JEE Main 2024 · 30 Jan 2024 · Shift 1

Q1.All surfaces shown in figure are assumed to be frictionless and the pulleys and the string are light. The acceleration of the block of mass 2 kg is :

Q2.A potential divider circuit is shown in figure. The output voltage ${\mathrm{V}}_0$ is

Q3.Young's modules of material of a wire of length ' L ' and cross-sectional area A is Y . If the length of the wire is doubled and cross-sectional area is halved then Young's modules will be :

Q4.The work function of a substance is 3.0 eV . The longest wavelength of light that can cause the emission of photoelectrons from this substance is approximately:

Q5.The ratio of the magnitude of the kinetic energy to the potential energy of an electron in the $5^{\text{th }}$ excited state of a hydrogen atom is :

Q6.A particle is placed at the point A of a frictionless track ABC as shown in figure. It is gently pushed toward right. The speed of the particle when it reaches the point $B$ is : (Take $g=10\mathrm{m}/{\mathrm{s}}^2$ ).

Q7.The electric field of an electromagnetic wave in free space is represented as $\vec{E}=E_0\cos (\omega t-kz)\hat{i}$.

The corresponding magnetic induction vector will be :

Q8.Two insulated circular loop A and B radius ' a ' carrying a current of ' I ' in the anti clockwise direction as shown in figure. The magnitude of the magnetic induction at the centre will be :

Q9.The diffraction pattern of a light of wavelength 400 nm diffracting from a slit of width 0.2 mm is focused on the focal plane of a convex lens of focal length 100 cm . The width of the $1^{\text{st }}$ secondary maxima will be :

Q10.Primary coil of a transformer is connected to 220 V ac. Primary and secondary turns of the transforms are 100 and 10 respectively. Secondary coil of transformer is connected to two series resistance shown in shown in figure. The output voltage ( ${\mathrm{V}}_0$ ) is :

Q11.The gravitational potential at a point above the surface of earth is $-5.12\times 10^7\mathrm{J}/\mathrm{kg}$ and the acceleration due to gravity at that point is $6.4\mathrm{m}/{\mathrm{s}}^2$. Assume that the mean radius of earth to be 6400 km . The height of thi …

Q12.An electric toaster has resistance of $60\Omega$ at room temperature $\left(27^{\circ }\mathrm{C}\right)$. The toaster is connected to a 220 V supply. If the current flowing through it reaches 2.75 A , the temperature attained by toaster is around : (if …

Q13.A Zener diode of breakdown voltage 10 V is used as a voltage regulator as shown in the figure. The current through the Zener diode is

Q14.Two thermodynamical process are shown in the figure. The molar heat capacity for process A and $B$ are $C_A$ and $C_B$. The molar heat capacity at constant pressure and constant volume are represented by ${\mathrm{C}}_{\mathrm{P}}$ and ${\mathrm{C}}_{\mathrm{V}}$, respe …

Q15.The electrostatic potential due to an electric dipole at a distance ' $r$ ' varies as :

Q16.A spherical body of mass 100 g is dropped from a height of 10 m from the ground. After hitting the ground, the body rebounds to a height of 5 m . The impulse of force imparted by the ground to the body is given by : (given $\mathrm{g}=9.8\mathrm{m}/{\mathrm{s}}^2$ )

Q17.A particle of mass m projected with a velocity ' u ' making an angle of $30^{\circ }$ with the horizontal. The magnitude of angular momentum of the projectile about the point of projection when the particle is at its maximum height $h$ is :

Q18.At which temperature the r.m.s. velocity of a hydrogen molecule equal to that of an oxygen molecule at $47^{\circ }\mathrm{C}$ ?

