JEE Main 2025 — 23 January, Morning Shift

The value of $\int_{e^{2}}^{e^{4}} \frac{1}{x}\left(\frac{e^{\left(\left(\log _{e} x\right)^{2}+1\right)^{-1}}}{e^{\left(\left(\log _{e} x\right)^{2}+1\right)^{-1}}+e^{\left(\left(6-\log _{e} x\right)^{2}+1\right)^{-1}}}\right) d x$ is

Let $\mathrm{I}(\mathrm{x})=\int \frac{\mathrm{dx}}{(\mathrm{x}-11)^{\frac{11}{13}}(\mathrm{x}+15)^{\frac{15}{13}}}$. If $\mathrm{I}(37)-\mathrm{I}(24)=\frac{1}{4}\left(\frac{1}{\mathrm{~b}^{\frac{1}{13}}}-\frac{1}{\mathrm{c}^{\frac{1}{13}}}\right), \mathrm{b}, \mathrm{c} \in \mathbb{N}$, then $3(b+c)$ is equal to

If the function

is continuous at $\mathrm{x}=0$, then $\mathrm{k}_{1}{ }^{2}+\mathrm{k}_{2}{ }^{2}$ is equal to

If the line $3 x-2 y+12=0$ intersects the parabola $4 y=3 x^{2}$ at the points $A$ and $B$, then at the vertex of the parabola, the line segment AB subtends an angle equal to

Let a curve $y=f(x)$ pass through the points $(0,5)$ and $\left(\log _{\mathrm{e}} 2, k\right)$. If the curve satisfies the differential

equation $2(3+y) e^{2 x} d x-\left(7+e^{2 x}\right) d y=0$, then $k$ is equal to

Let $f(x)=\log _{e} x$ and $g(x)=\frac{x^{4}-2 x^{3}+3 x^{2}-2 x+2}{2 x^{2}-2 x+1}$ . Then the domain of fog is

Let the arc AC of a circle subtend a right angle at the centre $O$. If the point $B$ on the arc $A C$, divides the arc AC

such that $\frac{\text { length of } \operatorname{arc} \mathrm{AB}}{\text { length of } \operatorname{arc} \mathrm{BC}}=\frac{1}{5}$, and $\overrightarrow{\mathrm{OC}}=\alpha \overrightarrow{\mathrm{OA}}+\beta \overrightarrow{\mathrm{OB}}$, then $\alpha+\sqrt{2}(\sqrt{3}-1) \beta$ is equal to

If the first term of an A.P. is 3 and the sum of its first four terms is equal to one-fifth of the sum of the next four terms, then the sum of the first 20 terms is equal to

Let P be the foot of the perpendicular from the point $Q(10,-3,-1)$ on the line $\frac{x-3}{7}=\frac{y-2}{-1}=\frac{z+1}{-2}$. Then the area of the right angled triangle PQR , where R is the point $(3,-2,1)$, is

Let $\left|\frac{\bar{z}-i}{2 \bar{z}+i}\right|=\frac{1}{3}, z \in \mathbb{C}$, be the equation of a circle with center at $C$. If the area of the triangle, whose vertices are at the points $(0,0), \mathrm{C}$ and $(\alpha, 0)$ is 11 square units, then $\alpha^{2}$ equals

Let $\mathrm{R}=\{(1,2),(2,3),(3,3)\}$ be a relation defined on the set $\{1,2,3,4\}$. Then the minimum number of

elements, needed to be added in R so the R becomes an equivalence relation, is :

The number of words, which can be formed using all the letters of the word "DAUGHTER", so that all the vowels never come together, is

Let the area of a $\triangle P Q R$ with vertices $P(5,4), Q(-2,4)$ and $R(a, b)$ be 35 square units. If its orthocenter and centroid are $O\left(2, \frac{14}{5}\right)$ and $C(c, d)$ respectively, then $\mathrm{c}+2 \mathrm{~d}$ is equal to

