JEE Main 2025 Jan 23 Shift 1 — Question Paper

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

Let $R = \{(1, 2), (2, 3), (3, 3)\}$ be a relation defined on the set $\{1, 2, 3, 4\}$. Then the minimum number of elements, needed to be added in $R$ so that $R$ becomes an equivalence relation, 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 $(0, 0)$, $C$ and $(\alpha, 0)$ is 11 square units, then $\alpha^2$ equals:

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

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

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

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

Let the area of a $\triangle PQR$ 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 $c + 2d$ is equal to

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

The value of $(\sin 70°)(\cot 10° \cot 70° - 1)$ is

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

Let the arc $AC$ of a circle subtend a right angle at the centre $O$. If the point $B$ on the arc $AC$, divides the arc $AC$ such that $\frac{\text{length of arc } AB}{\text{length of arc } BC} = \frac{1}{5}$, and $\vec{OC} = \alpha \vec{OA} + \beta \vec{OB}$, then $\alpha + \sqrt{2}(\sqrt{3} - 1)\beta$ is equal to

Let the position vectors of the vertices $A$, $B$ and $C$ of a tetrahedron $ABCD$ 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 $ABC$ at the point $E$. If the length of $AD$ 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 [Note: stem recovered from a low-resolution embedded thumbnail; the English and Hindi stem images were destroyed by PDF corruption — numeric values should be verified against another copy of this paper]

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

If the function

$$f(x) = \begin{cases} \frac{2}{x}\{\sin(k_1 + 1)x + \sin(k_2 - 1)x\}, & x < 0 \\ 4, & x = 0 \\ \frac{2}{x}\log_e\left(\frac{2 + k_1 x}{2 + k_2 x}\right), & x > 0 \end{cases}$$

is continuous at $x = 0$, then $k_1^2 + k_2^2$ is equal to

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

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

The value of

$$\int_{e^2}^{e^4} \frac{1}{x}\left(\frac{e^{((\log_e x)^2 + 1)^{-1}}}{e^{((\log_e x)^2 + 1)^{-1}} + e^{((6 - \log_e x)^2 + 1)^{-1}}}\right) dx$$ is

Let a curve $y = f(x)$ pass through the points $(0, 5)$ and $(\log_e 2, k)$. If the curve satisfies the differential equation $2(3 + y)e^{2x}dx - (7 + e^{2x})dy = 0$, then $k$ is equal to

If the equation $a(b - c)x^2 + b(c - a)x + c(a - b) = 0$ has equal roots, where $a + c = 15$ and $b = \frac{36}{5}$, then $a^2 + c^2$ is equal to____

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

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

A light hollow cube of side length 10 cm and mass 10g, is floating in water. It is pushed down and released to execute simple harmonic oscillations. The time period of oscillations is $y\pi \times 10^{-2}$ s, where the value of $y$ is

(Acceleration due to gravity, $g = 10\ \mathrm{m/s^2}$, density of water $= 10^3\ \mathrm{kg/m^3}$)

A point particle of charge $Q$ is placed at $P$ along the axis of an electric dipole 1 at a distance $r$, as shown in the figure. The point $P$ is also at a distance $r$ from the equatorial plane of a second electric dipole 2. The dipoles are made of opposite charges $q$ separated by a distance $2a$. For the charge particle at $P$ not to experience any net force, which of the following correctly describes the situation?

[Figure: dipole 1 (charges $+q$ and $-q$ separated by $2a$) lies along a horizontal axis with point P on this axis at distance $r$ from it; dipole 2 (charges $+q$ and $-q$ separated by $2a$) lies horizontally below P, with P at distance $r$ on its equatorial (perpendicular bisector) line]

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:

Consider a moving coil galvanomenter (MCG):

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

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

C. If we increase number of turns (N) to its double (2N), 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:

The electric field of an electromagnetic wave in free space is $$\vec{E} = 57 \cos\left[7.5\times10^{6}\,\mathrm{t} - 5\times10^{-3}\left(3x+4y\right)\right]\left(4\hat{i}-3\hat{j}\right)\ \mathrm{N/C}.$$ The associated magnetic field in Tesla is

Given a thin convex lens (refractive index $\mu_2$), kept in a liquid (refractive index $\mu_1$, $\mu_1 < \mu_2$) having radii of curvatures $|R_1|$ and $|R_2|$. Its second surface is silver polished. Where should an object be placed on the optic axis so that a real and inverted image is formed at the same place?

