JEE Main 2025 — 29 January, Morning Shift

Total number of nucleophiles from the following is :$\mathrm{NH}_{3}, \mathrm{PhSH},\left(\mathrm{H}_{3} \mathrm{C}\right)_{2} \mathrm{~S}, \mathrm{H}_{2} \mathrm{C}=\mathrm{CH}_{2}, \stackrel{\ominus}{\mathrm{O}} \mathrm{H}, \mathrm{H}_{3} \mathrm{O}^{\oplus}$, $\left.\left(\mathrm{CH}_{3}\right)_{2} \mathrm{CO},\right\rangle=\mathrm{NCH}_{3}$

The standard reduction potential values of some of the p-block ions are given below. Predict the one with the strongest oxidising capacity.

At temperature $T$, compound $\mathrm{AB}_{2(\mathrm{~g})}$ dissociates as $\mathrm{AB}_{2(\mathrm{~g})} \rightleftharpoons \mathrm{AB}_{(\mathrm{g})}+\frac{1}{2} \mathrm{~B}_{2(\mathrm{~g})}$ having degree of dissociation x (small compared to unity). The correct expression for x in terms of $\mathrm{K}_{\mathrm{p}}$ and p is

Choose the correct statements.

(A) Weight of a substance is the amount of matter present in it

(B) Mass is the force exerted by gravity on an object.

(C) Volume is the amount of space occupied by a substance

(D) Temperatures below $0^{\circ} \mathrm{C}$ are possible in Celsius scale, but in Kelvin scale negative temperature is not possible.

(E) Precision refers to the closeness of various measurements for the same quantity.

The correct increasing order of stability of the complexes based on $\Delta_{0}$ value is :

(I) $\left[\mathrm{Mn}(\mathrm{CN})_{6}\right]^{3-}$

(II) $\left[\mathrm{Co}(\mathrm{CN})_{6}\right]^{4-}$

(III) $\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]^{4-}$

(IV) $\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]^{3-}$

Match List-I with List-II.

In the following substitution reaction :

The correct option with order of melting points of the pairs $(\mathrm{Mn}, \mathrm{Fe}),(\mathrm{Tc}, \mathrm{Ru})$ and $(\mathrm{Re}, \mathrm{Os})$ is :

1.24 g of $\mathrm{AX}_{2}$ (molar mass $124 \mathrm{~g} \mathrm{~mol}^{-1}$ ) is dissolved in 1 kg of water to form a solution with boiling point of $100.0156^{\circ} \mathrm{C}$, while 25.4 g of $\mathrm{AY}_{2}$ (molar mass $250 \mathrm{~g} \mathrm{~mol}^{-1}$ ) in 2 kg of water constitutes a solution with a boiling point of $100.0260^{\circ} \mathrm{C}$. $\mathrm{K}_{\mathrm{b}}\left(\mathrm{H}_{2} \mathrm{O}\right)=0.52 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}$

Which of the following is correct?

500 J of energy is transferred as heat to 0.5 mol of Argon gas at 298 K and 1.00 atm . The final temperature and the change in internal energy respectively are: Given : $\mathrm{R}=8.3 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$

The reaction $\mathrm{A}_{2}+\mathrm{B}_{2} \rightarrow 2 \mathrm{AB}$ follows the mechanism

$\mathrm{A}+\mathrm{B}_{2} \xrightarrow{\mathrm{k}_{2}} \mathrm{AB}+\mathrm{B}$ (slow) $\mathrm{A}+\mathrm{B} \rightarrow \mathrm{AB}$ (fast) The overall order of the reaction is :

If $\mathrm{a}_{0}$ is denoted as the Bohr radius of hydrogen atom, then what is the de-Broglie wavelength ($\lambda$) of the electron present in the second orbit of hydrogen atom? [ n : any integer]

The product $(\mathrm{P})$ formed in the following reaction is :

An element $E$ has the ionisation enthalpy value of $374\,\mathrm{kJ\,mol^{-1}}$. $E$ reacts with elements $A$, $B$, $C$ and $D$ having electron gain enthalpy values of $-328,\ -349,\ -325\ \text{and}\ -295\,\mathrm{kJ\,mol^{-1}}$, respectively.

