JEE Main 2025 — 29 January, Evening Shift

For hydrogen-like species, which graph correctly represents $E$ vs $Z$ for constant $n$?

(Given: $E$ = energy of stationary state, $Z$ = atomic number)

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

Statement I: In partition chromatography, stationary phase is thin film of liquid present in the inert support.

Statement II: In paper chromatography, the material of paper acts as a stationary phase.

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

Identify the essential amino acids from below :

(A) Valine (B) Proline

(C) Lysine (D) Threonine

(E) Tyrosine

Choose the correct answer from the options given below:

Which among the following halides will generate the most stable carbocation in Nucleophillic substitution reaction?

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

Statement I: On nitration of m-xylene with $\mathrm{HNO}_{3}, \mathrm{H}_{2} \mathrm{SO}_{4}$ followed by oxidation, 4-nitrobenzene-1, 3-dicarboxylic acid is obtained as the major product.

Statement II: $\mathrm{CH}_{3}$ group is o/p-directing while$\mathrm{NO}_{2}$ group is m-directing group.

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

$0.1\,\mathrm{M}$ solution of KI reacts with excess of $\mathrm{H_2SO_4}$ and $\mathrm{KIO_3}$ solution. According to equation: $5\,\mathrm{I^-} + \mathrm{IO_3^-} + 6\,\mathrm{H^+} \rightarrow 3\,\mathrm{I_2} + 3\,\mathrm{H_2O}$ identify the correct statements:

(A) $200\,\mathrm{mL}$ of KI solution reacts with $0.004\,\mathrm{mol}$ of $\mathrm{KIO_3}$

(B) $200\,\mathrm{mL}$ of KI solution reacts with $0.006\,\mathrm{mol}$ of $\mathrm{H_2SO_4}$

(C) $0.5\,\mathrm{L}$ of KI solution produces $0.005\,\mathrm{mol}$ of $\mathrm{I_2}$

(D) Equivalent weight of $\mathrm{KIO_3}$ is $\dfrac{\text{Molecular\; weight}}{5}$

Choose the correct answer from the options given below:

Total number of sigma ( $\sigma$ ) and $\mathrm{pi}(\pi)$ bonds respectively present in hex-1-en-4-yne are :

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

Statement I: It is impossible to specify simultaneously with arbitrary precision, both the linear momentum and the position of a particle.

Statement II: If the uncertainty in the measurement of position and uncertainty in measurement of momentum are equal for an electron, then $\Delta v \ge \sqrt{\frac{h}{\pi}} \times \frac{1}{2m}$.

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

Which one of the following reaction sequences will give an azo dye ?

The type of oxide formed by the element among $\mathrm{Li},\;\mathrm{Na}, \;\mathrm{Be}, \;\mathrm{Mg}, \;\mathrm{B}$ and $\mathrm{Al}$ that has the least atomic radius is :

First ionisation enthalpy values of first four group 15 elements are given below.

Choose the correct value for the element that is a main component of apatite family:

Which one of the following, with HBr will give a phenol?

Isomeric hydrocarbons $\rightarrow$ negative Baeyer's test (Molecular formula $\mathrm{C}_{9} \mathrm{H}_{12}$ ) The total number of isomers from above with four different non-aliphatic substitution sites is -

In the Claisen–Schmidt reaction to prepare dibenzalacetone from $5.3\,\mathrm{g}$ benzaldehyde, a total of $3.51\,\mathrm{g}$ of product was obtained. The percentage yield in this reaction was

In the sulphur estimation, 0.20 g of a pure organic compound gave 0.40 g of barium sulphate. The percentage of sulphur in the compound is $\_\_\_\_\_$ $\times 10^{-1} \%$. (Molar mass : $\mathrm{O}=16, \mathrm{~S}=32, \mathrm{Ba}=137$ in $\mathrm{g} \mathrm{mol}^{-1}$ )

Total number of non bonded electrons present in $\mathrm{NO}_{2}^{-}$ion based on Lewis theory is $\qquad$ .

The difference of temperature in a material can convert heat energy into electrical energy. To harvest the heat energy, the material should have

Given below are two statements: One is labelled as Assertion (A) and the other is labelled as Reason (R)

Assertion (A): With the increase in the pressure of an ideal gas, the volume falls off more rapidly in an isothermal process in comparison to the adiabatic process.

Reason (R): In isothermal process, PV = constant, while in adiabatic process $\mathrm{PV}^{\gamma}=$ constant. Here $\gamma$ is the ratio of specific heats, P is the pressure and V is the volume of the ideal gas.

