JEE Main 2025 — 4 April, Evening Shift

The incorrect relationship in the following pairs in relation to ionisation enthalpies is

Consider the ground state of chromium atom ( $Z=$ 24). How many electrons are with Azimuthal quantum number $\mathrm{I}=1$ and $\mathrm{I}=2$ respectively?

The correct order of basicity for the following molecules is

The IUPAC name of the following compound is

$\mathrm{HC \equiv C - CH_2 - CH(OH) - CH_2 - CH = CH_2}$

$X$ ' is the number of electrons in $t_{2 g}$ orbitals of the most stable complex ion among $\left[\mathrm{Fe}\left(\mathrm{NH}_{3}\right)_{6}\right]^{3+}$,

$\left[\mathrm{FeCl}_{6}\right]^{3-}, \quad\left[\mathrm{Fe}\left(\mathrm{C}_{2} \mathrm{O}_{4}\right)_{3}\right]^{3-}$ and $\left[\mathrm{Fe}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{3+}$. The nature of oxide of vanadium of the type

$\mathrm{V}_{2} \mathrm{Ox}$ is:

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

Statement I: Molal depression constant $\mathrm{K}_{\mathrm{f}}$ is given by $\frac{M_{1} R T_{f}}{\Delta S_{f u s}}$, where symbols have their usual meaning.

Statement II: $\mathrm{K}_{\mathrm{f}}$ for benzene is less than the $\mathrm{K}_{\mathrm{f}}$ for water.

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

Half life of zero order reaction $A \rightarrow$ product is 1 hour, when initial concentration of reactant is $2.0 \mathrm{~mol} \mathrm{~L}^{-1}$. The time required to decrease concentration of $A$ from 0.50 to $0.25 \mathrm{~mol} \mathrm{~L}^{-1}$ is:

Which among the following compounds give yellow solid when reacted with $\mathrm{NaOI} / \mathrm{NaOH}$ ?

Choose the correct answer from the options given below

The correct order of $\left[\mathrm{FeF}_{6}\right]^{3-},\left[\mathrm{CoF}_{6}\right]^{3-},\left[\mathrm{Ni}(\mathrm{CO})_{4}\right]$ and $\left[\mathrm{Ni}(\mathrm{CN})_{4}\right]^{2-}$ complex species based on

the number of unpaired electrons present 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: The first ionisation enthalpy of group 14 elements is higher than the corresponding elements of group 13.

Statement II: Melting points and boiling points of group 13 elements are in general much higher than those of corresponding elements of group 14.

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

Consider the given data:

(a) $\mathrm{HCl}(\mathrm{g})+10 \mathrm{H}_{2} \mathrm{O}(\mathrm{I}) \rightarrow \mathrm{HCl} .10 \mathrm{H}_{2} \mathrm{O} \Delta \mathrm{H}=-69.01$ $\mathrm{kJ} \mathrm{mol}^{-1}$

(b) $\mathrm{HCl}(\mathrm{g})+40 \mathrm{H}_{2} \mathrm{O}(\mathrm{I}) \rightarrow \mathrm{HCl} .40 \mathrm{H}_{2} \mathrm{O} \Delta \mathrm{H}=-72.79$ $\mathrm{kJ} \mathrm{mol}^{-1}$

Choose the correct statement:

Sea water, which can be considered as a 6 molar ( 6 M ) solution of NaCl , has a density of $2 \mathrm{~g} \mathrm{~mL}{ }^{-1}$. The concentration of dissolved oxygen $\left(\mathrm{O}_{2}\right)$ in sea water is 5.8 ppm . Then the concentration of dissolved oxygen $\left(\mathrm{O}_{2}\right)$ in sea water, is $x \times 10^{-4} \mathrm{~m}$. $x=$ $\_\_\_\_\_$ (Nearest integer) Given: Molar mass of NaCl is $58.5 \mathrm{~g} \mathrm{~mol}^{-1}$ Molar mass of $\mathrm{O}_{2}$ is $32 \mathrm{~g} \mathrm{~mol}^{-1}$

A metal complex with a formula $\mathrm{MCl}_{4} \cdot 3 \mathrm{NH}_{3}$ is involved in $s p^{3} d^{2}$ hybridisation. It upon reaction with excess of $\mathrm{AgNO}_{3}$ solution gives ' x ' moles of AgCl . Consider ' $x$ ' is equal to the number of lone pairs of electron present in central atom of $\mathrm{BrF}_{5}$. Then the number of geometrical isomers exhibited by the complex is $\_\_\_\_\_$ -.