If $\frac{\pi}{2} \leq x \leq \frac{3 \pi}{4}$, then $\cos ^{-1}\left(\frac{12}{13} \cos x+\frac{5}{13} \sin x\right)$

is equal to

The value of $\left(\sin 70^{\circ}\right)\left(\cot 10^{\circ} \cot 70^{\circ}-1\right)$ is

Marks obtained by all the students of class 12 are presented in a frequency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval 12-18 and median class frequency 12 . If the number of students whose marks are less than 12 is 18 , then the total number of students is

Let the position vectors of the vertices $A, B$ and $C$ of a tetrahedron $A B C D$ be $\hat{i}+2 \hat{j}+\hat{k}, \hat{i}+3 \hat{j}-2 \hat{k}$ and $2 \hat{i}+\hat{j}-\hat{k}$ respectively. The altitude from the vertex D to the opposite face ABC meets the median line segment through $A$ of the triangle $A B C$ at the point $E$. If the length of $A D$ is $\frac{\sqrt{110}}{3}$ and the volume of the tetrahedron is $\frac{\sqrt{805}}{6 \sqrt{2}}$, then the position vector of E is

If $\mathrm{A}, \mathrm{B}$ and $\left(\operatorname{adj}\left(\mathrm{A}^{-1}\right)+\operatorname{adj}\left(\mathrm{B}^{-1}\right)\right)$ are non-singular matrices of same order, then the inverse of

$\mathrm{A}\left(\operatorname{adj}\left(\mathrm{A}^{-1}\right)+\operatorname{adj}\left(\mathrm{B}^{-1}\right)\right)^{-1} \mathrm{~B}$, is equal to

If the system of equations

$(\lambda-1) x+(\lambda-4) y+\lambda z=5$

$\lambda x+(\lambda-1) y+(\lambda-4) z=7$

$(\lambda+1) x+(\lambda+2) y-(\lambda+2) z=9$

has infinitely many solutions, then $\lambda^{2}+\lambda$ is equal to

One die has two faces marked 1 , two faces marked 2 , one face marked 3 and one face marked 4. Another die has one face marked 1, two faces marked 2, two faces marked 3 and one face marked 4. The probability of getting the sum of numbers to be 4 or 5 , when both the dice are thrown together, is

If the area of the larger portion bounded between the curves $x^{2}+y^{2}=25$ and $y=|x-1|$ is $\frac{1}{4}(b \pi+c)$, $\mathrm{b}, \mathrm{c} \in \mathbb{N}$, then $\mathrm{b}+\mathrm{c}$ is equal to $\_\_\_\_\_$

The sum of all rational terms in the expansion of $\left(1+2^{1 / 3}+3^{1 / 2}\right)^{6}$ is equal to $\_\_\_\_\_$

Let the circle C touch the line $\mathrm{x}-\mathrm{y}+1=0$, have the centre on the positive x -axis, and cut off a chord of length $\frac{4}{\sqrt{13}}$ along the line $-3 x+2 y=1$. Let $H$ be the hyperbola $\frac{x^{2}}{\alpha^{2}}-\frac{y^{2}}{\beta^{2}}=1$, whose one of the foci is the centre of $C$ and the length of the transverse axis is the diameter of C . Then $2 \alpha^{2}+3 \beta^{2}$ is equal to $\_\_\_\_\_$

If the set of all values of $a$, for which the equation $5 x^{3}-15 x-a=0$ has three distinct real roots, is the interval $(\alpha, \beta)$, then $\beta-2 \alpha$ is equal to $\_\_\_\_\_$

If the equation $a(b-c) x^{2}+b(c-a) x+c(a-b)=0$ has equal roots, where $\mathrm{a}+\mathrm{c}=15$ and

$\mathrm{b}=\frac{36}{5}$, then $\mathrm{a}^{2}+\mathrm{c}^{2}$ is equal to $\_\_\_\_\_$

The element that does not belong to the same period of the remaining elements (modern periodic table) is:

Heat treatment of muscular pain involves radiation of wavelength about $900\,\mathrm{nm}$. Which spectral line of hydrogen atom is suitable?