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 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 sub-atomic particle of mass $10^{-30}$ kg is moving with a velocity $2.21 \times 10^{6}$ m/s. Under the matter wave consideration, the particle will behave closely like _____. ($h = 6.63 \times 10^{-34}$ J.s)

The decay constant of a radioactive nucleus $\mathrm{n_2}$ is 3 times that of another radioactive nucleus $\mathrm{n_1}$. If the initial number of both nuclei is the same, what is the ratio of the number of nuclei of $\mathrm{n_2}$ to the number of nuclei of $\mathrm{n_1}$ after one half-life of $\mathrm{n_1}$?

Refer to the circuit diagram given in the figure. which of the following observations are correct?

A. Total resistance of circuit is 6 $\Omega$

B. Current in Ammeter is 1A

C. Potential across AB is 4 Volts.

D. Potential across CD is 4 Volts

E. Total resistance of the circuit is 8 $\Omega$.

Choose the correct answer from the options given below:

[Figure: circuit with battery E = 6 V in series with an ammeter leading to node C; a 4 $\Omega$ resistor from C down to node D; from D one branch has a 4 $\Omega$ resistor to the bottom rail and a second branch passes through a forward-biased diode to node A, then a 4 $\Omega$ resistor from A down to node B on the bottom rail]

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

A force $f = x^2y\hat{i} + y^2\hat{j}$ 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 _____ Joule (round off to the nearest integer)

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

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

[Figure: series circuit with a 12 V battery, a 3 H inductor, and a rheostat R (resistor with sliding contact)]

A positive ion A and a negative ion B has charges $6.67 \times 10^{-19}$ C and $9.6 \times 10^{-10}$ C, and masses $19.2 \times 10^{-27}$ kg and $9 \times 10^{-27}$ kg respectively. At an instant, the ions are separated by a certain distance. At that instant the ratio of the magnitudes of electrostat_____

[Stem truncated: the remainder of this question is lost due to PDF corruption; the surviving text ends mid-sentence]

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

Given : $\mathrm{N_A} = 6.02 \times 10^{23}\ \mathrm{mol^{-1}}$

Heat treatment of muscular pain involves radiation of wavelength of about 900 nm. Which spectral line of H atom is suitable for this?

Given : Rydberg constant $\mathrm{R_H} = 10^5\ \mathrm{cm^{-1}}$, $h = 6.6 \times 10^{-34}$ J s, $c = 3 \times 10^8$ m/s)

Match the LIST-I with LIST-II

	LIST-I (Classification of molecules based on octet rule)		LIST-II (Example)
A.	Molecules obeying octet rule	I.	$\mathrm{NO,\ NO_2}$
B.	Molecules with incomplete octet	II.	$\mathrm{BCl_3,\ AlCl_3}$
C.	Molecules with incomplete octet with odd electron	III.	$\mathrm{H_2SO_4,\ PCl_5}$
D.	Molecules with expanded octet	IV.	$\mathrm{CCl_4,\ CO_2}$

Choose the correct answer from the options given below:

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

$\mathrm{CrCl_3.xNH_3}$ can exist as a complex. 0.1 molal aqueous solution of this complex shows a depression in freezing point of $0.558°\mathrm{C}$. Assuming 100% ionisation of this complex and coordination number of Cr is 6, the complex will be

(Given $\mathrm{K_f} = 1.86$ K kg mol$^{-1}$)

Which of the following happens when $\mathrm{NH_4OH}$ is added gradually to the solution containing 1M $\mathrm{A^{2+}}$ and 1M $\mathrm{B^{3+}}$ ions?

Given : $\mathrm{K_{sp}[A(OH)_2]} = 9 \times 10^{-10}$ and $\mathrm{K_{sp}[B(OH)_3]} = 27 \times 10^{-18}$ at 298 K.