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

An electric dipole is placed at a distance of 2 cm from an infinite plane sheet having positive charge

density $\sigma_{0}$. Choose the correct option from the following.

In an experiment with photoelectric effect, the stopping potential.

A point charge causes an electric flux of $-2 \times 10^{4}$ $\mathrm{Nm}^{2} \mathrm{C}^{-1}$

to pass through a spherical Gaussian surface of 8.0 cm radius, centred on the charge.

The value of the point charge is :

(Given $\epsilon_{0}=8.85 \times 10^{-12} \mathrm{C}^{2} \mathrm{~N}^{-1} \mathrm{~m}^{-2}$ )

A poly-atomic molecule $\left(\mathrm{C}_{\mathrm{v}}=3 \mathrm{R}, \mathrm{C}_{\mathrm{p}}=4 \mathrm{R}\right.$, where $R$ is gas constant) goes from phase space point $\mathrm{A}\left(\mathrm{P}_{\mathrm{A}}=10^{5} \mathrm{~Pa}, \mathrm{~V}_{\mathrm{A}}=4 \times 10^{-6} \mathrm{~m}^{3}\right)$ to point $\mathrm{B}\left(\mathrm{P}_{\mathrm{B}}=5\right.$

$\left.\times 10^{4} \mathrm{~Pa}, \mathrm{~V}_{\mathrm{B}}=6 \times 10^{-6} \mathrm{~m}^{3}\right)$ to point $\mathrm{C}\left(\mathrm{P}_{\mathrm{C}}=10^{4} \mathrm{~Pa}\right.$, $\mathrm{V}_{\mathrm{c}}=8 \times 10^{-6} \mathrm{~m}^{3}$ ).

A to B is an adiabatic path and B to C is an isothermal path.

The net heat absorbed per unit mole by the system is :

Two identical symmetric double convex lenses of focal length $f$ are cut into two equal parts $\mathrm{L}_{1}, \mathrm{~L}_{2}$ by AB plane and $L_{3}, L_{4}$ by XY plane as shown in figure respectively. The ratio of focal lengths of lenses $L_{1}$ and $L_{3}$ is

A plane electromagnetic wave propagates along the $+x$ direction in free space. The components of the electric field, $\vec{E}$ and magnetic field, $\vec{B}$ vectors associated with the wave in Cartesian frame are :

Two concave refracting surfaces of equal radii of curvature and refractive index 1.5 face each other in air as shown in figure. A point object O is placed midway, between P and B. The separation between the images of O , formed by each refracting surface is :

Two bodies A and B of equal mass are suspended from two massless springs of spring constant

$\mathrm{k}_{1}$ and $\mathrm{k}_{2}$, respectively. If the bodies oscillate vertically such that their amplitudes are equal, the ratio of the maximum velocity of A to the maximum velocity of $B$ is

Given below are two statements: One is labelled as Assertion (A) and the other is labelled as Reason (R)

Assertion (A): Three identical spheres of same mass undergo one dimensional motion as shown in figure with initial velocities $\mathrm{v}_{\mathrm{A}}=5 \mathrm{~m} / \mathrm{s}, \mathrm{v}_{\mathrm{B}}=2 \mathrm{~m} / \mathrm{s}, \mathrm{v}_{\mathrm{C}}=4 \mathrm{~m} / \mathrm{s}$ If we wait sufficiently long for elastic collision to happen, then $\mathrm{v}_{\mathrm{A}}=4 \mathrm{~m} / \mathrm{s}, \mathrm{v}_{\mathrm{B}}=2 \mathrm{~m} / \mathrm{s}, \mathrm{v}_{\mathrm{C}}=5 \mathrm{~m} / \mathrm{s}$ will be the final velocities.

Reason (R): In an elastic collision between identical masses, two objects exchange their velocities.

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

A sand dropper drops sand of mass $m(t)$ on a conveyer belt at a rate proportional to the square root of speed $(\mathrm{v})$ of the belt, i.e. $\frac{\mathrm{dm}}{\mathrm{dt}} \propto \sqrt{\mathrm{v}}$. If P is the power delivered to run the belt at constant speed then which of the following relationship is true?

