JEE Main 2024 · 05 Apr 2024 · Shift 1

Q1.Light emerges out of a convex lens when a source of light kept at its focus. The shape of wavefront of the light is:

Q2.Following gates section is connected in a complete suitable circuit.

For which of the following combination, bulb will glow (ON):

Q3.If $G$ be the gravitational constant and $u$ be the energy density then which of the following quantity have the dimension as that the $\sqrt{\mathrm{uG}}$ :

Q4.Given below are two statements :

Statement-I: When a capillary tube is dipped into a liquid, the liquid neither rises nor falls in the capillary. The contact angle may be $0^{\circ }$.
Statement-II: The contact angle between a solid and a liquid is a property of the material of …

Q5.Given below are two statements:

Statement-I: Figure shows the variation of stopping potential with frequency ( $v$ ) for the two photosensitive materials ${\mathrm{M}}_1$ and ${\mathrm{M}}_2$. The slope gives value of $\frac{\mathrm{h}}{\mathrm{e}}$, where h is Planck's …

Q6.The angle between vector $\overrightarrow{\mathrm{Q}}$ and the resultant of $(2\overrightarrow{\mathrm{Q}}+2\overrightarrow{\mathrm{P}})$ and $(2\overrightarrow{\mathrm{Q}}-2\overrightarrow{\mathrm{P}})$ is:

Q7.In hydrogen like system the ratio of coulombian force and gravitational force between an electron and a proton is in the order of:

Q8.In a co-axial straight cable, the central conductor and the outer conductor carry equal currents in opposite directions. The magnetic field is zero.

Q9.An electron rotates in a circle around a nucleus having positive charge Ze . Correct relation between total energy (E) of electron to its potential energy ( U ) is:

Q10.If the collision frequency of hydrogen molecules in a closed chamber at $27^{\circ }\mathrm{C}$ is Z , then the collision frequency of the same system at $127^{\circ }\mathrm{C}$ is :

Q11.Ratio of radius of gyration of a hollow sphere to that of a solid cylinder of equal mass, for moment of Inertia about their diameter axis AB as shown in figure is $\sqrt{\frac{8}{x}}$. The value of $x$ is:

Q12.Two conducting circular loops A and B are placed in the same plane with their centres coinciding as shown in figure. The mutual inductance between them is:

Q13.Match list-I with list-II:

(Where a = radius of planet orbit, r = radius of planet, M = mass of Sun, m = mass of planet)
Choose the correct answer from the options given below:

Q14.A wooden block of mass 5 kg rests on soft horizontal floor. When an iron cylinder of mass 25 kg is placed on the top of the block, the floor yields and the block and the cylinder together go down with an acceleration of $0.1{\mathrm{ms}}^{-2}$. The action force of the system on t …

Q15.A simple pendulum doing small oscillations at a place $R$ height above earth surface has time period of ${\mathrm{T}}_1=4\mathrm{s}$. ${\mathrm{T}}_2$ would be it's time period if it is brought to a point which is at a height 2$R$ from earth surface. Choose the correct relation …

Q16.A body of mass 50 kg is lifted to a height of 20 m from the ground in the two different ways as shown in the figures. The ratio of work done against the gravity in both the respective cases, will be:

Q17.Time periods of oscillation of the same simple pendulum measured using four different measuring clocks were recorded as $4.62\mathrm{s}$, $4.632\mathrm{s}$, $4.6\mathrm{s}$ and $4.64$ s. The arithmetic mean of these reading in correct significant figure is.

Q18.The heat absorbed by a system in going through the given cyclic process is :

Q19.In the given figure ${\mathrm{R}}_1=10\Omega$, ${\mathrm{R}}_2=8\Omega$, ${\mathrm{R}}_3=4\Omega$ and ${\mathrm{R}}_4=8\Omega$. Battery is ideal with emf 12 V. Equivalent resistant of the circuit and current supplied by battery are respectively.