The correct order of the products $EA$, $EB$, $EC$ and $ED$ in terms of ionic character is:

Match List - I with List - II.

The steam volatile compounds among the following are :

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: The radii of isoelectronic species increases in the order $\mathrm{Mg}^{2+}<\mathrm{Na}^{+}<\mathrm{F}^{-}<\mathrm{O}^{2-}$.

Statement II: The magnitude of electron gain enthalpy of halogen decreases in the order $\mathrm{Cl}>\mathrm{F}>\mathrm{Br}>\mathrm{I}$.

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

Given below are some nitrogen containing compounds.

Each of them is treated with HCl separately. 1.0 g of the most basic compound will consume $\_\_\_\_\_$ mg of HCl . (Given molar mass in $\mathrm{g} \mathrm{mol}^{-1}, \mathrm{C}: 12, \mathrm{H}: 1, \mathrm{O}: 16, \mathrm{Cl} : 35.5)$.

The molar mass of the water insoluble product formed from the fusion of chromite ore $\mathrm{(FeCr_2O_4)}$ with $\mathrm{Na_2CO_3}$ in the presence of $\mathrm{O_2}$ is $\_\_\_\_\mathrm{g\,mol^{-1}}$.

The sum of sigma $(\sigma)$ and $\mathrm{pi}(\pi)$ bonds in

Hex-1,3-dien-5-yne is $\qquad$ .

If $A_{2} B$ is $30 \%$ ionised in an aqueous solution, then the value of van't Hoff factor (i) is $\qquad$ $\times 10^{-1}$.

$0.1$ mole of compound $\mathrm{S}$ will weigh $\_\_\_\_$ g. (Given molar masses in $\mathrm{g\,mol^{-1}}$: $\mathrm{C}=12$, $\mathrm{H}=1$, $\mathrm{O}=16$)

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

Assertion (A): Choke coil is simply a coil having a large inductance but a small resistance. Choke coils are used with fluorescent mercury-tube fittings. If household electric power is directly connected to a mercury tube, the tube will be damaged.

Reason (R): By using the choke coil, the voltage across the tube is reduced by a factor $\left(R / \sqrt{R^{2}+\omega^{2} L^{2}}\right)$, where $\omega$ is frequency of the supply across resistor $R$ and inductor $L$. If the choke coil were not used, the voltage across the resistor would be the same as the applied voltage.

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

Two projectiles are fired with same initial speed from same point on ground at angles of

$\left(45^{\circ}-\alpha\right)$ and $\left(45^{\circ}+\alpha\right)$,

respectively, with the horizontal direction. The ratio of their maximum heights attained is :

An electric dipole of mass $m$, charge $q$, and length $l$ is placed in a uniform electric field $\overrightarrow{\mathrm{E}}=\mathrm{E}_{0} \hat{\mathrm{i}}$. When the dipole is rotated slightly from its equilibrium position and released, the time period of its oscillations will be :

The pair of physical quantities not having same dimensions is :

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

Assertion (A): Time period of a simple pendulum is longer at the top of a mountain than that at the base of the mountain.

Reason (R): Time period of a simple pendulum decreases with increasing value of acceleration due to gravity and vice-versa.

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

The expression given below shows the variation of velocity (v) with time ( $t$ ),

$v=A t^{2}+\frac{B t}{C+t}$. The dimension of ABC is :

Consider $I_{1}$ and $I_{2}$ are the currents flowing simultaneously in two nearby coils

$1 \& 2$, respectively. If $L_{1}=$ self inductance of coil 1 , $\mathrm{M}_{12}=$

mutual inductance of coil 1 with respect to coil 2 , then the value of induced emf in coil 1 will be

Consider a long straight wire of a circular cross-section (radius a) carrying a steady current I. The current is uniformly distributed across this cross-section. The distances from the centre of the wire's cross-section at which the magnetic field [inside the wire, outside the wire] is half of the maximum possible magnetic field, any where due to the wire, will be

As shown below, bob A of a pendulum having massless string of length ' R ' is released from $60^{\circ}$ to the vertical. It hits another bob B of half the mass that is at rest on a friction less table in the centre. Assuming elastic collision, the magnitude of the velocity of bob A after the collision will be (take $g$ as acceleration due to gravity)

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

Assertion (A): Emission of electrons in photoelectric effect can be suppressed by applying a sufficiently negative electron potential to the photoemissive substance.