A convex lens mode of glass (refractive index $=$ 1.5) has focal length 24 cm in air. When it is totally immersed in water (refractive index $=1.33$ ), its focal length changes to

A capacitor, $\mathrm{C}_{1}=6 \mu \mathrm{~F}$ is charged to a potential difference of $\mathrm{V}_{0}=5 \mathrm{~V}$ using a 5 V battery. The battery is removed and another capacitor, $\mathrm{C}_{2}=12$ $\mu \mathrm{F}$ is inserted in place of the battery. When the switch ' S ' is closed, the charge flows between the capacitors for some time until equilibrium condition is reached. What are the charges ( $\mathrm{q}_{1}$ and $\mathrm{q}_{2}$ ) on the capacitors $\mathrm{C}_{1}$ and $\mathrm{C}_{2}$ when equilibrium condition is reached.

Three equal masses $m$ are kept at vertices (A, B, C) of an equilateral triangle of side $a$ in free space. At $\mathrm{t}=0$, they are given an initial velocity $\vec{V}_{A}=V_{0} \overrightarrow{A C}, \quad \vec{V}_{B}=V_{0} \overrightarrow{B A}$ and $\vec{V}_{C}=V_{0} \overrightarrow{C B}$. Here, $\overrightarrow{\mathrm{AC}}, \overrightarrow{\mathrm{CB}}$ and $\overrightarrow{\mathrm{BA}}$ are unit vectors along the edges of the triangle. If the three masses interact gravitationally, then the magnitude of the net angular momentum of the system at the point of collision is :

Match List-I with List-II.

Choose the correct answer from the options given below :

Match List-I with List-II.

Choose the correct answer from the options given below :

The truth table for the circuit given below is :

A cup of coffee cools from $90^{\circ} \mathrm{C}$ to $80^{\circ} \mathrm{C}$ in t minutes when the room temperature is $20^{\circ} \mathrm{C}$. The time taken by the similar cup of coffee to cool from $80^{\circ} \mathrm{C}$ to $60^{\circ} \mathrm{C}$ at the same room temperature is :

The number of spectral lines emitted by atomic hydrogen that is in the $4^{\text {th }}$ energy level, is

The magnetic field inside a 200 turns solenoid of radius 10 cm is $2.9 \times 10^{-4}$ Tesla. If the solenoid carries a current of 0.29 A , then the length of the solenoid is $\qquad$ $\pi \mathrm{cm}$.

A parallel plate capacitor consisting of two circular plates of radius 10 cm is being charged by a constant current of 0.15 A . If the rate of change of potential difference between the plates is $7 \times 10^{8}$

$\mathrm{V} / \mathrm{s}$ then the integer value of the distance between the parallel plates is - $\left(\right.$

Take, $\left.\epsilon_{0}=9 \times 10^{-12} \frac{\mathrm{~F}}{\mathrm{~m}}, \pi=\frac{22}{7}\right)$

$\qquad$ $\mu \mathrm{m}$.

A physical quantity Q is related to four observables $a, b, c, d$ as follows : $Q=\frac{a^{4}}{c d}$ where, $\mathrm{a}=(60 \pm 3) \mathrm{Pa} ; \mathrm{b}=(20 \pm 0.1) \mathrm{m} ;$ $c=(40 \pm 0.2) \mathrm{Nsm}^{-2}$ and $\mathrm{d}=(50 \pm 0.1) \mathrm{m}$, then the percentage error in $Q$ is $\frac{x}{1000}$, where $\mathrm{x}=$ $\qquad$ .

Two planets, $A$ and $B$ are orbiting a common star in circular orbits of radii $R_{A}$ and $R_{B}$, respectively, with $R_{B}=2 R_{A}$. The planet $B$ is $4 \sqrt{2}$ times more massive than planet A. The ratio $\left(\frac{L_{B}}{L_{A}}\right)$ of angular momentum $\left(\mathrm{L}_{\mathrm{B}}\right)$ of planet B to that of planet $\mathrm{A}\left(\mathrm{L}_{\mathrm{A}}\right)$ is closest to integer $\qquad$

Two cars P and Q are moving on a road in the same direction. Acceleration of car $P$ increases linearly with time whereas car Q moves with a constant acceleration. Both cars cross each other at time $t=0$, for the first time. The maximum possible number of crossing(s) (including the crossing at $\mathrm{t}=0$ ) is $\_\_\_\_\_$ .