Q19.A series L,R circuit connected with an ac source $\mathrm{E}=(25\sin 1000\mathrm{t})\mathrm{V}$ has a power factor of $\frac{1}{\sqrt{2}}$. If the source of emf is changed to $\mathrm{E}=(20\sin 2000$ t)V, the new power factor of the circuit will be :

Q20.The horizontal component of earth's magnetic field at a place is $3.5\times 10^{-5}T$. A very long straight conductor carrying current of $\sqrt{2}A$ in the direction from South east to North West is placed. The force per unit length experienced by the conductor is ___\ …

Q21.Two cells are connected in opposition as shown. Cell ${\mathrm{E}}_1$ is of 8 V emf and $2\Omega$ internal resistance; the cell ${\mathrm{E}}_2$ is of 2 V emf and $4\Omega$ internal resistance. The terminal potential difference of cell ${\mathrm{E}}_2$ is:

Q22.A electron of hydrogen atom on an excited state is having energy ${\mathrm{E}}_{\mathrm{n}}=-0.85\mathrm{eV}$. The maximum number of allowed transitions to lower energy level is ____ . .

Q23.Each of three blocks $\mathrm{P},\mathrm{Q}$ and R shown in figure has a mass of 3 kg . Each of the wire $A$ and $B$ has cross-sectional area $0.005{\mathrm{cm}}^2$ and Young's modulus $2\times 10^{11}\mathrm{N}{\mathrm{m}}^{-2}$. Neglecting friction, the longitud …

Q24.The distance between object and its two times magnified real image as produced by a convex lens is 45 cm. The focal length of the lens used is ______ cm.

Q25.The displacement and the increase in the velocity of a moving particle in the time interval of t to $(t+1)$ s are 125 m and 50 m/s, respectively. The distance travelled by the particle in $(t+2{)}^{\text{th}}$ s is ______ m.

Q26.A capacitor of capacitance C and potential V has energy E. It is connected to another capacitor of capacitance 2C and potential 2V. Then the loss of energy is $\frac{x}{3}E$, where x is ______.

Q27.Consider a Disc of mass 5 kg, radius 2 m, rotating with angular velocity of 10 rad/s about an axis perpendicular to the plane of rotation. An identical disc is kept gently over the rotating disc along the same axis. The energy dissipated so that both the discs continue to rotate …

Q28.In a closed organ pipe, the frequency of fundamental note is 30 Hz. A certain amount of water is now poured in the organ pipe so that the fundamental frequency is increased to 110 Hz. If the organ pipe has a cross-sectional area of 2 cm², the amount of water poured in the organ t …

Q29.A ceiling fan having 3 blades of length 80 cm each is rotating with an angular velocity of 1200 rpm. The magnetic field of earth in that region is 0.5 G and angle of dip is $30^{\circ }$. The emf induced across the blades is $N\pi \times 10^{-5}$ V. The value of N is ______.

Q1.Given below are two statements:

Statement-I: The gas liberated on warming a salt with dil ${\mathrm{H}}_2{\mathrm{SO}}_4$, turns a piece of paper dipped in lead acetate into black, it is a confirmatory test for sulphide ion.
Statement-II: In statement-I the colour of paper tu …

Q2.This reduction reaction is known as:

Q3.Sugar which does not give reddish brown precipitate with Fehling's reagent is:

Q4.Given below are the two statements: one is labeled as Assertion (A) and the other is labeled as Reason (R).

Assertion (A): There is a considerable increase in covalent radius from N to P. However from As to Bi only a small increase in covalent radius is observed.

Reason (R): co …

Q5.Which of the following molecule/species is most stable?

Q6.Diamagnetic Lanthanoid ions are:

Q7.Aluminium chloride in acidified aqueous solution forms an ion having geometry

Q8.Given below are two statements:

Statement-I: The orbitals having same energy are called as degenerate orbitals.
Statement-II: In hydrogen atom, 3p and 3d orbitals are not degenerate orbitals.
In the light of the above statements, choose the most appropriate answer from the optio …

Q9.Example of vinylic halide is

Q10.Structure of 4-Methylpent-2-enal is

Q11.Match List-I with List-II

List-I (Molecule)
(A) ${\mathrm{BrF}}_5$
(B) ${\mathrm{H}}_2\mathrm{O}$
(C) ${\mathrm{ClF}}_3$
(D) ${\mathrm{SF}}_4$

List-II (Shape)
(I) T-shape
(II) See saw
(III) Bent
(IV) Square pyramidal

Q12.In the given reactions identify the reagent A and reagent B

Q13.Given below are two statement one is labeled as Assertion (A) and the other is labeled as Reason (R).