Q20.An alternating voltage of amplitude 40 V and frequency 4 kHz is applied directly across the capacitor of $12\mu \mathrm{F}$. The maximum displacement current between the plates of the capacitor is nearly:

Q21.In Young's double slit experiment, carried out with light of wavelength $5000\mathring{A}$, the distance between the slits is 0.3 mm and the screen is at 200 cm from the slits. The central maximum is at $\mathrm{x}=0\mathrm{cm}$. The value of x for third maxima is ______ mm.

Q22.A 2A current carrying straight metal wire of resistance $1\Omega$, resistivity $2\times 10^{-6}\Omega \mathrm{m}$, area of cross-section $10{\mathrm{mm}}^2$ and mass 500 g is suspended horizontally in mid air by applying a uniform magnetic field …

Q23.The electric field between the two parallel plates of a capacitor of $1.5\mu \mathrm{F}$ capacitance drops to one third of its initial value in $6.6\mu \mathrm{s}$ when the plates are connected by a thin wire. The resistance of this wire is ______ $\Omega$. (Given, …

Q24.Three blocks ${\mathrm{M}}_1$, ${\mathrm{M}}_2$, ${\mathrm{M}}_3$ having masses $4\mathrm{kg}$, $6\mathrm{kg}$ and $10\mathrm{kg}$ respectively are hanging from a smooth pully using rope 1, 2 and 3 as shown in figure. The tension in the rope $1$, $T_1$ when they are movin …

Q25.The density and breaking stress of a wire are $6\times 10^4\mathrm{kg}/{\mathrm{m}}^3$ and $1.2\times 10^8\mathrm{N}/{\mathrm{m}}^2$ respectively. The wire is suspended from a rigid support on a planet where acceleration due to gravity is $\frac{1}{3}$ of the …

Q26.A body moves on a frictionless plane starting from rest. If ${\mathrm{S}}_{\mathrm{n}}$ is distance moved between $\mathrm{t}=\mathrm{n}-1$ and t $=n$ and $S_{n-1}$ is distance moved between $t=n-2$ and $\mathrm{t}=\mathrm{n}-1$, then the ratio …

Q27.If three helium nuclei combine to form a carbon nucleus then the energy released in this reaction is ______ $\times 10^{-2}\mathrm{MeV}$. (Given $1\mathrm{u}=931\mathrm{MeV}/{\mathrm{c}}^2$, atomic mass of helium $=4.002603\mathrm{u}$)

Q28.An ac source is connected in given series LCR circuit. The rms potential difference across the capacitor of $20\mu \mathrm{F}$ is ______ V.

Q29.In the experiment to determine the galvanometer resistance by half-deflection method, the plot of $\frac{1}{\theta }$ vs the resistance (R) of the resistance box is shown in the figure. The figure of merit of the galvanometer is ______ $\times 10^{-1}\mathrm{A}$/division. [T …

Q30.Three capacitors of capacitances $25\mu \mathrm{F}$, $30\mu \mathrm{F}$ and $45\mu \mathrm{F}$ are connected in parallel to a supply of 100 V. Energy stored in the above combination is E. When these capacitors are connected in series to the same supply, the stored ene …

Q1.The incorrect postulates of the Dalton's atomic theory are :
(A) Atoms of different elements differ in mass.
(B) Matter consists of divisible atoms.
(C) Compounds are formed when atoms of different element combine in a fixed ratio.
(D) All the atoms of given element have differen …

Q2.The following reaction occurs in the Blast furnance where iron ore is reduced to iron metal
${\mathrm{Fe}}_2{\mathrm{O}}_{3(\mathrm{s})}+3{\mathrm{CO}}_{(\mathrm{g})}\rightleftharpoons {\mathrm{Fe}}_{(\mathrm{l})}+3{\mathrm{CO}}_{2(\mathrm{g})}$
Using the Le-chatelier's principle …

Q3.Identify compound (Z) in the following reaction sequence.