If the set of all $a \in \mathbb{R}$, for which the equation$2x^2 + (a - 5)x + 15 = 3a$ has no real root, is the interval $(\alpha, \beta)$, and $X = \{x \in \mathbb{Z} : \alpha < x < \beta\},$ then$\sum_{x \in X} x^2 \text{ is }$

If $\sin x+\sin ^{2} x=1, x \in\left(0, \frac{\pi}{2}\right)$, then $\left(\cos ^{12} x+\tan ^{12} x\right)+3\left(\cos ^{10} x+\tan ^{10} x+\cos ^{8} x+\tan ^{8} x\right)$ $+\left(\cos ^{6} x+\tan ^{6} x\right)$ is equal to

Let the area enclosed between the curves $|y|=1-$ $x^{2}$ and $x^{2}+y^{2}=1$ be $\alpha$.

If $9 \alpha=\beta \pi+\gamma ; \beta, \gamma$ are integers, then the value of $|\beta-\gamma|$ equals

If the domain of the function $\log _{5}\left(18 \mathrm{x}-\mathrm{x}^{2}-77\right)$ is $(\alpha, \beta)$ and the domain of the function

$\log _{(x-1)}\left(\frac{2 x^{2}+3 \mathrm{x}-2}{\mathrm{x}^{2}-3 \mathrm{x}-4}\right)$ is $(\gamma, \delta)$, then $\alpha^{2}+\beta^{2}+\gamma^{2}$ is equal to

Let the function $f(x)=\left(x^{2}-1\right)\left|x^{2}-a x+2\right|+\cos |x|$ be not differentiable at the two points $x=\alpha=2$ and $x=\beta$.

Then the distance of the point $(\alpha, \beta)$ from the line $12 x+5 y+10=0$ is equal to

Let a straight line $L$ pass through the point $P(2,-1,3)$ and be perpendicular to the lines $\frac{\mathrm{x}-1}{2}=\frac{\mathrm{y}+1}{1}=\frac{\mathrm{z}-3}{-2} \quad$ and $\quad \frac{\mathrm{x}-3}{1}=\frac{\mathrm{y}-2}{3}=\frac{\mathrm{z}+2}{4}$. If the line $L$ intersects the yz-plane at the point $Q$, then the distance between the points P and Q is

Let $\mathrm{S}=\mathbf{N} \cup\{0\}$. Define a relation $\mathbf{R}$ from $\mathbf{S}$ to $\mathbf{R}$ by :

$\mathbf{R}=\left\{(\mathrm{x}, \mathrm{y}): \log _{\mathrm{e}} \mathrm{y}=\mathrm{x} \log _{\mathrm{e}}\left(\frac{2}{5}\right), \mathrm{x} \in \mathrm{~S}, \mathrm{y} \in \mathbf{R}\right\}$ Then, the sum of all the elements in the range of $\mathbf{R}$ is equal to

Let the line $x+y=1$ meet the axes of $x$ and $y$ at $A$ and B , respectively. A right angled triangle AMN is inscribed in the triangle OAB ,

where O is the origin and the points M and N lie on the lines OB and $A B$, respectively. If the area of the triangle AMN is $\frac{4}{9}$ of the area of the triangle OAB and $\mathrm{AN}: \mathrm{NB}=\lambda: 1$, then the sum of all possible value(s) of is $\lambda$

If $\alpha x+\beta y=109$ is the equation of the chord of the ellipse $\frac{\mathrm{x}^{2}}{9}+\frac{\mathrm{y}^{2}}{4}=1$, whose mid point is $\left(\frac{5}{2}, \frac{1}{2}\right)$, then $\alpha+\beta$ is equal to

If all the words with or without meaning made using all the letters of the word "KANPUR" are arranged as in a dictionary, then the word at $440^{\text {th }}$ position in this arrangement, is

Let $\alpha, \beta$ ($\alpha \neq \beta$) be the values of $m$ for which the equations $x + y + z = 1,\quad x + 2y + 4z = m,\quad x + 4y + 10z = m^2$ have infinitely many solutions. Then the value of$\sum_{n=1}^{10} \left( \alpha^{n} + \beta^{n} \right)$ is equal to:

Let $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]$ be a matrix of order $3 \times 3$, with $\mathrm{a}_{\mathrm{ij}}=(\sqrt{2})^{\mathrm{i}+\mathrm{j}}$.

If the sum of all the elements in the third row of $\mathrm{A}^{2}$ is $\alpha+\beta \sqrt{2}, \alpha, \beta \in \mathbf{Z}$, then $\alpha+\beta$ is equal to

Let $P$ be the foot of the perpendicular from the point $(1,2,2)$ on the line $L: \frac{x-1}{1}=\frac{y+1}{-1}=\frac{z-2}{2}$.

