JEE Main 2024 — 6 April, Shift 1

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

If $\mathrm{A}(3,1,-1), \mathrm{B}\left(\frac{5}{3}, \frac{7}{3}, \frac{1}{3}\right), \mathrm{C}(2,2,1)$ and D $\left(\frac{10}{3}, \frac{2}{3}, \frac{-1}{3}\right)$ are the vertices of a quadrilateral ABCD , then its

area is

$\int_{0}^{\pi / 4} \frac{\cos ^{2} x \sin ^{2} x}{\left(\cos ^{3} x+\sin ^{3} x\right)^{2}} d x$ is equal to

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

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

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

Let C be the circle of minimum area touching the parabola $y=6-x^{2}$ and the lines $y=\sqrt{3}|x|$. Then, which one of the following points lies on the circle C ?

For $\alpha, \beta \in \mathrm{R}$ and a natural number n , let ${{A}_{r}}=\left| \begin{matrix}r & 1 & \frac{{{n}^{2}}}{2}+\alpha \\2r & 2 & {{n}^{2}}-\beta \\3r-2 & 3 & \frac{n\left( 3n-1 \right)}{2} \\\end{matrix} \right|$.Then $2 A_{10}-A_{8}$ is

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

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 plant B are rated of the standard quality. A motorcycle picked up randomly from the total production is found to be of the standard quality. If $p$ is the probability that it was manufactured at plant $B$, then 126 p is

Let, $\alpha, \beta$ be the distinct roots of the equation $x^{2}-\left(t^{2}-5 t+6\right) x+1=0, t \in R$ and $a_{n}=\alpha^{n}+\beta^{n}$.

Then the minimum value of $\frac{a_{2023}+a_{2025}}{a_{2024}}$ is

Let the relations $R_{1}$ and $R_{2}$ on the set $\mathrm{X}=\{1,2,3, \ldots, 20\}$ be given by $\mathrm{R}_{1}=\{(\mathrm{x}, \mathrm{y}): 2 \mathrm{x}-3 \mathrm{y}=2\}$ and

$R_{2}=\{(x, y):-5 x+4 y=0\}$. 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 relations symmetric, then $\mathrm{M}+\mathrm{N}$ equals

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

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

A circle in 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 $m+\mathrm{n}^{2}$ is equal to

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

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

Let $y=y(x)$ be the solution of the differential equation $\left(2 x \log _{e} x\right) \frac{d y}{d x}+2 y=\frac{3}{x} \log _{e} x, x>0$

and $\mathrm{y}\left(\mathrm{e}^{-1}\right)=0$. Then, $\mathrm{y}(\mathrm{e})$ is equal to

Let the area of the region enclosed by the curves $y=3 x, 2 y=27-3 x$ and $y=3 x-x \sqrt{x}$ be A. Then 10 A is equal to

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

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 $x, y, z \in R, x y z \neq$ 0 , then $6 \alpha+4 \beta+\gamma$ is equal to $\qquad$

Let a conic $C$ pass through the point $(4,-2)$ and $P(x, y), x \geq 3$, 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 the point $(7,1)$ on C is d , then 12 d equals $\qquad$

Let $r_{k}=\frac{\int_{0}^{1}\left(1-x^{7}\right)^{k} d x}{\int_{0}^{1}\left(1-x^{7}\right)^{k+1} d x}, k \in N$. Then the value of $\sum_{\mathrm{k}=1}^{10} \frac{1}{7\left(\mathrm{r}_{\mathrm{k}}-1\right)}$ is equal to $\qquad$

Let $x_{1}, x_{2}, x_{3}, x_{4}$ be the solution of the equation $4 x^{4}+8 x^{3}-17 x^{2}-12 x+9=0$ and $\left(4+x_{1}^{2}\right)\left(4+x_{2}^{2}\right)\left(4+x_{3}^{2}\right)\left(4+x_{4}^{2}\right)=\frac{125}{16} m$ . Then the value of $m$ is $\qquad$

Let $\mathrm{L}_{1}, \mathrm{~L}_{2}$ be the lines passing through the point $\mathrm{P}(0,1)$ and touching the parabola $9 x^{2}+12 x+18 y-14=0$.