Reason (R): A negative electric potential, which stops the emission of electrons from the surface of a photoemissive substance, varies linearly with frequency of incident radiation.

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

A coil of area A and N turns is rotating with angular velocity $\omega$ in a uniform magnetic field $\overrightarrow{\mathrm{B}}$ about an axis perpendicular to

$\overrightarrow{\mathrm{B}}$. Magnetic flux $\varphi$ and induced emf

$\varepsilon$ across it, at an instant when $\overrightarrow{\mathrm{B}}$ is parallel to the plane of coil, are :

The fractional compression $\left(\frac{\Delta V}{V}\right)$ of water at the depth of 2.5 km below the sea level is $\_\_\_\_\_\_$ $\%$. Given, the Bulk modulus of water $=2 \times 10^{9} \mathrm{Nm}^{-2}$, density of water $=10^{3} \mathrm{~kg} \mathrm{~m}^{-3}$, acceleration due to gravity $=\mathrm{g}=10 \mathrm{~ms}^{-2}$.

A body of mass ' m ' connected to a massless and unsearchable string goes in vertical circle of radius ' R ' under gravity g . The other end of the string is fixed at the centre of circle. If velocity at top of circular path is $n \sqrt{g R}$, where, $n \geq 1$, then ratio of kinetic energy of the body at bottom to that at top of the circle is

Let $u$ and $v$ be the distances of the object and the image from a lens of focal length $f$. The correct graphical representation of u and v for a convex lens when $|\mathbf{u}|>f$, is

Match List-I with List-II.

Choose the correct answer from the options given below :

The workdone in an adiabatic change in an ideal gas depends upon only :

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

Assertion (A): Electromagnetic waves carry energy but not momentum.

Reason (R): Mass of a photon is zero.

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

The coordinates of a particle with respect to origin in a given reference frame is $(1,1,1)$ meters. If a force of $\overrightarrow{\mathrm{F}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}$ acts on the particle, then the magnitude of torque (with respect to origin) in $z$-direction is $\qquad$ .

A container of fixed volume contains a gas at $27^{\circ} \mathrm{C}$. To double the pressure of the gas, the temperature of gas should be raised to $\_\_\_\_\_\_$ ${ }^{\circ} \mathrm{C}$.

Two light beams fall on a transparent material block at point 1 and 2 with angle $\theta_{1}$

and $\theta_{2}$, respectively, as shown in figure.

After refraction, the beams intersect at point 3 which is exactly on the interface at other end of the block.

Given : the distance between 1 and $2, d=4 \sqrt{3} \mathrm{~cm}$ and $\theta_{1}=\theta_{2}=\cos ^{-1}\left(\frac{n_{2}}{2 n_{1}}\right)$,

where refractive index of the block $n_{2}>$

refractive index of the outside medium $n_{1}$, then the thickness of the block is $\qquad$ cm .

In a hydraulic lift, the surface area of the input piston is $6 \mathrm{~cm}^{2}$

and that of the output piston is $1500 \mathrm{~cm}^{2}$.

If 100 N force is applied to the input piston to raise the output piston by 20 cm ,

then the work done is $\qquad$ kJ.

The maximum speed of a boat in still water is $27 \mathrm{~km} / \mathrm{h}$.

Now this boat is moving downstream in a river flowing at $9 \mathrm{~km} / \mathrm{h}$.

A man in the boat throws a ball vertically upwards with speed of $10 \mathrm{~m} / \mathrm{s}$.

Range of the ball as observed by an observer at rest on the river bank, is $\qquad$ cm . (Take $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$ )

Let the line $x+y=1$ meet the circle $x^{2}+y^{2}=4$ at the points $A$ and $B$. If the line perpendicular to $A B$ and passing through the mid point of the chord AB intersects is the circle at C and D , then the area of the quadrilateral ADBC is equal to

Let $M$ and $m$ respectively be the maximum and the minimum values of $f(x)=\left|\begin{array}{ccc}1+\sin ^{2} x & \cos ^{2} x & 4 \sin 4 x \\ \sin ^{2} x & 1+\cos ^{2} x & 4 \sin 4 x \\ \sin ^{2} x & \cos ^{2} x & 1+4 \sin 4 x\end{array}\right|, x \in R$ Then $M^{4}-m^{4}$ is equal to

Two parabolas have the same focus $(4,3)$ and their directrices are the $x$-axis and the $y$-axis, respectively.

