat{i}-3\hat{j}+4\hat{k}$ are perpendicular to each other when $3a+2b=7$, the ratio of $a$ to $b$ is $\frac{x}{2}$. The value of $x$ is ______.

Q28.Assume that protons and neutrons have equal masses. Mass of a nucleon is $1.6\times 10^{-27}$ kg and radius of nucleus is $1.5\times 10^{-15}{A}^{1/3}$ m. The approximate ratio of the nuclear density and water density is $n\times 10^{13}$. The value of n is ______.

Q29.A stream of a positively charged particles having $\frac{q}{m}=2\times 10^{11}\frac{C}{kg}$ and velocity ${\vec{v}}_0=3\times 10^7\hat{i}$ m/s is deflected by an electric field $1.8\hat{j}$ kV/m. The electric field exists in a region of 10 cm along x direction. Due to the el …

Q1.Compound (X) undergoes the following sequence of reactions to give the Lactone (Y): (i) HCHO, KOH (ii) KCN (alc.) (iii) H3O+. The Lactone (Y) is a five-membered gamma-butyrolactone bearing a gem-dimethyl group and an alpha-hydroxy group (alpha-hydroxy-beta,beta-dimethyl-gamma-b …

Q2.Assertion A : Hydrolysis of an alkyl chloride is a slow reaction but in the presence of NaI, the rate of the hydrolysis increases.
Reason R : ${\mathrm{I}}^{-}$ is a good nucleophile as well as a good leaving group.
In the light of the above statements, choose the correct answer fr …

Q3.The magnetic moment of a transition metal compound has been calculated to be 3.87 B.M. The metal ion is

Q4.It is observed that characteristic X-ray spectra of elements show regularity. When frequency to the power 'n' i.e. $v^{\mathrm{n}}$ of X-rays emitted is plotted against atomic number 'Z', following graph is obtained.

The value of 'n' is

Q5.Which of the Phosphorus oxoacid can create silver mirror from ${\mathrm{AgNO}}_3$ solution ?

Q6.The primary and secondary valencies of cobalt respectively in $\left[\mathrm{Co}{\left({\mathrm{NH}}_3\right)}_5\mathrm{Cl}\right]{\mathrm{Cl}}_2$ are :

Q7.An ammoniacal metal salt solution gives a brilliant red precipitate on addition of dimethylglyoxime. The metal ion is :

Q8.'R' formed in the following sequence of reaction is:

Q9.Statement I: For colloidal particles, the values of colligative properties are of small order as compared to values shown by true solutions at same concentration.

Statement II : For colloidal particles, the potential difference between the fixed layer and the diffused layer of s …

Q10.Reaction of BeO with ammonia and hydrogen fluoride gives 'A' which on thermal decomposition gives ${\mathrm{BeF}}_2$ and ${\mathrm{NH}}_4\mathrm{F}$. What is 'A' ?

Q11.'A' and 'B' formed in the following set of reactions are :

Q12.In the following given reaction ' A ' is

Q13.Decreasing order of the hydrogen bonding in following forms of water is correctly represented by
A. Liquid water
B. Ice
C. Impure water

Q14.Given below are two statements :

Statement I : Noradrenaline is a neurotransmitter.
Statement II : Low level of noradrenaline is not the cause of depression in human.
In the light of the above statements, choose the correct answer from the options given below

Q15.In the depression of freezing point experiment
A. Vapour pressure of the solution is less than that of pure solvent
B. Vapour pressure of the solution is more than that of pure solvent
C. Only solute molecules solidify at the freezing point
D. Only solvent molecules solidify at t …

Q16.Which of the following is true about freons ?

Q17.The dissociation constant of acetic is $\mathrm{x}\times 10^{-5}$. When 25 mL of $0.2\mathrm{M}{\mathrm{CH}}_3\mathrm{COONa}$ solution is mixed with 25 mL of $0.02\mathrm{M}{\mathrm{CH}}_3\mathrm{COOH}$ solution, the pH of the resultant solution is found to be equal to 5. …

Q18.5 g of NaOH was dissolved in deionized water to prepare a 450 mL stock solution. What volume (in $\mathrm{mL}$) of this solution would be required to prepare 500 mL of 0.1 M solution?