(Given: $R_H = 10^5\,\mathrm{cm^{-1}}$)

The incorrect statements among the following is

$\mathrm{CrCl}_{3} \cdot \mathrm{xNH}_{3}$ can exist as a complex. 0.1 molal aqueous solution of this complex shows a depression in freezing point of $0.558^{\circ} \mathrm{C}$. Assuming $100 \%$ ionisation of this complex and coordination number of Cr is 6 , the complex will be (Given $\mathrm{K}_{\mathrm{f}}=1.86 \mathrm{~K} \mathrm{~kg} \mathrm{~mol}^{-1}$ )

$\mathrm{FeO_4^{2-}} \xrightarrow{+2.0V} \mathrm{Fe^{3+}} \xrightarrow{+0.8V} \mathrm{Fe^{2+}} \xrightarrow{-0.5V} \mathrm{Fe^{0}}$

In the above diagram, the standard electrode potentials are given in volts(over the arrow)

$\text{The value of } E^\ominus_{\mathrm{FeO_4^{2-}/Fe^{2+}}} \text{ is}$

Match the 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 : Fructose does not contain an aldehydic group but still reduces Tollen's reagent

Statement II : In the presence of base, fructose undergoes rearrangement to give glucose.

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

After removing $10^{21}$ molecules from an $x\,\mathrm{mg}$ sample of $\mathrm{CO_2}$, $2.8 \times 10^{-3}\,\mathrm{mol}$ of $\mathrm{CO_2}$ is left. The mass of $\mathrm{CO_2}$ taken initially is:

Ice at $-5^{\circ} \mathrm{C}$ is heated to become vapor with temperature of $110^{\circ} \mathrm{C}$ at atmospheric pressure. The entropy change associated with this process can be obtained from :

The d-electronic configuration of an octahedral Co (II) complex having magnetic moment of 3.95 BM is :

The complex that shows Facial - Meridional isomerism is

The correct stability order of the following species/molecules is :

Propane molecule on chlorination under photochemical condition gives two di-chloro products, " $x$ " and " $y$ ". Amongst " $x$ " and " $y$ ", " $x$ " is an optically active molecule. How many tri-chloro products (consider only structural isomers) will be obtained from " $x$ " when it is further treated with chlorine under the photochemical condition?

What amount of bromine will be required to convert 2 g of phenol into $2,4,6$-tribromophenol ? (Given molar mass in $\mathrm{g} \mathrm{mol}^{-1}$ of $\mathrm{C}, \mathrm{H}, \mathrm{O}, \mathrm{Br}$ are $12,1,16,80$ respectively)

The correct set of ions (aqueous solution) with same colour from the following is :

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

Statement I : In Lassaigne's test, the covalent organic molecules are transformed into ionic compounds.

Statement II : The sodium fusion extract of an organic compound having N and S gives prussian blue colour with $\mathrm{FeSO}_{4}$ and $\mathrm{Na}_{4}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]$

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

Which of the following happens when $\mathrm{NH}_{4} \mathrm{OH}$ is added gradually to the solution containing

$1 \mathrm{M} \mathrm{A}^{2+}$ and $1 \mathrm{M} \mathrm{B}^{3+}$ ions ?

Given : $\mathrm{K}_{\text {sp }}\left[\mathrm{A}(\mathrm{OH})_{2}\right]=9 \times 10^{-10}$ and

Match the LIST-I with LIST-II

Choose the correct answer from the options given below :

Which among the following react with Hinsberg's reagent?

