JEE Main 2025 — 4 April, Morning Shift

On charging the lead storage battery, the oxidation state of lead changes from $x_{1}$ to $y_{1}$ at the anode and from $x_{2}$ to $y_{2}$ at the cathode. The values of $x_{1}, y_{1}, x_{2}$, $\mathrm{y}_{2}$ are respectively

An organic compound ( $X$ ) with molecular formula $\mathrm{C}_{3} \mathrm{H}_{6} \mathrm{O}$ is not readily oxidised. On reduction it gives $\mathrm{C}_{3} \mathrm{H}_{8} \mathrm{O}(\mathrm{Y})$ which reacts with HBr to give a bromide $(Z)$ which is converted to Grignard reagent. This Grignard reagent on reaction with ( X ) followed by hydrolysis give 2, 3-dimethylbutan-2-ol. Compounds ( X ), ( Y ) and ( Z ) respectively are

One mole of an ideal gas expands isothermally and reversibly from $10 \mathrm{dm}^{3}$ to $20 \mathrm{dm}^{3}$ at $300 \mathrm{~K} . \Delta \mathrm{U}, \mathrm{q}$ and work done in the process respectively are Given : $\mathrm{R}=8.3 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$

In $10=2.3$

$\log 2=0.30$

$\log 3=0.48$

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

Statement I: The dipole moment of $\overset{4}{\text{CH}_3} - \overset{3}{\text{CH}} = \overset{2}{\text{CH}} - \overset{1}{\text{CH}} = \text{O}$ is greater than $\overset{4}{\text{CH}_3} - \overset{3}{\text{CH}_2} - \overset{2}{\text{CH}_2} - \overset{1}{\text{CH}} = \text{O}$.

Statement II: C$_1$ - C$_2$ bond length of $\overset{4}{\text{CH}_3} - \overset{3}{\text{CH}} = \overset{2}{\text{CH}} - \overset{1}{\text{CH}} = \text{O}$ is greater than C$_1$ - C$_2$ bond length of $\overset{4}{\text{CH}_3} - \overset{3}{\text{CH}_2} - \overset{2}{\text{CH}_2} - \overset{1}{\text{CH}} = \text{O}$.

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

Pair of transition metal ions having the same number of unpaired electrons is

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

Statement I: Nitrogen forms oxides with +1 to +5 oxidation states due to the formation of $p \pi-p \pi$ bond with oxygen.

Statement II: Nitrogen does not form halides with +5 oxidation state due to the absence of d-orbital in it.

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

Rate law for a reaction between $A$ and $B$ is given by $\mathrm{r}=\mathrm{k}[\mathrm{~A}]^{\mathrm{n}}[\mathrm{~B}]^{\mathrm{m}}$ If concentration of $A$ is doubled and concentration of $B$ is halved from their initial value, the ratio of new rate of reaction to the initial rate of reaction $\left(\frac{r_{2}}{r_{1}}\right)$ is

Which of the following molecule(s) show/s paramagnetic behavior?

A. $\mathrm{O}_{2}$

B. $\mathrm{N}_{2}$

C. $F_{2}$

D. $\mathrm{S}_{2}$

E. $\mathrm{Cl}_{2}$

Choose the correct answer from the options given below.

Number of stereoisomers possible for the complexes, $\left[\mathrm{CrCl}_{3}(\mathrm{py})_{3}\right]$ and $\left[\mathrm{CrCl}_{2}(\mathrm{Ox})_{2}\right]^{3-}$ are respectively (py = pyridine, ox = oxalate)

$X Y$ is the membrane/partition between two chambers 1 and 2 containing sugar solutions of concentration $\mathrm{C}_{1}$ and $\mathrm{C}_{2}\left(\mathrm{C}_{1}>\mathrm{C}_{2}\right) \mathrm{mol} \mathrm{L}^{-1}$. For the reverse osmosis to take place identify the correct condition. (Here $p_{1}$ and $p_{2}$ are pressures applied on chamber 1 and 2)

A. Membrane/Partition: Cellophane, $\mathrm{p}_{1}>\pi$

B. Membrane/Partition : Porous, $\mathrm{p}_{2}>\pi$

C. Membrane/Partition : Parchment paper, $\mathrm{p}_{1}>\pi$

D. Memberane/Partition: Cellophane, $\mathrm{p}_{2}>\pi$

Choose the correct answer from the option given below.