Assertion (A): ${\mathrm{CH}}_2=\mathrm{CH}-{\mathrm{CH}}_2-\mathrm{Cl}$ is an example of allyl halide

Reason (R): Allyl halides are the compounds in which the halogen atom i …

Q14.What happens to freezing point of benzene when small quantity of napthalene is added to benzene?

Q15.Match List-I with List-II

List-I (Species):
(A) ${\mathrm{Cr}}^{+2}$
(B) ${\mathrm{Mn}}^{+}$
(C) ${\mathrm{Ni}}^{+2}$
(D) ${\mathrm{V}}^{+}$

List-II (Electronic distribution):
(I) $3d^8$
(II) $3{\mathrm{d}}^{3}4{\mathrm{s}}^1$
(III) $3{\mathrm{d}}^4$
(IV) …

Q16.Compound A formed in the following reaction reacts with B gives the product C. Find out A and B.

Q17.Following is a confirmatory test for aromatic primary amines. Identify reagent (A) and (B)

Q18.The Lassiagne's extract is boiled with dil ${\mathrm{HNO}}_3$ before testing for halogens because,

Q19.Choose the correct Statements from the following:
(A) Ethane-1 2-diamine is a chelating ligand.
(B) Metallic aluminium is produced by elecrtrolysis of aluminium oxide in presence of cryolite.
(C) Cyanide ion is used as ligand for leaching of silver.
(D) Phosphine act as a ligand …

Q20.The rate of first order reaction is $0.04\mathrm{mol}{\mathrm{L}}^{-1}{\mathrm{s}}^{-1}$ at 10 minutes and $0.03\mathrm{mol}{\mathrm{L}}^{-1}{\mathrm{s}}^{-1}$ at 20 minutes after initiation. Half life of the reaction is ____ minutes. (Given …

Q21.The pH at which $\mathrm{Mg}(\mathrm{OH}{)}_2\left[{\mathrm{K}}_{\mathrm{sp}}=1\times 10^{-11}\right]$ begins to precipitate from a solution containing 0.10 M ${\mathrm{Mg}}^{2+}$ ions is ____

Q22.An ideal gas undergoes a cyclic transformation starting from the point A and coming back to the same point by tracing the path $\mathrm{A}\to \mathrm{B}\to \mathrm{C}\to \mathrm{A}$ as shown in the diagram. The total work done in the process is ___ …

Q23.If IUPAC name of an element is "Unununnium" then the element belongs to nth group of periodic table. The value of $n$ is ____

Q24.The total number of molecular orbitals formed from 2s and 2p atomic orbitals of a diatomic molecule

Q25.On a thin layer chromatographic plate, an organic compound moved by 3.5 cm, while the solvent moved by 5 cm. The retardation factor of the organic compound is ______ $\times 10^{-1}$

Q26.The compound formed by the reaction of ethanal with semicarbazide contains ______ number of nitrogen atoms.

Q27.0.05 cm thick coating of silver is deposited on a plate of $0.05{\mathrm{m}}^2$ area. The number of silver atoms deposited on plate are ______ $\times 10^{23}$. (At mass Ag = 108, $\mathrm{d}=7.9{\mathrm{g}\mathrm{cm}}^{-3}$)

Q28.$2{\mathrm{MnO}}_4^{-}+\mathrm{b}{\mathrm{I}}^{-}+\mathrm{c}{\mathrm{H}}_2\mathrm{O}\to \mathrm{x}{\mathrm{I}}_2+\mathrm{y}{\mathrm{MnO}}_2+\mathrm{z}{\mathrm{OH}}^{-}$

If the above equation is balanced with integer coefficients, the value of z is ______

Q29.The mass of sodium acetate $\left({\mathrm{CH}}_3\mathrm{COONa}\right)$ required to prepare 250 mL of 0.35 M aqueous solution is ______ g. (Molar mass of ${\mathrm{CH}}_3\mathrm{COONa}$ is $82.02{\mathrm{g}\mathrm{mol}}^{-1}$)