x mg of $\mathrm{Mg}(\mathrm{OH})_{2}$ ( molar mass $=58$ ) is required to be dissolved in 1.0 L of water to produce a pH of 10.0 at 298 K . The value of $x$ is $\_\_\_\_\_$ mg. (Nearest integer) (Given: $\mathrm{Mg}(\mathrm{OH})_{2}$ is assumed to dissociate completely in $\mathrm{H}_{2} \mathrm{O}$ ]

The amount of calcium oxide produced on heating $150\,\mathrm{kg}$ limestone $(75\%$ pure$)$ is $\underline{\hspace{1cm}}$ kg. (Nearest integer)

Given: Molar mass (in g mol$^{-1}$): $\mathrm{Ca}=40,\ \mathrm{O}=16,\ \mathrm{C}=12$

The molar conductance of an infinitely dilute solution of ammonium chloride was found to be $185 \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}$ and the ionic conductance of hydroxyl and chloride ions are 170 and $70 \mathrm{~S} \mathrm{~cm}^{2}$ $\mathrm{mol}^{-1}$, respectively. If molar conductance of 0.02 M solution of ammonium hydroxide is $85.5 \mathrm{~S} \mathrm{~cm}^{2} \mathrm{~mol}^{-1}$, its degree of dissociation is given by $x \times 10^{-1}$. The value of $x$ is $\_\_\_\_\_$ . (Nearest integer)

Let $f$ be a differentiable function on $\mathbf{R}$ such that $f(2)=$ 1, $f^{\prime}(2)=4$. Let

$\lim _{x \rightarrow 0}(f(2+x))^{3 / x}=e^{\alpha}$. Then the number of times the curve

$y=4 x^{3}-4 x^{2}-4(\alpha-7)$ $x-\alpha$ meets $x$-axis is:

Let the matrix $A=\left[\begin{array}{lll}1 & 0 & 0\\ 1 & 0 & 1\\ 0 & 1 & 0\end{array}\right]$ satisfy $A^{n}=A^{n-2}+A^{2}-l$ for $n \geq 3$. Then the sum of all the elements of $A^{50}$ is:

Consider two sets $A$ and $B$, each containing three numbers in A.P. let the sum and the product of the elements of $A$ be 36 and $p$ respectively and the sum and the product of the elements of $B$ be 36 and $q$ respectively. Let $d$ and $D$ be the common differences of AP's in $A$ and $B$ respectively such that $D=d+3$, $d>0$. If $\frac{p+q}{p-q}=\frac{19}{5}$, then $p-q$ is equal to

Let the product of $\omega_{1}=(8+i) \sin \theta+(7+4 i) \cos \theta$ and $\omega_{2}=(1+8 i) \sin \theta+(4+7 i) \cos \theta$ be

$\alpha+i \beta, i=\sqrt{-1}$. Let $p$ and $q$ be the maximum and the minimum values of $\alpha+\beta$ respectively. Then $p+$ $q$ is

equal to:

Let for two distinct values of $p$ the lines $y=x+p$ touch the ellipse $E: \frac{x^{2}}{4^{2}}+\frac{y^{2}}{3^{3}}=1$ at the points $A$ and

$B$. Let the line $y=x$ intersect $E$ at the points $C$ and $D$. Then the area of the quadrilateral $A B C D$ is equal to :

If $1^{2} \cdot\left({ }^{15} C_{1}\right)+2^{2} \cdot\left({ }^{15} C_{2}\right)+3^{2} \cdot\left({ }^{15} C_{3}\right)+\ldots+15^{2} \cdot\left({ }^{15} C_{15}\right)$ $=2^{m} .3^{n}$. $5^{k}$, where

$m, n, k \in \mathbb{N}$, then $m+n+k$ is equal to :

If a curve $y=y(x)$ passes through the point $\left(1, \frac{\pi}{2}\right)$ and satisfies the differential equation

$\left(7 x^{4} \cot y-e^{x}\right.$ cosecy) $\frac{d x}{d y}=x^{5}, x \geq 1$, then at $x=2$, the value of cosy is :

Let $f(x)+2 f\left(\frac{1}{x}\right)=x^{2}+5$ and $2 g(x)-3 g\left(\frac{1}{2}\right)=$ $x, x>0$. If $\alpha=\int_{1}^{2} f(x) d x$, and