Given : Molar Mass of $\mathrm{Na}$, $\mathrm{O}$ and H is 23, 16 and 1 …

Q19.If wavelength of the first line of the Paschen series of hydrogen atom is 720 nm, then the wavelength of the second line of this series is ______ nm.
(Nearest integer)

Q20.The number of correct statement/s from the following is ______.
A. Larger the activation energy, smaller is the value of the rate constant.
B. The higher is the activation energy, higher is the value of the temperature coefficient.
C. At lower temperatures, increase in temperatur …

Q21.At 298 K, a 1 litre solution containing 10 mmol of ${\mathrm{Cr}}_2{\mathrm{O}}_7{}^{2-}$ and 100 mmol of ${\mathrm{Cr}}^{3+}$ shows a pH of 3.0.

Given : ${\mathrm{Cr}}_2{\mathrm{O}}_7{}^{2-}\to {\mathrm{Cr}}^{3+}$ ; ${\mathrm{E}}^0=1.330\mathrm{V}$ and …

Q22.When ${\mathrm{Fe}}_{0.93}\mathrm{O}$ is heated in presence of oxygen, it converts to ${\mathrm{Fe}}_2{\mathrm{O}}_3$. The number of correct statement/s from the following is ______.
A. The equivalent weight of ${\mathrm{Fe}}_{0.93}\mathrm{O}$ is …

Q23.The d-electronic configuration of ${\left[{\mathrm{CoCl}}_4\right]}^{2-}$ in tetrahedral crystal field is $e^m{t}_2{}^n$. Sum of ' $m$ ' and 'number of unpaired electrons is ______.

Q24.For independent process at 300 K.

The number of non-spont …

Q25.Uracil is base present in RNA with the following structure. % of N in uracil is ______.

Given:
Molar mass …

Q26.Number of moles of AgCl formed in the following reaction is ______.

Q27.Order of Covalent bond;
A. KF > KI; LiF > KF
B. KF < KI; LiF > KF
C. SnCl4 > SnCl2; CuCl > NaCl
D. LiF > KF; CuCl < NaCl
E. KF < KI; CuCl > NaCl
Choose the correct answer from the options given below:

Q1.Let the six numbers $a_1,a_2,a_3,a_4,a_5,a_6$ be in A.P. and $a_1+a_3=10$. If the mean of these six numbers is $\frac{19}{2}$ and their variance is ${\sigma }^2$, then $8{\sigma }^2$ is equal to

Q2.Let $f(x)$ be a function such that $f(x+y)=f(x)\cdot f(y)$ for all $x,y\in N$. If $f(1)=3$ and ${\sum }_{k=1}^{n}f(k)=3279$, then the value of n is

Q3.The number of real solutions of the equation $3\left(x^2+\frac{1}{x^2}\right)-2\left(x+\frac{1}{x}\right)+5=0$, is

Q4.If $f(x)=\frac{2^{2x}}{2^{2x}+2}$, $x\in R$, then $f\left(\frac{1}{2023}\right)+f\left(\frac{2}{2023}\right)+\dots +f\left(\frac{2022}{2023}\right)$ is equal to

Q5.If $f(x)=x^3-x^2{f}^{\prime }(1)+x{f}^{\prime \prime }(2)-f^{\prime \prime \prime }(3)$, $x\in R$, then

Q6.The number of integers, greater than 7000 that can be formed, using the digits $3,5,6,7,8$ without repetition, is

Q7.If the system of equations
$x+2y+3z=3$
$4x+3y-4z=4$
$8x+4y-\lambda z=9+\mu$
has infinitely many solutions, then the ordered pair $(\lambda ,\mu )$ is equal to

Q8.The value of ${\left(\frac{1+\sin \frac{2\pi }{9}+i\cos \frac{2\pi }{9}}{1+\sin \frac{2\pi }{9}-i\cos \frac{2\pi }{9}}\right)}^3$ is

Q9.The set of all values of a for which ${\lim }_{x\to a}([x-5]-[2x+2])=0$, where $[\alpha ]$ denotes the greatest integer less than or equal to $\alpha$ is equal to

Q10.If ${\left({}^{30}{C}_1\right)}^2+2{\left({}^{30}{C}_2\right)}^2+3{\left({}^{30}{C}_3\right)}^2+\dots +30{\left({}^{30}{C}_{30}\right)}^2=\frac{\alpha \cdot 60!}{(30!{)}^2}$, then $\alpha$ is equal to

Q11.Let the plane containing the line of intersection of the planes
P1: $x+(\lambda +4)y+z=1$ and
P2: $2x+y+z=2$ pass through the points $(0,1,0)$
and $(1,0,1)$. Then the distance of the point
$(2\lambda ,\lambda ,-\lambda )$ from the plane P2 is