Let $Q$ and $R$ be the points on the lines $L_{1}$ and $L_{2}$ such that the $\triangle P Q R$ is an isosceles triangle with base QR .

If the slopes of the lines $Q R$ are $m_{1}$ and $m_{2}$. then $16\left(m_{1}^{2}+m_{2}^{2}\right)$ is equal to $\qquad$

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

Let the first term of a series be $\mathrm{T}_{1}=6$ and its $\mathrm{r}^{\text {th }}$ term $T_{r}=3 T_{r-1}+6^{r}, r=2,3, \ldots ., n$. If the sum of the

first n terms of this series is $\frac{1}{5}\left(\mathrm{n}^{2}-12 \mathrm{n}+39\right)$ $\left(4.6^{\mathrm{n}}-5.3^{\mathrm{n}}+1\right)$. Then n is equal to $\qquad$

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

Let $P$ be the point $(10,-2,-1)$ and $Q$ be the foot of the perpendicular drawn from the point $\mathrm{R}(1,7,6)$ 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 $\qquad$

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 to $\qquad$

To find the spring constant $(\mathrm{k})$ 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 :

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 :

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

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

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 :

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 :

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 $\frac{x}{50} \mathrm{~mm}$. The value of $x$ is :

$\sigma$ 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 :

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

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

Match List I with List II

Choose the correct answer from the options given below :

Given below are two statements : one is labelled as Statement I and the other is labelled as Statement II : 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 statements, choose the correct from the options given below :

The correct truth table for the following logic circuit is :

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

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 $\frac{\mathrm{m}_{1}}{\mathrm{~m}_{2}}$ is :

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

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 3 t . The average speed of the train for this duration of journey will be (in $\mathrm{km} / \mathrm{h}$ ) :

An element $\Delta l=\Delta x \hat{i}$ is placed at the origin and carries a large current $I=10 \mathrm{~A}$. 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 :

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 :

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 :

Three infinitely long charged thin sheets are placed as shown in figure. The magnitude of electric field at the point P is $\frac{\mathrm{x} \sigma}{\epsilon_{0}}$. The value of x is $\qquad$

(all quantities are measured in SI units).

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 $\qquad$ .

When a dc voltage of 100 V is applied to an inductor, a de 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 $\qquad$ W.

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 $\qquad$ ${ }^{\circ}$.

A wire of resistance $R$ and radius $r$ is stretched till its radius became $\mathrm{r} / 2$. If new resistance of the stretched wire is $x R$, then value of $x$ is $\qquad$ .

Radius of a certain orbit of hydrogen atom is 8.48 Å. If energy of electron in this orbit is $\mathrm{E} / \mathrm{x}$, then $\mathrm{x}=$ $\_\_\_\_\_\_\_\_$ . (Given ${{a}_{0}}=0.529\overset{0}{\mathop{A}}\,,\,\,\,E=$ energy of electron in ground state)

A circular coil having 200 turns, $2.5 \times 10^{-4} \mathrm{~m}^{2}$ area and carrying $100 \mu \mathrm{~A}$ current is placed in a uniform magnetic field of 1 T . Initially the magnetic dipole moment ( $\overrightarrow{\mathrm{M}}$ ) was directed along $\overrightarrow{\mathrm{B}}$. Amount of work, required to rotate the coil through $90^{\circ}$ from its initial orientation such that $\vec{M}$ becomes perpendicular to $\vec{B}$, is $\_\_\_\_\_\_\_$ $\mu \mathrm{J}$.

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 $\qquad$ $\mathrm{cm} / \mathrm{s}$.

For three vectors $\vec{A}=(-x \hat{i}-6 \hat{j}-2 \hat{k})$, $\vec{B}=(-\hat{i}+4 \hat{j}+3 \hat{k}) \quad$ and $\quad \vec{C}=(-8 \hat{i}-\hat{j}+3 \hat{k}), \quad$

if $\overrightarrow{\mathrm{A}} \cdot(\overrightarrow{\mathrm{B}} \times \overrightarrow{\mathrm{C}})=0$, them value of x is $\qquad$ .

Functional group present in sulphonic acid is :

Match the List I with List II and choose the correct answer from the options given below :

Given below are two statements, one is labelled as Statement I and the other is labelled as Statement II.