Which one of the following complexes will have $\Delta_{o}=0$ and $\mu=5.96$ B.M?

For a reversible reaction at temperature $T$, both $\Delta H$ and $\Delta S$ are positive. If the equilibrium temperature is $T_e$, then the reaction becomes spontaneous at:

For $\mathrm{A}_{2}+\mathrm{B}_{2} \rightleftharpoons 2 \mathrm{AB}$ $E_{a}$ for forward and backward reaction are 180 and $200 \mathrm{~kJ} \mathrm{~mol}^{-1}$ respectively. If catalyst lowers $\mathrm{E}_{\mathrm{a}}$ for both reaction by $100 \mathrm{~kJ} \mathrm{~mol}^{-1}$. Which of the following statement is correct?

Which one of the following about an electron occupying the 1 s orbital in a hydrogen atom is incorrect? (Bohr's radius is represented by $\mathrm{a}_{0}$ )

$\mathrm{KMnO}_{4}$ acts as an oxidising agent in acidic medium. " $X$ " is the difference between the oxidation states of $M n$ in reactant and product, " $Y$ " is the number of ' $d$ ' electrons present in the brown red precipitate formed at the end of the acetate ion test with neutral ferric chloride. The value of $X+Y$ is $\_\_\_\_\_$

The pH of a 0.01 M weak acid $\mathrm{HX}\left(\mathrm{K}_{\mathrm{a}}=4 \times 10^{-10}\right)$ is found to be 5 . Now the acid solution is diluted with excess of water so that the pH of the solution changes to 6 . The new concentration of the diluted weak acid is given as $x \times 10^{-4} \mathrm{M}$. The value of x is $\_\_\_\_\_$ (nearest integer)

Fortification of food with iron is done using $\mathrm{FeSO}_{4} \cdot 7 \mathrm{H}_{2} \mathrm{O}$. The mass in grams of the $\mathrm{FeSO}_{4} \cdot 7 \mathrm{H}_{2} \mathrm{O}$ required to achieve 12 ppm of iron in 150 kg of wheat is $\_\_\_\_\_$ (Nearest integer) [Given : Molar mass of Fe, S and O respectively are 56, 32 and $16 \mathrm{~g} \mathrm{~mol}^{-1}$ ]

The total number of hydrogen bonds of a DNAdouble Helix strand whose one strand has the following sequence of bases is $\_\_\_\_\_$

5'-G-G-C-A-A-A-T-C-G-G-C-T-A-3'

In Dumas' method for estimation of nitrogen 1 g of an organic compound gave 150 mL of nitrogen collected at 300 K temperature and 900 mm Hg pressure. The percentage composition of nitrogen in the compound is $\_\_\_\_\_$ % (nearest integer) (Aqueous tension at $300 \mathrm{~K}=15 \mathrm{~mm} \mathrm{Hg}$ )

$1+3+5^{2}+7+9^{2}+, \ldots$ upto 40 terms is equal to

Considering the principal values of the inverse trigonometric functions, $\sin ^{-1}\left(\frac{\sqrt{3}}{2} x+\frac{1}{2} \sqrt{1-x^{2}}\right)$, $-\frac{1}{2} < \frac{1}{\sqrt2}$

Let $A=\{1,6,11,16, \ldots\}$ and $B\{9,16,23,30, \ldots\}$ be the sets consisting of the first 2025 terms of two

arithmetic progressions. Then $n(A \cup B)$ is

Let the three sides of a triangle are on the lines $4 x-7 y+10=0, x+y=5$ and $7 x+4 y=15$. Then

the distance of its orthocentre from the orthocentre of the triangle formed by the lines $x=0, y=0$ and

$x+y=1$ is

Consider two vectors $\vec{u}=3 \hat{i}-\hat{j}$ and $\vec{v}=2 \hat{i}+\hat{j}-\lambda \hat{k}, \lambda>0$. The angle between them is given by

$\cos ^{-1}\left(\frac{\sqrt{5}}{2 \sqrt{7}}\right)$. Let $\vec{v}=\vec{v}_{1}+\vec{v}_{2}$, where $\vec{v}_{1}$ is parallel to $\vec{u}$ and $\vec{v}_{2}$ is perpendicular to $\vec{u}$. Then the value $\left|\vec{v}_{1}\right|^{2}+\left|\vec{v}_{2}\right|^{2}$ is equal to

