JEE Main 2024 · 06 Apr 2024 · Shift 1

Q1.To find the spring constant (k) of a spring experimentally, a student commits $2\%$ positive error in the measurement of time and $1\%$ negative error in measurement of mass. The percentage error in determining value of k is :

Q2.A bullet of mass 50 g is fired with a speed $100\mathrm{m}/\mathrm{s}$ on a plywood and emerges with $40\mathrm{m}/\mathrm{s}$. The percentage loss of kinetic energy is :

Q3.The ratio of the shortest wavelength of Balmer series to the shortest wavelength of Lyman series for hydrogen atom is :

Q4.To project a body of mass m from earth's surface to infinity, the required kinetic energy is (assume, the radius of earth is ${\mathrm{R}}_{\mathrm{E}},\mathrm{g}=$ acceleration due to gravity on the surface of earth) :

Q5.Electromagnetic waves travel in a medium with speed of $1.5\times 10^8{\mathrm{ms}}^{-1}$. The relative permeability of the medium is 2.0 . The relative permittivity will be :

Q6.Which of the following phenomena does not explain by wave nature of light.
(A) reflection
(B) diffraction
(C) photoelectric effect
(D) interference
(E) polarization

Choose the most appropriate answer from the options given below :

Q7.While measuring diameter of wire using screw gauge the following readings were noted. Main scale reading is 1 mm and circular scale reading is equal to 42 divisions. Pitch of screw gauge is 1 mm and it has 100 divisions on circular scale. The diameter of the wire is …

Q8.$\sigma$ is the uniform surface charge density of a thin spherical shell of radius R . The electric field at any point on the surface of the spherical shell is :

Q9.The value of unknown resistance ( x ) for which the potential difference between B and D will be zero in the arrangement shown, is :

Q10.The specific heat at constant pressure of a real gas obeying ${\mathrm{PV}}^2=\mathrm{RT}$ equation is:

Q11.Match List I with List II

Q12.Given below are two statements :

Statement I: In an LCR series circuit, current is maximum at resonance.

Statement II : Current in a purely resistive circuit can never be less than that in a series LCR circuit when connected to same voltage source.

In the light of the above st …

Q13.The correct truth table for the following logic circuit is :

Q14.A sample contains mixture of helium and oxygen gas. The ratio of root mean square speed of helium and oxygen in the sample, is :

Q15.A light string passing over a smooth light pulley connects two blocks of masses ${\mathrm{m}}_1$ and ${\mathrm{m}}_2$ (where ${\mathrm{m}}_2>{\mathrm{m}}_1$). If the acceleration of the system is $\frac{\mathrm{g}}{\sqrt{2}}$, then the ratio of the masses …

Q16.Four particles $\mathrm{A},\mathrm{B},\mathrm{C},\mathrm{D}$ of mass $\frac{\mathrm{m}}{2},\mathrm{m},2\mathrm{m},4\mathrm{m}$, have same momentum, respectively. The particle with maximum kinetic energy is :

Q17.A train starting from rest first accelerates uniformly up to a speed of $80\mathrm{km}/\mathrm{h}$ for time t, then it moves with a constant speed for time 3t. The average speed of the train for this duration of journey will be (in $\mathrm{km}/\mathrm{h}$) :

Q18.An element $\Delta l=\Delta \mathrm{x}\hat{\mathrm{i}}$ is placed at the origin and carries a large current $\mathrm{I}=10\mathrm{A}$. The magnetic field on the y-axis at a distance of 0.5 m from the elements $\Delta \mathrm{x}$ of 1 cm length is :

Q19.A small ball of mass $m$ and density $\rho$ is dropped in a viscous liquid of density ${\rho }_0$. After sometime, the ball falls with constant velocity. The viscous force on the ball is :

Q20.In photoelectric experiment energy of 2.48 eV irradiates a photo sensitive material. The stopping potential was measured to be 0.5 V. Work function of the photo sensitive material is :

Q21.If the radius of earth is reduced to three-fourth of its present value without change in its mass then value of duration of the day of earth will be ______ hours 30 minutes.

Q22.Three infinitely long charged thin sheets are placed as shown in figure. The magnitude of electric field at the point P is $\frac{\mathrm{x}\sigma }{{ϵ}_0}$. The value of x is ______ (all quantities are measured in SI units).

Q23.A big drop is formed by coalescing 1000 small droplets of water. The ratio of surface energy of 1000 droplets to that of energy of big drop is $\frac{10}{x}$. The value of $x$ is ______.

Q24.When a dc voltage of 100 V is applied to an inductor, a dc current of 5A flows through it. When an ac voltage of 200 V peak value is connected to inductor, its inductive reactance is found to be $20\sqrt{3}\Omega$. The power dissipated in the circuit is ______ W.