If these parabolas intersects at the points $A$ and $B$, then $(A B)^{2}$ is equal to

Let ABC be a triangle formed by the lines $7 x-6 y+3=0, x+2 y-31=0$ and $9 x-2 y-19=0$, Let the point $(h, k)$

be the image of the centroid of $\Delta A B C$ in the line $3 x+6 y-53=0$. Then $h^{2}+k^{2}+h k$ is equal to

Let $\overrightarrow{\mathrm{a}}=2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+3 \hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}-5 \hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}}$ be a vector such that $\vec{a} \times \vec{c}=\vec{c} \times \vec{b}$

and $(\vec{a}+\vec{c}) \cdot(\vec{b}+\vec{c})=168$. Then the maximum value of $|\vec{c}|^{2}$ is

Let $P$ be the set of seven digit numbers with sum of their digits equal to 11 . If the numbers in P are formed by using the digits 1,2 and 3 only, then the number of elements in the set P is

Let the area of the region $\left\{(x, y): 2 y \leq x^{2}+3\right.$, $y+|x| \leq 3, y \geq|x-1|\}$ be A. Then $6 A$ is equal to:

The least value of $n$ for which the number of integral terms in the Binomial expansion of $(\sqrt[3]{7}+\sqrt[12]{11})^{\mathrm{n}}$ is 183 , is :

The number of solutions of the equation $\left(\frac{9}{x}-\frac{9}{\sqrt{x}}+2\right)\left(\frac{2}{x}-\frac{7}{\sqrt{x}}+3\right)=0$ is

Let $y=y(x)$ be the solution of the differential equation $\cos x\left(\log _{e}(\cos x)\right)^{2} d y+\left(\sin x-3 y \sin x \log _{e}(\cos x)\right) d x=0$ $\mathrm{x} \in\left(0, \frac{\pi}{2}\right)$. If $\mathrm{y}\left(\frac{\pi}{4}\right)=\frac{-1}{\log _{\mathrm{e}} 2}$, then $\mathrm{y}\left(\frac{\pi}{6}\right)$ is

Define a relation R on the interval $\left[0, \frac{\pi}{2}\right)$ by x R y if and only if $\sec ^{2} x-\tan ^{2} y=1$. Then $R$ is

Let the ellipses $E_1:\ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,\quad a > b$ and $E_2:\ \frac{x^2}{A^2} + \frac{y^2}{B^2} = 1,\quad A < B$ have the same eccentricity $\dfrac{1}{\sqrt{3}}$. Let the product of their lengths of latus rectum be $\frac{32}{\sqrt{3}},$ and the distance between the foci of $E_1$ be $4$. If $E_1$ and $E_2$ meet at points $A, B, C,$ and $D$, then the area of the quadrilateral $ABCD$ equals:

Consider an A.P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800 .

Then its $11^{\text {th }}$ term is

Let $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{b}}=2 \hat{\mathrm{i}}+7 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$. Let $L_{1}: \overrightarrow{\mathrm{r}}=(-\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}})+\lambda \overrightarrow{\mathrm{a}}, \lambda \in \mathrm{R}$ and $\mathrm{L}_{2}: \overrightarrow{\mathrm{r}} (\hat{\mathrm{j}}+\hat{\mathrm{k}})+\mu \overrightarrow{\mathrm{b}}, \mu \in \mathrm{R}$ be two lines. If the line $L_{3}$ passes through the point of intersection of $L_{1}$ and $L_{2}$, and is parallel to $\vec{a}+\vec{b}$, then $L_{3}$ passes through the point

The value of $\lim _{n \rightarrow \infty}\left(\sum_{K=1}^{n} \frac{k^{3}+6 k^{2}+11 k+5}{(k+3)!}\right)$ is