Q4.Given below are two statements : One is labelled as Assertion (A) and the other is labelled as Reason (R)
Assertion (A): Enthalpy of neutralisation of strong monobasic acid with strong monoacidic base is always $-57\mathrm{kJ}{\mathrm{mol}}^{-1}$
Reason (R): Enthalpy of neutr …

Q5.The statement(s) that are correct about the species ${\mathrm{O}}^{2-},{\mathrm{F}}^{-},{\mathrm{Na}}^{+}$and ${\mathrm{Mg}}^{2+}$.
(A) All are isoelectronic
(B) All have the same nuclear charge
(C) ${\mathrm{O}}^{2-}$ has the largest ionic radii
(D) ${\mathrm{Mg}}^{2+}$ has the sm …

Q6.For the compounds:
(A) ${\mathrm{H}}_3\mathrm{C}-{\mathrm{CH}}_2-\mathrm{O}-{\mathrm{CH}}_2-{\mathrm{CH}}_2-{\mathrm{CH}}_3$
(B) ${\mathrm{H}}_3\mathrm{C}-{\mathrm{CH}}_2-{\mathrm{CH}}_2-{\mathrm{CH}}_2-{\mathrm{CH}}_3$
(C) (see structure)
(D) (see structure)

The increasin …

Q7.Given below are two statements :

Statement I: In group 13, the stability of +1 oxidation state increases down the group.
Statement II: The atomic size of gallium is greater than that of aluminium.
In the light of the above statements, choose the most appropriate answer from the …

Q8.Number of $\sigma$ and $\pi$ bonds present in ethylene molecule is respectively :

Q9.Identify ' A ' in the following reaction :

Q10.The reaction at cathode in the cells commonly used in clocks involves.

Q11.Which one of the following complexes will exhibit the least paramagnetic behaviour ?
[Atomic number, $\mathrm{Cr}=24,\mathrm{Mn}=25,\mathrm{Fe}=26,\mathrm{Co}=27$ ]

Q12.Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): Cis form of alkene is found to be more polar than the trans form
Reason (R): Dipole moment of trans isomer of 2-butene is zero.

In the light of the above sta …

Q13.Given below are two statements :

Statement I: Nitration of benzene involves the following step -

Statement II: Use of Lewis base promotes the electrophilic substitution of benzene.

In the light of the above statements, choose the most appropriate answer from the options given …

Q14.The correct order of ligands arranged in increasing field strength.

Q15.Which of the following gives a positive test with ninhydrin?

Q16.The metal that shows highest and maximum number of oxidation state is:

Q17.Ail organic compound has $42.1\%$ carbon, $6.4\%$ hydrogen and remainder is oxygen. If its molecular weight is 342 , then its molecular formula is :

Q18.Given below are two statement :

Statement I: Bromination of phenol in solvent with low polarity such as ${\mathrm{CHCl}}_3$ or ${\mathrm{CS}}_2$ requires Lewis acid catalyst.
Statement II : The lewis acid catalyst polarises the bromine to generate ${\mathrm{Br}}^{+}$.
In the …

Q19.Molar ionic conductivities of divalent cation and anion are $57\mathrm{S}{\mathrm{cm}}^2{\mathrm{mol}}^{-1}$ and $73\mathrm{S}{\mathrm{cm}}^2{\mathrm{mol}}^{-1}$ respectively. The molar conductivity of solution of an electrolyte with the above cation and anion will be …

Q20.The number of neutrons present in the more abundant isotope of boron is ' x '. Amorphous boron upon heating with air forms a product, in which the oxidation state of boron is ' y '. The value of $\mathrm{x}+\mathrm{y}$ is ...

Q21.The value of Rydberg constant $({\mathrm{R}}_{\mathrm{H}})$ is $2.18\times 10^{-18}\mathrm{J}$. The velocity of electron having mass $9.1\times 10^{-31}\mathrm{kg}$ in Bohr's first orbit of hydrogen atom $=\dots \dots \dots \times 10^5{\mathrm{ms}}^{-1}$ (nearest …

Q22.In a borax bead test under hot condition, a metal salt (one from the given) is heated at point B of the flame, resulted in green colour salt bead. The spin-only magnetic moment value of the salt is
____ BM (Nearest integer)
[Given atomic number of …