Q12.Let $\vec{\alpha }=4\hat{i}+3\hat{j}+5\hat{k}$ and $\vec{\beta }=\hat{i}+2\hat{j}-4\hat{k}$. Let ${\vec{\beta }}_1$ be parallel to $\vec{\alpha }$ and ${\vec{\beta }}_2$ be perpendicular to $\vec{\alpha }$. If $\vec{\beta }={\vec{\beta }}_1+{\vec{\beta }}_2$, then the value of …

Q13.The locus of the mid points of the chords of the circle $C_1:(x-4{)}^2+(y-5{)}^2=4$ which subtend an angle ${\theta }_i$ at the centre of the circle $C_1$, is a circle of radius $r_i$. If ${\theta }_1=\frac{\pi }{3},{\theta }_3=\frac{2\pi }{3}$ and …

Q14.If the foot of the perpendicular drawn from $(1,9,7)$ to the line passing through the point $(3,2,1)$ and parallel to the planes $x+2y+z=0$ and $3y-z=3$ is $(\alpha ,\beta ,\gamma )$, then $\alpha +\beta +\gamma$ is equal to

Q15.Let $y=y(x)$ be the solution of the differential equation $(x^2-3y^2)dx+3xydy=0,y(1)=1$. Then $6y^2(e)$ is equal to

Q16.Let p and q be two statements.

Then $\sim (p\wedge (p\Rightarrow \sim q))$ is equivalent to

Q17.The number of square matrices of order 5 with entries from the set $\{0,1\}$, such that the sum of all the elements in each row is 1 and the sum of all the elements in each column is also 1, is

Q18.${\int }_{\frac{3\sqrt{2}}{4}}^{\frac{3\sqrt{3}}{4}}\frac{48}{\sqrt{9-4x^2}}dx$ is equal to

Q19.Let A be a $3\times 3$ matrix such that $|\mathrm{adj}(\mathrm{adj}(\mathrm{adj}A))|=12^4$. Then $\left|A^{-1}\mathrm{adj}A\right|$ is equal to

Q20.The urns A, B and C contain 4 red, 6 black; 5 red, 5 black and $\lambda$ red, 4 black balls respectively. One of the urns is selected at random and a ball is drawn. If the ball drawn is red and the probability that it is drawn from urn C is 0.4 then the square of the length of th …

Q21.If $\frac{1^3+2^3+3^3+\dots \text{upto }n\text{ terms}}{1\cdot 3+2\cdot 5+3\cdot 7+\dots \text{upto }n\text{ terms}}=\frac{9}{5}$, then the value of $n$ is

Q22.Let f be a differentiable function defined on $\left[0,\frac{\pi }{2}\right]$ such that $f(x)>0$ and $f(x)+{\int }_0^{x}f(t)\sqrt{1-({\log }_{e}f(t){)}^2}dt=e,\forall x\in \left[0,\frac{\pi }{2}\right]$. Then ${\left(6{\log }_{e}f\left(\frac{\pi }{6}\right)\right)}^2$ is equal to

Q23.The minimum number of elements that must be added to the relation $\mathrm{R}=\{(\mathrm{a},\mathrm{b}),(\mathrm{b},\mathrm{c}),(\mathrm{b},\mathrm{d})\}$ on the set $\{\mathrm{a},\mathrm{b},\mathrm{c},\mathrm{d}\}$ so that it is an equivalence relation, is ____.

Q24.Let $a=\hat{i}+2\hat{j}+\lambda \hat{k},b=3\hat{i}-5\hat{j}-\lambda \hat{k},\vec{a}\cdot \vec{c}=7$, $2\vec{b}\cdot \vec{c}+43=0,\vec{a}\times \vec{c}=\vec{b}\times \vec{c}$. Then $|\vec{a}\cdot \vec{b}|$ is equal to

Q25.Let the sum of the coefficients of the first three terms in the expansion of ${\left(x-\frac{3}{x^2}\right)}^n,x\ne 0,n\in N$, be 376. Then the coefficient of $x^4$ is ____.

Q26.If the shortest between the lines
$\frac{x+\sqrt{6}}{2}=\frac{y-\sqrt{6}}{3}=\frac{z-\sqrt{6}}{4}$ and
$\frac{x-\lambda }{3}=\frac{y-2\sqrt{6}}{4}=\frac{z+2\sqrt{6}}{5}$ is 6, then the square of sum of all possible values of $\lambda$ is

Q27.Let $\mathrm{S}=\{\theta \in [0,2\pi ):\tan (\pi \cos \theta )+\tan (\pi \sin \theta )=0\}$.

Then ${\sum }_{\theta \in \mathrm{S}}{\sin }^2\left(\theta +\frac{\pi }{4}\right)$ is equal to