Let $f, g:(1, \infty) \rightarrow \mathbb{R}$ be defined as $f(x)=\frac{2 x+3}{5 x+2}$ and $g(x)=\frac{2-3 x}{1-x}$. If the range of the function

$f \circ g:[2,4] \rightarrow \mathbb{R}$ is $[\alpha, \beta]$, then $\frac{1}{\beta-\alpha}$ is equal to

Let $f:[0, \infty) \rightarrow \mathbb{R}$ be a differentiable function such that $f(x)=1-2 x+\int_{0}^{x} e^{x-t} f(t) d t$ for all

$x \in[0, \infty)$. Then the area of the region bounded by $y=f(x)$ and the coordinate axes is

In the expansion of $\left(\sqrt[3]{2}+\frac{1}{\sqrt[3]{3}}\right)^{n}, n \in \mathbb{N}$, if the ratio of $15^{\text {th }}$ term from the beginning to the $15^{\text {th }}$ term from the end is $\frac{1}{6}$, then the value of ${ }^{n} C_{3}$ is

The probability, of forming a 12 persons committee from 4 engineers, 2 doctors and 10 professors containing at least 3 engineers and at least 1 doctor, is

If $10 \sin ^{4} \theta+15 \cos ^{4} \theta=6$, then the value of $\frac{27 \operatorname{cosec}^{6} \theta+8 \sec ^{6} \theta}{16 \sec ^{8} \theta}$ is

Consider the equation $x^{2}+4 x-n=0$ where $n \in[20$, 100] is a natural number. Then the number of all distinct values of $n$, for which the given equation has integral roots, is equal to

A box contains 10 pens of which 3 are defective. A sample of 2 pens is drawn at random and let $X$ denote the number of defective pens. Then the variance of $X$ is

The value of $\int_{-1}^{1} \frac{(1+\sqrt{|x|-x}) e^{x}+(\sqrt{|x|-x}) e^{-x}}{e^{x}+e^{-x}} d x$ is equal to

If $\lim _{x \rightarrow 1^{+}} \frac{(x-1)(6+\lambda \cos (x-1))+\mu \sin (1-x)}{(x-1)^{3}}=-1$, where $\lambda, \mu \in \mathbb{R}$, then $\lambda+\mu$ is equal to

For an integer $n \geq 2$, if the arithmetic mean of all coefficients in the binomial expansion of $(x+y)^{2 n-3}$ is 16 , then the distance of the point $P(2 n-1$, $n^{2}-4 n$ ) from the line $x+y=8$ is

Find the length of latus rectum of an ellipse if foci are $(2,5)$ and $(2,-3)$ and the eccentricity of the ellipse is $\frac{4}{5}$

Let the shortest distance between the lines $\frac{x-3}{3}=\frac{y-\alpha}{-1}=\frac{z-3}{1}$ and $\frac{x+3}{-3}=\frac{y+7}{2}=\frac{z-\beta}{4}$ be

$3 \sqrt{30}$. Then the positive value of $5 \alpha+\beta$ is

Let $A$ and $B$ be two distinct points on the line $L: \frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}$. Both $A$ and $B$ are at a distance

$2 \sqrt{17}$ from the foot of perpendicular drawn from the point $(1,2,3)$ on the line $L$. If $O$ is the origin, then

$\overrightarrow{O A} \cdot \overrightarrow{O B}$ is equal to

Consider the sets $A=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x^{2}+y^{2}=25\right\}$,

$B=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x^{2}+9 y^{2}=144\right\}, C=\{(x, y) \in \mathbb{Z}$ $\left.\times \mathbb{Z}: x^{2}+y^{2} \leq 4\right\}$ and

$D=A \cap B$. The total number of one-one functions from the set $D$ to the set $C$ is:

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function satisfying $f(0)=1$ and $f(2 x)-f(x)=x$ for all $x \in \mathbb{R}$. If

$\lim _{n \rightarrow \infty}\left\{f(x)-f\left(\frac{x}{2^{n}}\right)\right\}=G(x)$, then $\sum_{r=1}^{10} G\left(r^{2}\right)$ is equal to