Q23.The heat of combustion of solid benzoic acid at constant volume is -321.30 kJ at $27^{\circ }\mathrm{C}$. The heat of combustion at constant pressure is ( $-321.30-x\mathrm{R}$ ) kJ , the value of x is ____

Q24.Consider the given chemical reaction sequence :

Total sum of oxygen atoms in Product A and Product B are ____

Q25.The spin only magnetic moment value of the ion among ${\mathrm{Ti}}^{2+},{\mathrm{V}}^{2+},{\mathrm{Co}}^{3+}$ and ${\mathrm{Cr}}^{2+}$, that acts as strong oxidising agent in aqueous solution is
____ BM (Near integer).
(Given atomic numbers : Ti : 22, V : 23, Cr : 24, Co : 27)

Q26.During Kinetic study of reaction $2\mathrm{A}+\mathrm{B}\to \mathrm{C}+\mathrm{D}$, the following results were obtained :
| I | 0.1 | 0.1 | …

Q27.An artificial cell is made by encapsulating 0.2 M glucose solution within a semipermeable membrane. The osmotic pressure developed when the artificial cell is placed within a 0.05 M solution of NaCl at 300 K is ____ $\times 10^{-1}$ bar.
(Nearest Integer)
[Given : …

Q28.The number of halobenzenes from the following that can be prepared by Sandmeyer's reaction is ____

Q29.In the lewis dot structure for ${\mathrm{NO}}_2{}^{-}$, total number of valence electrons around nitrogen is ____

Q30.9.3 g of pure aniline is treated with bromine water at room temperature to give a white precipitate of the product 'P'. The mass of product 'P' obtained is 26.4 g. The percentage yield is ____ %.

Q1.Suppose $\theta \in \left[0,\frac{\pi }{4}\right]$ is a solution of $4\cos \theta -3\sin \theta =1$. Then $\cos \theta$ is equal to:

Q2.If $\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\dots +\frac{1}{\sqrt{99}+\sqrt{100}}=m$ and $\frac{1}{1.2}+\frac{1}{2.3}+\dots +\frac{1}{99.100}=n$, then the point $(m,n)$ lies on the line

Q3.The integral ${\int }_0^{\pi /4}\frac{136\sin x}{3\sin x+5\cos x}dx$ is equal to:

Q4.Let a circle C of radius 1 and closer to the origin be such that the lines passing through the point $(3,2)$ and parallel to the coordinate axes touch it. Then the shortest distance of the circle C from the point $(5,5)$ is:

Q5.If the function $f(x)=\frac{\sin 3x+\alpha \sin x-\beta \cos 3x}{x^3},x\in R$, is continuous at $x=0$, then $f(0)$ is equal to:

Q6.If $y=y(x)$ is the solution of the differential equation $\frac{dy}{dx}+2y=\sin (2x),y(0)=\frac{3}{4}$, then $y\left(\frac{\pi }{8}\right)$ is equal to:

Q7.Let $A$ and $B$ be two square matrices of order 3 such that $|A|=3$ and $|B|=2$. Then $\left|A^{T}A(\mathrm{adj}(2A){)}^{-1}(\mathrm{adj}(4B))(\mathrm{adj}(AB){)}^{-1}A{A}^T\right|$ is equal to:

Q8.If $A(1,-1,2),B(5,7,-6),C(3,4,-10)$ and $D(-1,-4,-2)$ are the vertices of a quadrilateral ABCD, then its area is

Q9.For the function $f(x)=\sin x+3x-\frac{2}{\pi }(x^2+x)$, where $x\in [0,\frac{\pi }{2}]$, consider the following two statements: (I) f is increasing in $(0,\frac{\pi }{2})$; (II) $f^{\prime }$ is decreasing in $(0,\frac{\pi }{2})$. Between the above two statements,

Q10.If the line $\frac{2-x}{3}=\frac{3y-2}{4\lambda +1}=4-z$ makes a right angle with the line $\frac{x+3}{3\mu }=\frac{1-2y}{6}=\frac{5-z}{7}$, then $4\lambda +9\mu$ is equal to:

Q11.Let the line $2x+3y-k=0,k>0$, intersect the x-axis and y-axis at the points A and B, respectively. If the equation of the circle having the line segment AB as a diameter is $x^2+y^2-3x-2y=0$ and the length of the latus rectum of the ellipse $x^2+9y^2=k^2$ is …

Q12.Consider the following two statements: Statement I: For any two non-zero complex numbers $z_1,z_2$, $(|z_1|+|z_2|)\left|\frac{z_1}{|z_1|}+\frac{z_2}{|z_2|}\right|\le 2(|z_1|+|z_2|)$; Statement II: if $x,y,z$ are three distinct complex numbers and $a,b,c$ are three …

Q13.The value of ${\int }_{-\pi }^{\pi }\frac{2y(1+\sin y)}{1+{\cos }^{2}y}dy$ is

Q14.If the system of equations $11x+y+\lambda z=-5$, $2x+3y+5z=3$, $8x-19y-39z=\mu$ has infinitely many solutions, then ${\lambda }^4-\mu$ is equal to

Q15.Let $d$ be the distance of the point of intersection of the lines $\frac{x+6}{3}=\frac{y}{2}=\frac{z+1}{1}$ and $\frac{x-7}{4}=\frac{y-9}{3}=\frac{z-4}{2}$ from the point $(7,8,9)$. Then $d^2+6$ is equal to

Q16.Let $f(x)=x^5+2x^3+3x+1,x\in R$, and $g(x)$ be a function such that $f(g(x))=x$ for all $x\in R$. Then $\frac{g(7)}{g^{\prime }(7)}$ is equal to:

Q17.The coefficients $a,b,c$, in the quadratic equation $a{x}^2+bx+c=0$ are chosen from the set $\{1,2,3,4,5,6,7,8\}$. The probability of this equation having repeated roots is

Q18.If the constant term in the expansion of $(1+2x-3x^3){\left(\frac{3}{2}{x}^2-\frac{1}{3x}\right)}^9$ is $p$, then $108p$ is equal to ____

Q19.From a lot of 10 items, which include 3 defective items, a sample of 5 items is drawn at random. Let the random variable $X$ denote the number of defective items in the sample. If the variance of $X$ is ${\sigma }^2$, then $96{\sigma }^2$ is equal to ____.

Q20.Let $a_1,a_2,a_3,\dots$ be in an arithmetic progression of positive terms. Let $A_k=a_1^2-a_2^2+a_3^2-a_4^2+\dots +a_{2k-1}^2-a_{2k}^2$. If $A_3=-153,A_5=-435$ and $a_1^2+a_2^2+a_3^2=66$, then $a_{17}-A_7$ is equal to ____.

Q21.The area of the region enclosed by the parabolas $y=x^2-5x$ and $y=7x-x^2$ is ____.

Q22.Let $f$ be a differentiable function in the interval $(0,\infty )$ such that $f(1)=1$ and ${\lim }_{t\to x}\frac{t^{2}f(x)-x^{2}f(t)}{t-x}=1$ for each $x>0$. Then $2f(2)+3f(3)$ is equal to ____.

Q23.Let $\vec{a}=\hat{i}-3\hat{j}+7\hat{k},\vec{b}=2\hat{i}-\hat{j}+\hat{k}$ and $\vec{c}$ be a vector such that $(\vec{a}+2\vec{b})\times \vec{c}=3(\vec{c}\times \vec{a})$. If $\vec{a}\cdot \vec{c}=130$, then $\vec{b}\cdot \vec{c}$ is equal to ____.

Q24.The number of ways of getting a sum 16 on throwing a dice four times is ____.

Q25.If $S=\{a\in R:|2a-1|=3[a]+2\{a\}\}$, where $[t]$ denotes the greatest integer less than or equal to $t$ and $\{t\}$ represents the fractional part of $t$, then $72{\sum }_{a\in S}a$ is equal to ____.

Q26.The number of distinct real roots of the equation $|x||x+2|-5|x+1|-1=0$ is ____.