Q25.The refractive index of prism is $\mu =\sqrt{3}$ and the ratio of the angle of minimum deviation to the angle of prism is one. The value of angle of prism is ______ ${}^{\circ }$.

Q26.A wire of resistance R and radius r is stretched till its radius became $r/2$. If new resistance of the stretched wire is x R, then value of x is ______.

Q27.Radius of a certain orbit of hydrogen atom is $8.48\text{ A˚}$. If energy of electron in this orbit is $E/x$, then $x=$ ______.
(Given $a_0=0.529\text{ A˚}$, $E=$ energy of electron in ground state)

Q28.A circular coil having 200 turns, $2.5\times 10^{-4}{\text{ m}}^2$ area and carrying $100\mu \text{A}$ current is placed in a uniform magnetic field of 1 T. Initially the magnetic dipole moment ($\vec{M}$) was directed along $\vec{B}$. Amount of work, required to rotate t …

Q29.A particle is doing simple harmonic motion of amplitude 0.06 m and time period 3.14 s. The maximum velocity of the particle is ______ cm/s.

Q30.For three vectors $\vec{A}=(-x\hat{i}-6\hat{j}-2\hat{k})$, $\vec{B}=(-\hat{i}+4\hat{j}+3\hat{k})$ and $\vec{C}=(-8\hat{i}-\hat{j}+3\hat{k})$, if $\vec{A}\cdot (\vec{B}\times \vec{C})=0$, then value of x is ______.

Q1.Functional group present in sulphonic acid is :

Q2.Match List I with List II :

Choose …

Q3.Given below are two statements :

Statement I: Picric acid is 2, 4, 6-trinitrotoluene.
Statement II : Phenol-2, 4-disulphuric acid is treated with conc. ${\mathrm{HNO}}_3$ to get picric acid.

In the light of the above statement, choose the most appropriate answer from the option …

Q4.Which of the following is metamer of the given compound (X) ?

Q5.DNA molecule contains 4 bases whoes structure are shown below. One of the structure is not correct, identify the incorrect base structure.

Q6.Match List I with List II :
| D. …

Q7.Given below are two statements :

Statement I : Gallium is used in the manufacturing of thermometers.

Statement II : A thermometer containing gallium is useful for measuring the freezing point ( 256 K ) of brine solution.

In the light of the above statement, choose the correct …

Q8.Which of the following statements are correct?
A. Glycerol is purified by vacuum distillation because it decomposes at its normal boiling point.
B. Aniline can be purified by steam distillation as aniline is miscible in water.
C. Ethanol can be separated from ethanol water mixtur …

Q9.Match List I with List II :

Ch …

Q10.In Reimer - Tiemann reaction, phenol is converted into salicylaldehyde through an intermediate. The structure of intermediate is ______ .

Q11.Which of the following material is not a semiconductor.

Q12.Consider the following complexes.
${\left[\mathrm{CoCl}{\left({\mathrm{NH}}_3\right)}_5\right]}^{2+}$,
${\left[\mathrm{Co}(\mathrm{CN}{)}_6\right]}^{3-}$,
(A)
(B)
${\left[\mathrm{Co}{\left({\mathrm{NH}}_3\right)}_5\left({\mathrm{H}}_2\mathrm{O}\right)\right]}^{3+}$, …

Q13.Match List I with List II :
| B. | …

Q14.The electron affinity value are negative for :
A. $\mathrm{Be}\to {\mathrm{Be}}^{-}$
B. $\mathrm{N}\to {\mathrm{N}}^{-}$
C. $\mathrm{O}\to {\mathrm{O}}^{2-}$
D. $\mathrm{Na}\to {\mathrm{Na}}^{-}$
E. $\mathrm{Al}\to {\mathrm{Al}}^{-}$ …

Q15.The number of element from the following that do not belong to lanthanoids is :
$\mathrm{Eu},\mathrm{Cm},\mathrm{Er},\mathrm{Tb},\mathrm{Yb}$ and Lu

Q16.The density of ' x ' M solution (' x ' molar) of NaOH is $1.12\mathrm{g}{\mathrm{mL}}^{-1}$. while in molality, the concentration of the solution is 3 m ( 3 molal). Then x is
(Given : Molar mass of NaOH is $40\mathrm{g}/\mathrm{mol}$ )

Q17.Which among the following aldehydes is most reactive towards nucleophilic addition reactions?