Let $m$ and $n$ be the number of points at which the function $f(x)=\max \left\{x, x^{3}, x^{5}, \ldots x^{21}\right\}, x \in \mathbb{R}$, is

not differentiable and not continuous, respectively. Then $m+n$ is equal to $\_\_\_\_$

If the area of the region $\{(x, y):|x-5| \leq y \leq 4 \sqrt{x}\}$ is $A$, then $3 A$ is equal to $\qquad$

Let $A=\{z \in C:|z-2-i|=3\}$, $B=\{z \in C: \operatorname{Re}(z-i z)=2\}$ and $S=A \cap B$. Then

$\sum_{z \in S}|z|^{2}$ is equal to $\qquad$ -.

Let $C$ be the circle $x^{2}+(y-1)^{2}=2, E_{1}$ and $E_{2}$ be two ellipses whose centres lie at the origin and major

axes lie on $x$-axis and $y$-axis respectively. Let the straight-line $x+y=3$ touch the curves $C, E_{1}$ and $E_{2}$ at

$P\left(x_{1}, y_{1}\right), Q\left(x_{2}, y_{2}\right)$ and $R\left(x_{3}, y_{3}\right)$ respectively. Given that $P$ is the midpoint of the line segment

$Q R$ and $P Q=\frac{2 \sqrt{2}}{3}$, the value of $9\left(x_{1} y_{1}+x_{2} y_{2}+x_{3} y_{3}\right)$ is equal to $\qquad$

Let $A=\left[\begin{array}{ccc}\cos \theta & 0 & -\sin \theta\\ 0 & 1 & 0\\ \sin \theta & 0 & \cos \theta\end{array}\right]$. If for some $\theta \in(0, \pi), A^{2}=A^{T}$, then the sum of the diagonal

elements of the matrix $(A+I)^{3}+(A-l)^{3}-6 A$ is equal to $\_\_\_\_$ .

When an object is placed 40 cm away from a spherical mirror an image of magnification $\frac{1}{2}$ is produced. To obtain an image with magnification of $\frac{1}{3}$, the object is to be moved

Considering the Bohr model of hydrogen like atoms, the ratio of the radius of $5^{\text {th }}$ orbit of the

electron in $\mathrm{Li}^{2+}$ and $\mathrm{He}^{+}$is

The Boolean expression $Y=A \bar{B} C+\bar{A} \bar{C}$ can be realised with which of the following gate configurations.

A. One 3-input AND gate, 3 NOT gates and one 2-input OR gate, One 2-input AND gate,

B. One 3-input AND gate, 1 NOT gate, One 2-input NOR gate and one 2-input OR gate

C. 3-input OR gate, 3 NOT gates and one 2-input AND gate Choose the correct answer from the given below.

Consider the sound wave travelling in ideal gases of $\mathrm{He}, \mathrm{CH}_{4}$ and

$\mathrm{CO}_{2}$. All the gases have the same ratio $\frac{P}{\rho}$, where $P$ is the pressure

and $\rho$ is the density. The ratio of the speed of sound through the gases

$v_{\mathrm{He}}: v_{\mathrm{CH}_{4}}: v_{\mathrm{CO}_{2}}$ is given by

The mean free path and the average speed of oxygen molecules at 300 K and 1 atm are

$3 \times 10^{-7} \mathrm{~m}$ and $600 \mathrm{~m} / \mathrm{s}$, respectively.

Find the frequency of its collisions.

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.

Assertion A : In photoelectric effect, on increasing the intensity of incident light the stopping potential increases. Reason

Reason R : Increase in intensity of light increases the rate of photoelectrons emitted, provided the frequency of incident light is greater than threshold frequency.

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

In a Young's double slit experiment, the slits are separated by 0.2 mm . If the slits separation is increased to 0.4 mm , the percentage change of the fringe width is

A small mirror of mass $m$ is suspended by a massless thread of length $I$. Then the small angle

through which the thread will be deflected when a short pulse of laser of energy $E$ falls normal

on the mirror ( $c=$ speed of light in vacuum and $g=$ acceleration due to gravity)

Which of the following are correct expression for torque acting on a body?