$\beta=\int_{1}^{2} g(x) d x$, then the value of $9 \alpha+\beta$ is :

The axis of a parabola is the line $y=x$ and its verte $x$ and focus are in the first quadrant at distances $\sqrt{2}$ and

$2 \sqrt{2}$ units from the origin, respectively. If the point $(1, k)$ lies on the parabola, then a possible value of $k$ is:

Let $A=\{-3,-2,-1,0,1,2,3\}$ and $R$ be a relation on $A$ defined by $x R y$ if and only if

$2 x-y \in\{0,1\}$. Let $I$ be the number of elements in $R$. Let $m$ and $n$ be the minimum number of elements

required to be added in $R$ to make it reflexive and symmetric relations, respectively. Then $I+m+n$ is equal to:

The centre of a circle $C$ is at the centre of the ellipse $E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, a>b$. Let $C$ pass through the

foci $F_{1}$ and $F_{2}$ of $E$ such that the circle $C$ and the ellipse $E$ intersect at four points. Let $P$ be one of these four

points. If the area of the triangle $P F_{1} F_{2}$ is 30 and the length of the major axis of $E$ is 17 , then the distance

between the foci of $E$ is:

A line passing through the point $A(-2,0)$, touches the parabola $P: y^{2}=x-2$ at the point $B$ in the first quadrant. The area, of the region bounded by the line $A B$, parabola $P$ and the $x$-axis, is:

Let $A$ be the point of intersection of the lines $L_{1}: \frac{x-7}{1}=\frac{y-5}{0}=\frac{z-3}{-1}$ and

$L_{2}: \frac{x-1}{3}=\frac{y+3}{4}=\frac{z+7}{5}$. Let $B$ and $C$ be the points on the lines $L_{1}$ and $L_{2}$ respectively such that

$A B=A C=\sqrt{15}$. Then the square of the area of the triangle $A B C$ is:

If the lines $\vec{r}=(\hat{i}-\hat{j}+\hat{k})+\lambda(3 \hat{j}-\hat{k})$ and $\vec{r}=(\alpha \hat{i}-\hat{j})+\mu(2 \hat{j}-3 \hat{k})$ are co-planar, then the

distance of the plane containing these two lines from the point $(\alpha, 0,0)$ is :

If the sum of the first 20 terms of the series

$\frac{4 \cdot 1}{4+3 \cdot 1^{2}+1^{4}}+\frac{4 \cdot 2}{4+3 \cdot 2^{2}+2^{4}}+\frac{4 \cdot 3}{4+3 \cdot 3^{2}+3^{4}}+$ $\frac{4 \cdot 4}{4+3 \cdot 4^{2}+4^{4}}+\ldots$ is $\frac{m}{n}$, where $m$ and $n$ are

coprime, then $m+n$ is equal to :

Let the mean and the standard deviation of the observation $2,3,3,4,5,7, a, b$ be 4 and $\sqrt{2}$ respectively. Then

the mean deviation about the mode of these observations is :

Let the sum of the focal distances of the point $P(4$, 3) on the hyperbola $H: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ be $\sqrt[8]{\frac{5}{3}}$. If for

$H$, the length of the latus rectum is $I$ and the product of the focal distance of the point $P$ is $m$, then $9l^2+$ 6 m is equal to:

Let $a>0$. If the function $f(x)=6 x^{3}-45 a x^{2}+108 a^{2} x$ +1 attains its local maximum and minimum

values at the points $x_{1}$ and $x_{2}$ respectively such $x_{1} x_{2}=54$, then $a+x_{1}+x_{2}$ is equal to

Let the domains of the functions $f(x)=\log _{4} \log _{3} \log _{7}(8$ $\left.-\log _{2}\left(x^{2}+4 x+5\right)\right)$ and

$g(x)=\sin ^{-1}\left(\frac{7 x+10}{x-2}\right)$ be $(\alpha, \beta)$ and $[\gamma, \delta]$, respectively. Then $\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}$ is equal to:

The sum of the infinite series $\cot ^{-1}\left(\frac{7}{4}\right)+\cot ^{-1}\left(\frac{19}{4}\right)+\cot ^{-1}\left(\frac{39}{4}\right)+\cot ^{-1} \frac{67}{4}+\ldots$ is:

If $\alpha$ is a root of the equation $x^{2}+x+1=0$ and $\sum_{k=1}^{n}\left(\alpha^{k}+\frac{1}{a^{k}}\right)^{2}=20$, then $n$ is equal to $\_\_\_\_$ .