Q18.At $-20^{\circ }\mathrm{C}$ and 1 atm pressure, a cylinder is filled with equal number of ${\mathrm{H}}_2$. ${\mathrm{I}}_2$ and HI molecules for the reaction
${\mathrm{H}}_2(\mathrm{g})+{\mathrm{I}}_2(\mathrm{g})\rightleftharpoons 2\mathrm{HI}(\mathrm{g})$, the …

Q19.Match List I with List II :

Choose the correct answer fro …

Q20.A conductivity cell with two electrodes (dark side) are half filled with infinitely dilute aqueous solution of a weak electrolyte. If volume is doubled by adding more water at constant temperature, the molar conductivity of the cell will -

Q21.Consider the dissociation of the weak acid HX as given below
$\mathrm{HX}(\mathrm{aq})\rightleftharpoons {\mathrm{H}}^{+}(\mathrm{aq})+{\mathrm{X}}^{-}(\mathrm{aq}),\mathrm{Ka}=1.2\times 10^{-5}$
[ ${\mathrm{K}}_{\mathrm{a}}$ : dissociation constant]
The osmotic pressure of 0.03 …

Q22.The difference in the 'spin-only' magnetic moment values of ${\mathrm{KMnO}}_4$ and the manganese product formed during titration of ${\mathrm{KMnO}}_4$ against oxalic acid in acidic medium is ____ BM. (nearest integer)

Q23.Time required for $99.9\%$ completion of a first order reaction is ____ time the time required for completion of $90\%$ reaction.(nearest integer).

Q24.Number of molecules from the following which can exhibit hydrogen bonding is ____. (nearest integer)

$\mathrm{HF},{\mathrm{NH}}_3$

Q25.9.3 g of pure aniline upon diazotisation followed by coupling with phenol gives an orange dye. The mass of orange dye produced (assume 100% yield/ conversion) is ______ g. (nearest integer)

Q26.The major product of the following reaction is P.

Number of oxygen atoms present in product 'P' is ______ (nearest integer).

Q27.Frequency of the de-Broglie wave of electron in Bohr's first orbit of hydrogen atom is ______ × 10¹³ Hz (nearest integer).
[Given: Rₕ (Rydberg constant) = 2.18 × 10⁻¹⁸ J. h (Plank's constant) = 6.6 × 10⁻³⁴ J.s.]

Q28.The major products from the following reaction sequence are product A and product B.

The total sum of π electrons in product A and product B are ______ (nearest integer)

Q29.Among CrO, Cr₂O₃ and CrO₃, the sum of spin-only magnetic moment values of basic and amphoteric oxides is ______ × 10⁻² BM (nearest integer).
(Given atomic number of Cr is 24)

Q30.An ideal gas, ${\overline{C}}_V=\frac{5}{2}R$, is expanded adiabatically against a constant pressure of 1 atm until it doubles in volume. If the initial temperature and pressure is 298 K and 5 atm, respectively then the final temperature is ______ K (nearest integer).
[ …

Q1.Let $y=y(x)$ be the solution of the differential equation $(2x{\log }_{e}x)\frac{dy}{dx}+2y=\frac{3}{x}{\log }_{e}x$, $x>0$ and $y(e^{-1})=0$. Then, $y(e)$ is equal to

Q2.A company has two plants A and B to manufacture motorcycles. 60% motorcycles are manufactured at plant A and the remaining are manufactured at plant B. 80% of the motorcycles manufactured at plant A are rated of the standard quality, while 90% of the motorcycles manufactured at p …

Q3.Let $A=\{n\in [100,700]\cap N:n\text{ is neither a multiple of 3 nor a multiple of 4}\}$. Then the number of elements in $A$ is

Q4.Let a variable line of slope $m>0$ passing through the point $(4,-9)$ intersect the coordinate axes at the points $A$ and $B$. The minimum value of the sum of the distances of $A$ and $B$ from the origin is

Q5.If $f(x)=\{\begin{array}{ll}{x}^3\sin \left(\frac{1}{x}\right),& x\ne 0\\ 0,& x=0\end{array}$, then

Q6.A circle is inscribed in an equilateral triangle of side of length 12. If the area and perimeter of any square inscribed in this circle are $m$ and $n$, respectively, then $m+n^2$ is equal to

Q7.Let $y=y(x)$ be the solution of the differential equation $(1+x^2)\frac{dy}{dx}+y=e^{{\tan }^{-1}x}$, $y(1)=0$. Then $y(0)$ is

Q8.For $\alpha ,\beta \in R$ and a natural number $n$, let $A_r=\left|\begin{array}{ccc}r& 1& \frac{n^2}{2}+\alpha \\ 2r& 2& n^2-\beta \\ 3r-2& 3& \frac{n(3n-1)}{2}\end{array}\right|$. Then $2A_{10}-A_8$ is