Choose the correct answer from the options given below :

If 1 mM solution of ethylamine produces $\mathrm{pH}=9$, then the ionization constant $\left(\mathrm{K}_{\mathrm{b}}\right)$ of ethylamine is $10^{-x}$. The value of x is___ (nearest integer).[0pt] [The degree of ionization of ethylamine can be neglected with respect to unity.]

During "S" estimation, 160 mg of an organic compound gives 466 mg of barium sulphate. The percentage of Sulphur in the given compound is___$\%$.

(Given molar mass in $\mathrm{g} \mathrm{mol}^{-1}$ of $\mathrm{Ba}: 137, \mathrm{~S}: 32$, O:16)

Consider the following sequence of reactions to produce major product (A)

Molar mass of product (A) is $\qquad$ $\mathrm{g} \mathrm{mol}^{-1}$.

(Given molar mass in $\mathrm{g} \mathrm{mol}^{-1}$ of $\mathrm{C}: 12, \mathrm{H}: 1$, $\mathrm{O}: 16, \mathrm{Br}: 80, \mathrm{~N}: 14, \mathrm{P}: 31$ )

The standard enthalpy and standard entropy of decomposition of $\mathrm{N}_{2} \mathrm{O}_{4}$ to $\mathrm{NO}_{2}$ are $55.0 \mathrm{~kJ} \mathrm{~mol}^{-1}$ and $175.0 \mathrm{~J} / \mathrm{K} / \mathrm{mol}$ respectively. The standard free energy change for this reaction at $25^{\circ} \mathrm{C}$ in $\mathrm{J} \mathrm{mol}^{-1}$ is____ (Nearest integer)

Regarding self-inductance : A : The self-inductance of the coil depends on its geometry. B : Self-inductance does not depend on the permeability of the medium. C : Self-induced e.m.f. opposes any change in the current in a circuit. D : Self-inductance is electromagnetic analogue of mass in mechanics. E : Work needs to be done against self-induced e.m.f. in establishing the current. Choose the correct answer from the options given below:

A light hollow cube of side length 10 cm and mass 10 g , is floating in water. It is pushed down and released to execute simple harmonic oscillations. The time period of oscillations is $\mathrm{y} \pi \times 10^{-2} \mathrm{~s}$, where the value of y is\ (Acceleration due to gravity, $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$, density of water $=10^{3} \mathrm{~kg} / \mathrm{m}^{3}$ )

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

Statement-I : The hot water flows faster than cold water.

Statement-II : Soap water has higher surface tension as compared to fresh water.

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

. A sub-atomic particle of mass $10^{-30} \mathrm{~kg}$ is moving with

a velocity $2.21 \times 10^{6} \mathrm{~m} / \mathrm{s}$. Under the matter

wave consideration, the particle will behave closely like

$\qquad$ . $\left(\mathrm{h}=6.63 \times 10^{-34} \mathrm{~J} . \mathrm{s}\right)$

A spherical surface of radius of curvature R, separates air from glass (refractive index =1.5 ). The centre of curvature is in the glass medium. A point object ' O ' placed in air on the optic axis of the surface, so that its real image is formed at ' I ' inside glass. The line OI intersects the spherical surface at P and PO=PI. The distance PO equals to-

A radioactive nucleus n_2 has 3 times the decay constant as compared

to the decay constant of another radioactive nucleus n_1. If initial number

of both nuclei are the same, what is the ratio of number of nuclei .

of n_2 to the number of nuclei of n_1, after one half-life of n_1 ?

Identify the valid statements relevant to the given circuit at the instant when the key is closed. IMAGES A. There will be no current through resistor R . B. There will be maximum current in the connecting wires. C. Potential difference between the capacitor plates A and B is minimum. D. Charge on the capacitor plates is minimum. Choose the correct answer from the options given below :

. The position of a particle moving on $x$-axis is given by

$x(t)=A \sin t+B \cos ^{2} t+\mathrm{Ct}^{2}+D$, where $t$ is time.