Q1.A line passing through the point $\mathrm{A}(9,0)$ makes an angle of $30^{\circ }$ with the positive direction of x-axis. If this line is rotated about A through an angle of $15^{\circ }$ in the clockwise direction, then its equation in the new position is

Q2.Let ${\mathrm{S}}_n$ denote the sum of first n terms of an arithmetic progression. If ${\mathrm{S}}_{20}=790$ and ${\mathrm{S}}_{10}=145$, then ${\mathrm{S}}_{15}-{\mathrm{S}}_5$ is :

Q3.If $\mathrm{z}=\mathrm{x}+\mathrm{iy},\mathrm{xy}\ne 0$, satisfies the equation $z^2+i\bar{z}=0$, then $\left|z^2\right|$ is equal to :

Q4.Let $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ and $\vec{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ be two vectors such that $|\vec{a}|=1$; $\vec{a}\cdot \vec{b}=2$ and $|\vec{b}|=4$. If $\vec{c}=2(\vec{a}\times \vec{b})-3\vec{b}$, then the angle betwee …

Q5.The maximum area of a triangle whose one vertex is at $(0,0)$ and the other two vertices lie on the curve $\mathrm{y}=-2{\mathrm{x}}^2+54$ at points $(\mathrm{x},\mathrm{y})$ and $(-\mathrm{x},\mathrm{y})$ where $\mathrm{y}>0$ is :

Q6.The value of ${\lim }_{n\to \infty }{\sum }_{k=1}^n\frac{n^3}{\left(n^2+k^2\right)\left(n^2+3k^2\right)}$ is :

Q7.Let $\mathrm{g}:\mathrm{R}\to \mathrm{R}$ be a non constant twice differentiable such that $g^{\prime }\left(\frac{1}{2}\right)=g^{\prime }\left(\frac{3}{2}\right)$. If a real valued function f is defined as …

Q8.The area (in square units) of the region bounded by the parabola ${\mathrm{y}}^2=4(\mathrm{x}-2)$ and the line $\mathrm{y}=2\mathrm{x}-8$

Q9.Let $y=y(x)$ be the solution of the differential equation $\sec xdy+\{2(1-x)\tan x+x(2-x)\}dx=0$ such that $\mathrm{y}(0)=2$. Then $\mathrm{y}(2)$ is equal to :

Q10.Let $(\alpha ,\beta ,\gamma )$ be the foot of perpendicular from the point (1,2,3) on the line $\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}$. Then $19(\alpha +\beta +\gamma )$ is equal to :

Q11.Two integers x and y are chosen with replacement from the set $\{0,1,2,3,\dots ,10\}$. Then the probability that $|\mathrm{x}-\mathrm{y}|>5$ is :

Q12.If the domain of the function $f(x)={\cos }^{-1}\left(\frac{2-|x|}{4}\right)+{\left({\log }_e(3-x)\right)}^{-1}$ is $[-\alpha ,\beta )-\{\gamma \}$, then $\alpha +\beta +\gamma$ is equal to :

Q13.Consider the system of linear equation $x+y+z=4\mu$, $x+2y+2\lambda z=10\mu$, $x+3y+4{\lambda }^{2}z={\mu }^2+15$, where $\lambda ,\mu \in \mathrm{R}$. Which one of the following statements is NOT correct?

Q14.If the circles $(x+1{)}^2+(y+2{)}^2=r^2$ and $x^2+y^2-4x-4y+4=0$ intersect at exactly two distinct points, then

Q15.If the length of the minor axis of ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is:

Q16.Let M denote the median of the following frequency distribution.