Q9.The mean and standard deviation of 20 observations are found to be 10 and 2, respectively. On rechecking, it was found that an observation by mistake was taken 8 instead of 12. The correct standard deviation is

Q10.${\int }_0^{\pi /4}\frac{{\cos }^{2}x{\sin }^{2}x}{({\cos }^{3}x+{\sin }^{3}x{)}^2}dx$ is equal to

Q11.Let the relations $R_1$ and $R_2$ on the set $X=\{1,2,3,\dots ,20\}$ be given by $R_1=\{(x,y):2x-3y=2\}$ and $R_2=\{(x,y):-5x+4y=0\}$. If $M$ and $N$ be the minimum number of elements required to be added in $R_1$ and $R_2$, respectively, in order to make the …

Q12.The interval in which the function $f(x)=x^x$, $x>0$, is strictly increasing is

Q13.The function $f(x)=\frac{x^2+2x-15}{x^2-4x+9}$, $x\in R$ is

Q14.The shortest distance between the line $\frac{x-3}{2}=\frac{y+15}{-7}=\frac{z-9}{5}$ and $\frac{x+1}{2}=\frac{y-1}{1}=\frac{z-9}{-3}$ is

Q15.Let $\alpha ,\beta$ be the distinct roots of the equation $x^2-(t^2-5t+6)x+1=0$, $t\in R$ and $a_n={\alpha }^n+{\beta }^n$. Then the minimum value of $\frac{a_{2023}+a_{2025}}{a_{2024}}$ is

Q16.The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is

Q17.Let $f:(-\infty ,\infty )-\{0\}\to R$ be a differentiable function such that $f^{\prime }(1)={\lim }_{a\to \infty }{a}^{2}f\left(\frac{1}{a}\right)$. Then ${\lim }_{a\to \infty }\frac{a(a+1)}{2}{\tan }^{-1}\left(\frac{1}{a}\right)+a^2-2{\log }_{e}a$ is equal to

Q18.Let the first term of a series be $T_1=6$ and its $r^{th}$ term $T_r=3T_{r-1}+6^r$, $r=2,3,\dots ,n$. If the sum of the first $n$ terms of this series is $\frac{1}{5}(n^2-12n+39)(4\cdot 6^n-5\cdot 3^n+1)$, then $n$ is equal to ____.

Q19.If the second, third and fourth terms in the expansion of $(x+y{)}^n$ are 135, 30 and $\frac{10}{3}$, respectively, then $6(n^3+x^2+y)$ is equal to ____.

Q20.Let $x_1,x_2,x_3,x_4$ be the solution of the equation $4x^4+8x^3-17x^2-12x+9=0$ and $(4+x_1^2)(4+x_2^2)(4+x_3^2)(4+x_4^2)=\frac{125}{16}m$. Then the value of $m$ is ____.

Q21.Let $r_k=\frac{{\int }_0^1(1-x^7{)}^{k}dx}{{\int }_0^1(1-x^7{)}^{k+1}dx}$, $k\in N$. Then the value of ${\sum }_{k=1}^{10}\frac{1}{7(r_k-1)}$ is equal to ____.

Q22.Let $\vec{a}=2\hat{i}-3\hat{j}+4\hat{k}$, $\vec{b}=3\hat{i}+4\hat{j}-5\hat{k}$ and a vector $\vec{c}$ be such that $\vec{a}\times (\vec{b}+\vec{c})+\vec{b}\times \vec{c}=\hat{i}+8\hat{j}+13\hat{k}$. If $\vec{a}\cdot \vec{c}=13$, then $(24-\vec{b}\cdot \vec{c})$ is equal t …

Q23.For $n\in N$, if ${\cot }^{-1}3+{\cot }^{-1}4+{\cot }^{-1}5+{\cot }^{-1}n=\frac{\pi }{4}$, then $n$ is equal to ____.

Q24.Let $P$ be the point $(10,-2,-1)$ and $Q$ be the foot of the perpendicular drawn from the point $R(1,7,6)$ on the line passing through the points $(2,-5,11)$ and $(-6,7,-5)$. Then the length of the line segment $PQ$ is equal to ____.

Q25.Let a conic $C$ pass through the point $(4,-2)$ and $P(x,y)$, $x\ge 3$, be any point on $C$. Let the slope of the line touching the conic $C$ only at a single point $P$ be half the slope of the line joining the points $P$ and $(3,-5)$. If the focal distance of …

Q26.Let $\alpha \beta \gamma =45$; $\alpha ,\beta ,\gamma \in R$. If $x(\alpha ,1,2)+y(1,\beta ,2)+z(2,3,\gamma )=(0,0,0)$ for some $x,y,z\in R$, $xyz\ne 0$, then $6\alpha +4\beta +\gamma$ is equal to ____.