The integral $80 \int_{0}^{\frac{\pi}{4}}\left(\frac{\sin \theta+\cos \theta}{9+16 \sin 2 \theta}\right) d \theta$ is equal to

Let $\mathrm{L}_{1}: \frac{\mathrm{x}-1}{1}=\frac{\mathrm{y}-2}{-1}=\frac{\mathrm{z}-1}{2}$ and $L_{2}: \frac{\mathrm{x}+1} {-1}=\frac{\mathrm{y}-2}{2}=\frac{\mathrm{z}}{1}$ be two lines. Let $L_{3}$ be a line passing through the point $(\alpha, \beta, \gamma)$ and be perpendicular to both $L_{1}$ and $L_{2}$. If $L_{3}$ intersects $L_{1}$, then $|5 \alpha-11 \beta-8 \gamma|$ equals

Let $\mathrm{x}_{1}, \mathrm{x}_{2}, \ldots \ldots . \mathrm{x}_{10}$ be ten observations such that

$\sum_{i=1}^{10}\left(x_{i}-2\right)=30, \sum_{i=1}^{10}\left(x_{i}-\beta\right)^{2}=98, \beta>2$ and

their variance is $\frac{4}{5}$. If $\mu$ and $\sigma^{2}$ are respectively the mean and the variance of

$2\left(x_{1}-1\right)+4 \beta, 2\left(x_{2}-1\right)+$ $4 \beta, \ldots . ., 2\left(x_{10}-1\right)+4 \beta$,

then $\frac{\beta \mu}{\sigma^{2}}$ is equal to

Let $\left|z_{1}-8-2 i\right| \leq 1$ and $\left|z_{2}-2+6 i\right| \leq 2$, $z_{1}, z_{2} \in C$. Then the minimum value of $\left|z_{1}-z_{2}\right|$ is

Let $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]=\left[\begin{array}{cc}\log _{5} 128 & \log _{4} 5 \\ \log _{5} 8 & \log _{4} 25\end{array}\right]$.

If $\mathrm{A}_{\mathrm{ij}}$ is the cofactor of

$\mathrm{a}_{\mathrm{ij}}, \mathrm{C}_{\mathrm{ij}}=\sum_{\mathrm{k}=1}^{2} \mathrm{a}_{\mathrm{ik}} \mathrm{A}_{\mathrm{jk}}, 1 \leq \mathrm{i}$, $\mathrm{j} \leq 2$,

and $\mathrm{C}=\left[\mathrm{C}_{\mathrm{ij}}\right]$, then $8|\mathrm{C}|$ is equal to

Let $\mathrm{f}:(0, \infty) \rightarrow \mathrm{R}$ be a twice differentiable function. If for some $a \neq 0, \int_{0}^{1} f(\lambda x) d \lambda=\operatorname{af}(x)$, $f(1)=1$ and $f(16)=\frac{1}{8}$, then $16-f^{\prime}\left(\frac{1}{16}\right)$ is equal to $\qquad$ :

Let $S=\left\{m \in Z: A^{m^{2}}+A^{m}=3 I-A^{-6}\right\}$, where $A=\left[\begin{array}{cc}2 & -1 \\ 1 & 0\end{array}\right]$. Then $n(S)$ is equal to $\qquad$ -

Let $[t]$ be the greatest integer less than or equal to $t$.

Then the least value of $p \in N$ for which $\lim _{x \rightarrow 0^{+}}\left(x\left(\left[\frac{1}{x}\right]+\left[\frac{2}{x}\right]+\ldots . .+\left[\frac{p}{x}\right]\right)-x^{2}\left(\left[\frac{1}{x^{2}}\right]+\left[\frac{2^{2}}{x^{2}}\right]+\ldots .+\left[\frac{9^{2}}{x^{2}}\right]\right)\right) \geq 1$

is equal to $\qquad$ .

The number of 6 -letter words, with or without meaning, that can be formed using the letters of the word MATHS

such that any letter that appears in the word must appear at least twice, is 4 $\qquad$

Let $S=\left\{x: \cos ^{-1} x=\pi+\sin ^{-1} x+\sin ^{-1}(2 x+1)\right\}$.

Then $\sum_{x \in S}(2 x-1)^{2}$ is equal to $\qquad$ .