The dimension of $\frac{A B C}{D}$ is-

Match the List-I with List-II

Choose the correct answer from the options given below :

Consider a moving coil galvanometer (MCG) :

A : The torsional constant in moving coil galvanometer has dimensions [ $\mathrm{ML}^{2} \mathrm{~T}^{-2}$ ]

B : Increasing the current sensitivity may not necessarily increase the voltage sensitivity.

C : If we increase number of turns $(\mathrm{N})$ to its double $(2 \mathrm{~N})$, then the voltage sensitivity doubles.

D : MCG can be converted into an ammeter by introducing a shunt resistance of large value in parallel with galvanometer.

E: Current sensitivity of MCG depends inversely on number of turns of coil.

Choose the correct answer from the options given below :

A gun fires a lead bullet of temperature 300 K into a wooden block. The bullet having melting temperature of 600 K penetrates into the block and melts down. If the total heat required for the process is 625 J , then the mass of the bullet is $\qquad$ grams.

(Latent heat of fusion of lead $=2.5 \times 10^{4} \mathrm{JKg}^{-1}$ and specific heat capacity of lead $=125 \mathrm{JKg}^{-1} \mathrm{~K}^{-1}$ )

What is the lateral shift of a ray refracted through a parallel-sided glass slab of thickness ' $h$ ' in terms of the angle of incidence ' $i$ ' and angle of refraction ' $r$ ', if the glass slab is placed in air medium ?

A solid sphere of mass ' $m$ ' and radius ' $r$ ' is allowed to roll without slipping from the highest point of an inclined plane of length 'L' and makes an angle $30^{\circ}$ with the horizontal. The speed of the particle at the bottom of the plane is $\mathrm{v}_{1}$. If the angle of inclination is increased to $45^{\circ}$ while keeping $L$ constant. Then the new speed of the sphere at the bottom of the plane is $v_{2}$. The ratio of $v_{1}{ }^{2}: v_{2}{ }^{2}$ is

The electric field of an electromagnetic wave in free space is ${{E}^{\to }}=57cos[7.5\text{ }\!\!\times\!\!\text{ }{{10}^{6}}t-5\text{ }\!\!\times\!\!\text{ }{{10}^{(-3)}}(3x+4y)](4{{i}^{\wedge }}-3{{j}^{\wedge }})N/C.$ The associated magnetic field in Tesla is-

Consider a circular disc of radius 20 cm with centre located at the origin. A circular hole of a radius 5 cm is cut from this disc in such a way that the edge of the hole touches the edge of the disc. The distance of centre of mass of residual or remaining disc from the origin will be-

Two particles are located at equal distance from origin. The position vectors of those are represented by $\overrightarrow{\mathrm{A}}=2 \hat{\mathrm{i}}+3 \mathrm{n} \hat{\mathrm{j}}+2 \hat{\mathrm{k}} \quad$ and $\vec{B}=2 \hat{i}-2 \hat{j}+4 \hat{p}$, respectively. If both the vectors are at right angle to each other, the value of $\mathrm{n}^{-1}$ is $\qquad$ .

item An ideal gas initially at $0^{\circ} \mathrm{C}$ temperature, is compressed suddenly to one fourth of its volume. If the ratio of specific heat at constant pressure to that at constant volume is $3 / 2$, the change in temperature due to the thermodynamics process is $\qquad$ K.

A force $f=x^{2} y \hat{i}+y^{2} \hat{j}$ acts on a particle in a plane $x+y=10$. The work done by this force during a displacement from $(0,0)$ to $(4 \mathrm{~m}, 2 \mathrm{~m})$ is $\qquad$ Joule (round off to the nearest integer)

In the given circuit the sliding contact is pulled outwards such that electric current in the circuit changes at the rate of $8 \mathrm{~A} / \mathrm{s}$. At an instant when R is $12 \Omega$, the value of the current in the circuit will be $\qquad$ A.