A card from a pack of 52 cards is lost. From the remaining 51 cards, $n$ cards are drawn and are found to be spades. If the probability of the lost card to be a spade is $\frac{11}{50}$, then $n$ is equal to $\qquad$ -

If \int \frac{(\sqrt{1+x^2} + x)^{10}}{(\sqrt{1+x^2} - x)^{9}} dx $$ = \frac{1}{m} \left( (\sqrt{1+x^2} + x)^n (\sqrt{1+x^2} - x)^n \right) + C where $C$ is the constant of integration and $m$, $n \in \mathbb{N}$, then $m+n$ is equal to $\_\_\_\_$ -.

Let the three sides of a triangle $A B C$ be given by the vectors $2 \hat{i}-\hat{j}+\hat{k}, \hat{i}-3 \hat{j}-5 \hat{k}$ and $3 \hat{i}-4 \hat{j}-4 \hat{k}$ .

Let $G$ be the centroid of the triangle $A B C$. Then $6\left(|\overrightarrow{A G}|^{2}+|\overrightarrow{B G}|^{2}+|\overrightarrow{C G}|^{2}\right)$ is equal to $\_\_\_\_$ -.

Let $m$ and $n,(m < n),$ be two 2-digit numbers. Then the total number of pairs (m, n), such that gcd (m,n) = 6, is $\_\_\_\_$

A finite size object is placed normal to the principal axis at a distance of 30 cm from a convex mirror of focal length 30 cm . A plane mirror is now placed in such a way that the image produced by both the mirrors coincide with each other. The distance between the two mirrors is :

Displacement of a wave is expressed as $x(t)=5 \cos \left(628 t+\frac{\pi}{2}\right) \mathrm{m}$.

The wavelength of the wave when its velocity is $300 \mathrm{~m} / \mathrm{s}$ is : ( $\pi=3.14$ )

In an electromagnetic system, a quantity defined as the ratio of electric dipole moment and magnetic

dipole moment has dimension of $\left[M^{P} L^{Q} T^{R} A^{S}\right]$. The value of $P$ and $Q$ are :

Consider a $n$-type semiconductor in which $n_{e}$ and $n_{h}$ are number of electrons and holes, respectively.

(A) Holes are minority carriers

(B) The dopant is a pentavalent atom

(C) $n_{e} n_{h} \neq n_{i}^{2}$ (where $n_{i}$ is number of electrons or holes in semiconductor when it is intrinsic form) (D) $n_{e} n_{h} \geq n_{i}^{2}$

(E) The holes are not generated due to the donors Choose the correct answer from the options given below:

The displacement $x$ versus time graph is shown below.

(A) The average velocity during 0 to 3 s is $10 \mathrm{~m} / \mathrm{s}$

(B) The average velocity during 3 to 5 s is $0 \mathrm{~m} / \mathrm{s}$

(C) The instantaneous velocity at $t=2 \mathrm{~s}$ is $5 \mathrm{~m} / \mathrm{s}$

(D) The average velocity during 5 to 7 s and instantaneous velocity at $t=6.5 \mathrm{~s}$ are equal

(E) The average velocity from $t=0$ to $t=9 \mathrm{~s}$ is zero

Choose the correct answer from the options given below

For the determination of refractive index of glass slab, a travelling microscope is used whose main scale contains 300 equal divisions equals to 15 cm . The vernier scale attached to the microscope has 25 divisions equals to 24 divisions of main scale. The least count (LC) of the travelling microscope is (in cm )

An object is kept at rest at a distance of $3 R$ above the earth's surface where $R$ is earth's radius. The minimum speed with which it must be projected so that it does not return to earth is (Assume $M$ = mass of earth, $G=$ Universal gravitational constant)

A wheel is rolling on a plane surface. The speed of a particle on the highest point of the rim is $8 \mathrm{~m} / \mathrm{s}$. The speed of the particle on the rim of the wheel at the same level as the centre of wheel, will be

Two polarisers $P_{1}$ and $P_{2}$ are placed in such a way that the intensity of the transmitted light will be zero. A third polariser $P_{3}$ is inserted in between $P_{1}$ and $P_{2}$, at particular angle between $P_{2}$ and $P_{3}$. The transmitted intensity of the light passing the through all three polarisers is maximum. The angle between the polarisers $P_{2}$ and $P_{3}$ is

Match List - I with List - II.