Then 20M is equal to:

Q17.If $f(x)=\left|\begin{array}{ccc}2{\cos }^{4}x& 2{\sin }^{4}x& 3+{\sin }^{2}2x\\ 3+2{\cos }^{4}x& 2{\sin }^{4}x& {\sin }^{2}2x\\ 2{\cos }^{4}x& 3+2{\sin }^{4}x& {\sin }^{2}2x\end{array}\right|$ then $\frac{1}{5}{f}^{\prime }(0)$ is equal to

Q18.Let $A(2,3,5)$ and $C(-3,4,-2)$ be opposite vertices of a parallelogram ABCD. If the diagonal $\overrightarrow{BD}=\hat{i}+2\hat{j}+3\hat{k}$, then the area of the parallelogram is equal to

Q19.If $2{\sin }^{3}x+\sin 2x\cos x+4\sin x-4=0$ has exactly 3 solutions in the interval $\left[0,\frac{n\pi }{2}\right]$, $n\in \mathrm{N}$, then the roots of the equation $x^2+nx+(n-3)=0$ belong to:

Q20.Let $f:\left[-\frac{\pi }{2},\frac{\pi }{2}\right]\to \mathrm{R}$ be a differentiable function such that $f(0)=\frac{1}{2}$. If ${\lim }_{x\to 0}\frac{x{\int }_0^{x}f(t)dt}{e^{x^2}-1}=\alpha$, then $8{\alpha }^2$ is equal to:

Q21.A group of 40 students appeared in an examination of 3 subjects - Mathematics, Physics & Chemistry. It was found that all students passed in at least one of the subjects, 20 students passed in Mathematics, 25 students passed in Physics, 16 students passed in Chemistry, at most 11 …

Q22.If $d_1$ is the shortest distance between the lines $x+1=2y=-12z$, $x=y+2=6z-6$ and $d_2$ is the shortest distance between the lines $\frac{x-1}{2}=\frac{y+8}{-7}=\frac{z-4}{5}$, $\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-6}{-3}$, then the value of $\frac{32\sqrt{3}{d}_1}{d_2}$ i …

Q23.Let the latus rectum of the hyperbola $\frac{x^2}{9}-\frac{y^2}{b^2}=1$ subtend an angle of $\frac{\pi }{3}$ at the centre of the hyperbola. If $b^2$ is equal to $\frac{l}{m}(1+\sqrt{n})$, where $l$ and $m$ are co-prime numbers, then $l^2+m^2+n^2$ is equal to

Q24.Let $A=\{1,2,3,\dots ,7\}$ and let $P(A)$ denote the power set of $A$. If the number of functions $f:A\to P(A)$ such that $a\in f(a)$, $\forall a\in A$ is $m^n$, $m$ and $n\in \mathbb{N}$ and $m$ is least, then $m+n$ is equal to

Q25.The value $9{\int }_0^9\left[\sqrt{\frac{10x}{x+1}}\right]dx$, where $[t]$ denotes the greatest integer less than or equal to $t$, is ______.

Q26.Number of integral terms in the expansion of ${\left\{7^{\left(\frac{1}{2}\right)}+11^{\left(\frac{1}{6}\right)}\right\}}^{824}$ is equal to ______.

Q27.Let $y=y(x)$ be the solution of the differential equation $(1-x^2)dy=[xy+(x^3+2)\sqrt{3(1-x^2)}]dx$, $-1<x<1$, $y(0)=0$. If $y\left(\frac{1}{2}\right)=\frac{m}{n}$, $m$ and $n$ are coprime numbers, then $m+n$ is equal to ______.

Q28.Let $\alpha ,\beta \in \mathbb{N}$ be roots of equation $x^2-70x+\lambda =0$, where $\frac{\lambda }{2},\frac{\lambda }{3}\notin \mathbb{N}$. If $\lambda$ assumes the minimum possible value, then $\frac{(\sqrt{\alpha -1}+\sqrt{\beta -1})(\lambda +35)}{|\alpha -\beta |}$ is equal to _ …

Q29.If the function $f(x)=\{\begin{array}{cl}\frac{1}{|x|}& ,|x|\ge 2\\ a{x}^2+2b,& |x|<2\end{array}$ is differentiable on $\mathbb{R}$, then $48(a+b)$ is equal to ______.

Q30.Let $\alpha =1^2+4^2+8^2+13^2+19^2+26^2+\dots$ upto 10 terms and $\beta ={\sum }_{n=1}^{10}{n}^4$. If $4\alpha -\beta =55k+40$, then $k$ is equal to ______.