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

In the above diagram, the standard electrode potentials are given in volts (over the arrow). The value of $E^{\ominus}_{\mathrm{FeO_4^{2-}}/\mathrm{Fe^{2+}}}$ is

The element, which does not occur in the same period as the other elements (modern periodic table):

The incorrect statement among the following is

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

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:

On chlorination under photochemical conditions, propane molecule gives two dichloro products "x" and "y". Among "x" and "y", "x" is an optically active molecule. How many trichloro products (consider structural isomers only) will be obtained from "x" when it is treated with chlorine again under photochemical conditions?

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

[Figure: three ring structures — p: cyclopropenyl anion (three-membered ring with one C=C and a negative charge), q: cyclopentadienyl anion (five-membered ring with two C=C, an H and a negative charge on the sp³ carbon), r: cyclooctatetraene (eight-membered ring with four alternating C=C)]

Match the LIST-I with LIST-II

	LIST-I (Name reaction)		LIST-II (Product obtainable)
A.	Swarts reaction	I.	Ethyl benzene
B.	Sandmeyer's reaction	II.	Ethyl iodide
C.	Wurtz Fittig reaction	III.	Cyanobenzene
D.	Finkelstein reaction	IV.	Ethyl fluoride

Choose the correct answer from the options given below:

What amount of bromine will be required to convert 2 g of phenol into 2,4,6-tribromophenol?

(Given molar masses in g mol$^{-1}$ of C, H, O, Br are 12, 1, 16 and 80 respectively)

The major product of the following reaction is:

$$\mathrm{CH_3CH_2CH{=}O} \xrightarrow[\text{reflux}]{\substack{\text{excess HCHO} \\ \text{alkali}}} ?$$

Which among the following react with Hinsberg's reagent?

A. $\mathrm{C_6H_5{-}NH_2}$ (aniline)

B. $\mathrm{C_6H_5{-}N(CH_3)_2}$ (N,N-dimethylaniline)

C. $\mathrm{CH_3{-}NH_2}$

D. $\mathrm{N(CH_3)_3}$

E. $\mathrm{(C_6H_5)_2NH}$ (diphenylamine)

Choose the correct answer from the options given below:

Given below are two statements:

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

Given below are two statements:

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 $\mathrm{Na_4[Fe(CN)_6]}$

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

The standard enthalpy and standard entropy of decomposition of $\mathrm{N_2O_4}$ to $\mathrm{NO_2}$ are 55.0 kJ mol$^{-1}$ and 175.0 J/K/mol respectively. The standard free energy change for this reaction at 25°C in J mol$^{-1}$ is _____ (Nearest integer)

If 1 mM solution of ethylamine produces $\mathrm{pH} = 9$, then the ionization constant ($\mathrm{K_b}$) of ethylamine is $10^{-x}$. The value of x is _____ (nearest integer).

[The degree of ionization of ethylamine can be neglected with respect to unity.]

For the thermal decomposition reaction of $\mathrm{N_2O_5(g)}$ mentioned below, at constant volume, the following table can be created.

$$\mathrm{2N_2O_5(g) \rightarrow 2N_2O_4(g) + O_2(g)}$$

S.No.	Time (s)	Total pressure (atm)
1	0	0.6
2	100	'x'

$x =$ _____ $\times 10^{-3}$ atm (nearest integer)

(Given : rate constant for the reaction is $4.606 \times 10^{-2}\ \mathrm{s^{-1}}$.)

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

[Figure: 3-nitrotoluene (benzene ring with $\mathrm{CH_3}$ group and $\mathrm{NO_2}$ at the meta position)]

$$\xrightarrow{\substack{\text{i) } \mathrm{Br_2},\ \mathrm{Fe} \\ \text{ii) } \mathrm{Sn}, \mathrm{HCl} \\ \text{iii) } \mathrm{NaNO_2}, \mathrm{HCl}, 273\,\mathrm{K} \\ \text{iv) } \mathrm{H_3PO_2}, \mathrm{H_2O}}} \text{(A) Major Product}$$

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

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

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 g mol$^{-1}$ of Ba : 137, S : 32, O : 16)