$\Delta Q=$ Heat supplied $\Delta W=$ Work done by the system $\Delta U=$ Change in internal energy $P=$ Pressure of the system $\Delta V=$ Change in volume of the system

Choose the correct answer from the options given below:

Consider a rectangular sheet of solid material of length $/=9 \mathrm{~cm}$ and width $d=4 \mathrm{~cm}$. The coefficient of linear expansion is $\alpha=3.1 \times 10^{-5} \mathrm{~K}^{-1}$ at room temperature and one atmospheric pressure. The mass of sheet $m=0.1 \mathrm{~kg}$ and the specific heat capacity $C_{v}=900 \mathrm{~J} \mathrm{~kg}^{-1} \mathrm{~K}^{-1}$. If the amount of heat supplied to the material is $8.1 \times 10^{2} \mathrm{~J}$ then change in area of the rectangular sheet is

A radioactive material $P$ first decays into $Q$ and then $Q$ decays to non-radioactive material $R$. Which of the following figure represents time dependent mass of $P$, $Q$ and $R$ ?

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

Statement (I): The dimensions of Planck's constant and angular momentum are same.

Statement (II) : In Bohr's model electron revolve around the nucleus only in those orbits for which angular momentum is integral multiple of Planck's constant.

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

There are $n$ number of identical electric bulbs, each is designed to draw a power $p$ independently from the mains supply. They are now joined in series across the mains supply. The total power drawn by the combination is

Three parallel plate capacitors $C_{1}, C_{2}$ and $C_{3}$ each of capacitance $5 \mu \mathrm{~F}$ are connected as shown in figure. The effective capacitance between points $A$ and $B$, when the space between the parallel plates of $C_{1}$ capacitor is filled with a dielectric medium having dielectric constant of 4 , is:

A cylindrical rod of length 1 m and radius 4 cm is mounted vertically. It is subjected to a shear force of

$10^{5} \mathrm{~N}$ at the top. Considering infinitesimally small displacement in the upper edge,

the angular displacement $\theta$ of the rod axis from its original position would be

(shear moduli, $G=10^{10} \mathrm{~N} / \mathrm{m}^{2}$ )

A block of mass 25 kg is pulled along a horizontal surface by a force at an angle $45^{\circ}$ with the horizontal. The friction coefficient between the block and the surface is 0.25 . The block travels at a uniform velocity. The work done by the applied force during a displacement of 5 m of the block is :

There are two vessels filled with an ideal gas where volume of one is double the volume of other. The large vessel contains the gas at 8 kPa at 1000 K while the smaller vessel contains the gas at 7 kPa at 500 K . If the vessels are connected to each other by a thin tube allowing the gas to flow and the temperature of both vessels is maintained at 600 K , at steady state the pressure in the vessels will be (in kPa ).

An inductor of self inductance 1 H is connected in series with a resistor of $100 \pi$ ohm and an ac supply of $100 \pi$ volt, 50 Hz . Maximum current flowing in the circuit is $\qquad$ A.

A particle of charge $1.6 \mu \mathrm{C}$ and mass $16 \mu \mathrm{~g}$ is present in a strong magnetic field of 6.28 T . The particle is then fired perpendicular to magnetic field. The time required for the particle to return to original location for the first time is $\qquad$ s. $(\pi=3.14)$

If an optical medium possesses a relative permeability of $\frac{10}{\pi}$ and relative permittivity of

$\frac{1}{0.0885}$, then the velocity of light is greater in vacuum than that in this medium by

$\qquad$ times. $\left(\mu_{0}=4 \pi \times 10^{-7} \mathrm{H} / \mathrm{m}, \in_{0}=8.85 \times 10^{-12} \mathrm{~F} / \mathrm{m}\right.$,

$c=3 \times 10^{8} \mathrm{~m} / \mathrm{s}$ )

A solid sphere with uniform density and radius $R$ is rotating initially with constant angular velocity $\left(\omega_{1}\right)$ about its diameter. After some time during the rotation its starts loosing mass at a uniform rate, with no change in its shape. The angular velocity of the sphere when its radius become $R / 2$ is $x \omega_{1}$. The value of $x$ is $\qquad$ -.

In a Young's double slit experiment, two slits are located 1.5 mm apart. The distance of screen from slits is 2 m and the wavelength of the source is 400 nm . If the 20 maxima of the double slit pattern are contained within the central maximum of the single slit diffraction pattern, then the width of each slit is $x \times 10^{-3} \mathrm{~cm}$, where $x$-value is $\qquad$ -.