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 options given below :

Match List I with List II :

Choose the correct answer from the options given below :

Given below are two statements, one is labelled as Statement I and the other is labelled as Statement II.

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

Statement II : A thermometer containing gallium is useful for measuring the freezing point $(256 \mathrm{~K})$ of brine solution.

In the light of the above statement, choose the correct answer from the options given below :

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 mixture by azeotropic distillation because it forms azeotrope.

D. An organic compound is pure, if mixed M.P. is remained same.

Choose the most appropriate answer from the options given below :

Match the List I with List II and choose the correct answer from the options given below:

Which of the following material is not a semiconductor.

Consider the following complexes.

The correct order of $\mathrm{A}, \mathrm{B}, \mathrm{C}$ and D in terms of wavenumber of light absorbed is :

Match List I with List II :

Choose the correct answer from the options given below :

The electron affinity values are negative for:

(A) $\mathrm{Be \rightarrow Be^-}$

(B) $\mathrm{N \rightarrow N^-}$

(C) $\mathrm{O \rightarrow O^{2-}}$

(D) $\mathrm{Na \rightarrow Na^-}$

(E) $\mathrm{Al \rightarrow Al^-}$

Choose the most appropriate answer from the options given below:

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

The density of $x\ \mathrm{M}$ solution of $\mathrm{NaOH}$ is $1.12\ \mathrm{g\ mL^{-1}}$ while in molality the concentration of the solution is $3\ \mathrm{m}$. Then $x$ is $\_\_\_\_$.

[Given: Molar mass of $\mathrm{NaOH} = 40\ \mathrm{g\ mol^{-1}}$]

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 $\mathrm{K}_{\mathrm{P}}$ for the process is $\mathrm{x} \times 10^{-1} \cdot \mathrm{x}=$ $\qquad$ .

[Given : $\mathrm{R}=0.082 \mathrm{~L} \mathrm{~atm} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ ]

Match List I with List II :

Choose the correct answer from the options given below :

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 -

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 M aqueous solution of HX at 300 K is $\qquad$ $\times 10^{-2}$ bar (nearest integer)

$\text{[Given : R = 0.083 L bar Mol}^{-1} \text{K}^{-1} \text{]}$

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 $\qquad$ BM. (nearest integer)

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

$9.3\ \mathrm{g}$ of pure aniline upon diazotisation followed by coupling with phenol gives an orange dye. The mass of orange dye produced (assuming $100\%$ yield) is $\_\_\_$ g (nearest integer).

The major product of the following reaction is P . $\mathrm{CH}_{3} \mathrm{C} \equiv \mathrm{C}-\mathrm{CH}_{3} \xrightarrow[\substack{\text { (ii) dil. } 273 \mathrm{~K}}]{\text { (i) } \mathrm{NMO} / \mathrm{MH}_{3}} \mathrm{I}^{\prime}$ Number of oxygen atoms present in product ' P ' is $\qquad$ (nearest integer).

Frequency of the de-Broglie wave of electron in Bohr's first orbit of hydrogen atom is $\_\_\_\_$ $\times 10^{13}\ \mathrm{Hz}$ (nearest integer).

[Given: $\mathrm{R_H}$ (Rydberg constant) $= 2.18 \times 10^{-18}\ \mathrm{J}$, $\mathrm{h}$ (Planck's constant) $= 6.6 \times 10^{-34}\ \mathrm{J\,s}$.]

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

The total sum of $\pi$ electrons in product $A$ and product B are $\qquad$ (nearest integer)

Among $\mathrm{CrO}, \mathrm{Cr}_{2} \mathrm{O}_{3}$ and $\mathrm{CrO}_{3}$, the sum of spin-only magnetic moment values of basic and amphoteric oxides is $\qquad$ $10^{-2} \mathrm{BM}$ (nearest integer). (Given atomic number of Cr is 24 )

An ideal gas, $\overline{\mathrm{C}}_{\mathrm{V}}=\frac{5}{2} \mathrm{R}$, is expanded adiabatically against a constant pressure of 1 atm untill it doubles in volume. If the initial temperature and pressure is 298 K and 5 atm , respectively then the final temperature is $\qquad$ K (nearest integer). [ $\overline{\mathrm{C}}_{\mathrm{V}}$ is the molar heat capacity at constant volume]