A. $\vec{\tau}=\vec{r} \times \vec{L}$

B. $\vec{\tau}=\frac{d}{d t}(\vec{r} \times \vec{p})$

C. $\vec{\tau}=\vec{r} \times \frac{d \vec{p}}{d t}$

D. $\vec{\tau}=I \vec{\alpha}$ E. $\vec{\tau}=\vec{r} \times \vec{F}$ ( $\vec{r}=$ position vector; $\vec{p}=$ linear momentum; $\vec{L}=$ angular momentum; $\vec{\alpha}=$ angular acceleration; $I=$ moment of inertia; $\vec{F}=$ force; $t=$ time) Choose the correct answer from the options given below:

If $\vec{L}$ and $\vec{P}$ represent the angular momentum and linear momentum respectively of a

particle of mass ' $m$ ' having position vector as $\vec{r}=a(\hat{i} \cos \omega t+\hat{j} \sin \omega t)$.

The direction of force is

Two small spherical balls of mass 10 g each with charges $-2 \mu \mathrm{C}$ and $2 \mu \mathrm{C}$,are attached to two ends of very light rigid rod of length 20 cm . The arrangement is now placed near an infinite nonconducting charge sheet with uniform charge density of $100 \mu \mathrm{C} / \mathrm{m}^{2}$ such that length of rod makes an angle of $30^{\circ}$ with electric field generated by charge sheet. Net torque acting on the rod is:

(Take $\varepsilon_{0}=8.85 \times 10^{-12} \mathrm{C}^{2} / \mathrm{Nm}^{2}$ )

Two simple pendulums having lengths $I_{1}$ and $I_{2}$ with negligible string mass undergo angular displacements $\theta_{1}$ and $\theta_{2}$, from their mean positions, respectively. If the angular accelerations of both pendulums are same, then which expression is correct?

Current passing through a wire as function of time is given as $I(t)=0.02 t+0.01 \mathrm{~A}$.

The charge that will flow through the wire from $t=1 \mathrm{~s}$ to $t=2 \mathrm{~s}$ is

Given below are two statements: one is labelled as Assertion $\mathbf{A}$ and the other is labelled as Reason $\mathbf{R}$

Assertion A : The kinetic energy needed to project a body of mass $m$ from earth surface to infinity is $\frac{1}{2} m g R$, where $R$ is the radius of earth.

Reason R: The maximum potential energy of a body is zero when it is projected to infinity from earth surface.

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

A body of mass $m$ is suspended by two strings making angles $\theta_{1}$ and $\theta_{2}$ with

the horizontal ceiling with tensions $T_{1}$ and $T_{2}$ simultaneously. $T_{1}$ and $T_{2}$ are

related by $T_{1}=\sqrt{3} T_{2}$, the angles $\theta_{1}$ and $\theta_{2}$ are

An alternating current is represented by the equation, $i=100 \sqrt{2} \sin (100 \pi t)$ ampere.

The RMS value of current and the frequency of the given alternating current are

Two liquids $A$ and $B$ have $\theta_{A}$ and $\theta_{B}$ as contact angles in a capillary tube.

If $K=\frac{\cos \theta_{A}}{\cos \theta_{B}}$, then identify the correct statement:

In an electromagnetic system, the quantity representing the ratio of electric flux and magnetic flux has

dimension of $M^{P} L Q T^{R} A^{\text {S }}$, where value of ' $Q$ ' and ' $R$ ' is

In an experiment with a closed organ pipe, it is filled with water by $\left(\frac{1}{5}\right)$ th of its volume.

The frequency of the fundamental note will change by

Four capacitors each of capacitance $16 \mu \mathrm{~F}$ are connected as shown in the figure.

The capacitance between points $A$ and $B$ is : $\qquad$ (in $\mu F$ ).

Conductor wire $A B C D E$ with each arm 10 cm in length is placed in magnetic field of $\frac{1}{\sqrt{2}}$ Tesla, perpendicular to its plane.

When conductor is pulled towards right with constant velocity of $10 \mathrm{~cm} / \mathrm{s}$,

induced emf between points $A$ and $E$ is $\qquad$ mV .

Distance between object and its image (magnified by $\left.-\frac{1}{3}\right)$ is 30 cm .

The focal length of the mirror used is $\left(\frac{x}{4}\right) \mathrm{cm}$, where magnitude of value of

$x$ is $\qquad$