Let the line $\overrightarrow{\mathrm{r}}=(-\hat{\mathrm{i}}+\hat{\mathrm{j}}-2 \hat{\mathrm{k}})+\lambda(\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}), \lambda \in \mathbf{R}$, intersect the line L at Q . Then $2(\mathrm{PQ})^{2}$ is equal to:

Let a circle $C$ pass through the points $(4,2)$ and $(0$, 2 ), and its centre lie on $3 x+2 y+2=0$.

Then the length of the chord, of the circle C , whose midpoint is $(1,2)$, is

Let $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]$ be a $2 \times 2$ matrix such that $\mathrm{a}_{\mathrm{ij}} \in\{0,1\}$ for all i and j .

Let the random variable X denote the possible values of the determinant of the matrix $A$. Then, the variance of X is:

Bag 1 contains 4 white balls and 5 black balls, and Bag 2 contains $n$ white balls and 3 black balls.

One ball is drawn randomly from Bag 1 and transferred to Bag 2. A ball is then drawn randomly from Bag 2.

If the probability, that the ball drawn is white, is $29 / 45$, then n is equal to

The remainder, when $7^{103}$ is divided by 23 , is equal to

Let $f(x)=\int_{0}^{x} t\left(t^{2}-9 t+20\right) d t, \quad 1 \leq x \leq 5$. If the range of $f$ is $[\alpha, \beta]$, then $4(\alpha+\beta)$ equals:

Let â be a unit vector perpendicular to the vectors $\vec{b}=\hat{i}-2 \hat{j}+3 \hat{k}$ and $\vec{c}=2 \hat{i}+3 \hat{j}-\hat{k}$, and makes an angle of $\cos ^{-1}\left(-\frac{1}{3}\right)$ with the vector $\hat{i}+\hat{j}+\hat{k}$. If $\hat{a}$ makes an angle of $\frac{\pi}{3}$ with the vector $\hat{i}+\alpha \hat{j}+\hat{k}$, then the value of $\alpha$ is

If for the solution curve $y=f(x)$ of the differential

equation $\frac{d y}{d x}+(\tan x) y=\frac{2+\sec x}{(1+2 \sec x)^{2}}$, $x \in\left(\frac{-\pi}{2}, \frac{\pi}{2}\right), f\left(\frac{\pi}{3}\right)=\frac{\sqrt{3}}{10}$, then $\mathrm{f}\left(\frac{\pi}{4}\right)$ is equal to

If $24 \int_{0}^{\frac{\pi}{4}}\left(\sin \left|4 x-\frac{\pi}{12}\right|+[2 \sin x]\right) d x=2 \pi+\alpha$, where [.] denotes the greatest integer function, then $\alpha$ is equal to $\qquad$

If \mathop {\lim }\limits_{t \to 0} {\left( {\mathop \smallint \limits_0^1 \left( {3x + 5} \right)dx} \right)^{1/t}}=$\frac{\alpha }{{5e}}{\left( {\frac{8}{5}} \right)^{\frac{2}{3}}}$ , then $\alpha \;is\;equal\;to\;\_\_\_\_\_$

Let $a_1, a_2, \ldots, a_{2024}$ be an arithmetic progression such that $a_1 + (a_5 + a_{10} + a_{15} + \cdots + a_{2020}) + a_{2024} = 2233.$ Then $a_1 + a_2 + a_3 + \cdots + a_{2024}$ is equal to:

Let integers a, b ∈ [-3 , 3] be such that a+b ≠ 0.

Then the number of all possible ordered pairs (a,b) for which

\left| {\frac{{z - a}}{{z + b}}} \right| = 1\;and\;\left| {\begin{array}{*{20}{c}}{z + 1}&\omega &{{\omega ^2}}\\\omega &{z + {\omega ^2}}&1\\{{\omega ^2}}&1&{z + \omega }\end{array}} \right|=1, Z∈ C, where $\omega \;and\;{\omega ^2}$ are the roots ${x^2} + x + 1 = 0,\;is\;equal\;to\;$

Let $y^{2}=12 x$ the parabola and $S$ be its focus. Let PQ be a focal chord of the parabola such that (SP) $(S Q)=\frac{147}{4}$.

Let C be the circle described taking PQ as a diameter. If the equation of a circle C is $64 x^{2}+64 y^{2}-\alpha x-64 \sqrt{3} y=\beta$,

then $\beta-\alpha$ is equal to $\qquad$ .