JEE Main 2025 Jan 22 Shift 1 — Question Paper

Radius of the first excited state of Helium ion is given as :

$a_0 \rightarrow$ radius of first stationary state of hydrogen atom.

A liquid when kept inside a thermally insulated closed vessel at $25°\mathrm{C}$ was mechanically stirred from outside. What will be the correct option for the following thermodynamic parameters ?

Arrange the following solutions in order of their increasing boiling points.

(i) $10^{-4}$ M NaCl (ii) $10^{-4}$ M Urea (iii) $10^{-3}$ M NaCl (iv) $10^{-2}$ M NaCl

A vessel at 1000 K contains $\mathrm{CO_2}$ with a pressure of 0.5 atm. Some of $\mathrm{CO_2}$ is converted into CO on addition of graphite. If total pressure at equilibrium is 0.8 atm, then Kp is :

A solution of aluminium chloride is electrolysed for 30 minutes using a current of 2A. The amount of the aluminium deposited at the cathode is ________.

[Given : molar mass of aluminium and chlorine are $27\ \mathrm{g\ mol^{-1}}$ and $35.5\ \mathrm{g\ mol^{-1}}$ respectively. Faraday constant $= 96500\ \mathrm{C\ mol^{-1}}$]

Which of the following electrolyte can be used to obtain $\mathrm{H_2S_2O_8}$ by the process of electrolysis ?

Which of the following statement is not true for radioactive decay ?

Which of the following electronegativity order is incorrect ?

Match List-I with List-II.

	List-I		List-II
(A)	$\mathrm{Al^{3+}} < \mathrm{Mg^{2+}} < \mathrm{Na^{+}} < \mathrm{F^{-}}$	(I)	Ionisation Enthalpy
(B)	B < C < O < N	(II)	Metallic character
(C)	B < Al < Mg < K	(III)	Electronegativity
(D)	Si < P < S < Cl	(IV)	Ionic radii

Choose the correct answer from the options given below :

Lanthanoid ions with $4f^7$ configuration are :

(A) $\mathrm{Eu^{2+}}$ (B) $\mathrm{Gd^{3+}}$ (C) $\mathrm{Eu^{3+}}$ (D) $\mathrm{Tb^{3+}}$ (E) $\mathrm{Sm^{2+}}$

Choose the correct answer from the options given below :

From the magnetic behaviour of $\mathrm{[NiCl_4]^{2-}}$ (paramagnetic) and $\mathrm{[Ni(CO)_4]}$ (diamagnetic), choose the correct geometry and oxidation state.

In which of the following complexes the CFSE, $\Delta_o$ will be equal to zero ?

The IUPAC name of the following compound is :

$$\mathrm{CH_3-CH(COOH)-CH_2-CH_2-CH(COOCH_3)-CH_3}$$

(COOH on the second carbon and $\mathrm{COOCH_3}$ on the fifth carbon of the hexane chain)

How many different stereoisomers are possible for the given molecule ?

$$\mathrm{CH_3-CH(OH)-CH=CH-CH_3}$$

The incorrect statements regarding geometrical isomerism are :

(A) Propene shows geometrical isomerism. (B) Trans isomer has identical atoms/groups on the opposite sides of the double bond. (C) Cis-but-2-ene has higher dipole moment than trans-but-2-ene. (D) 2-methylbut-2-ene shows two geometrical isomers. (E) Trans-isomer has lower melting point than cis isomer.

Choose the correct answer from the options given below :

Given below are two statements :

Statement I : One mole of propyne reacts with excess of sodium to liberate half a mole of $\mathrm{H_2}$ gas.

Statement II : Four g of propyne reacts with $\mathrm{NaNH_2}$ to liberate $\mathrm{NH_3}$ gas which occupies 224 mL at STP.

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

Given below are two statements :

Statement I : $\mathrm{CH_3-O-CH_2-Cl}$ will undergo $S_N1$ reaction though it is a primary halide.

Statement II : $\mathrm{(CH_3)_3C-CH_2-Cl}$ will not undergo $S_N2$ reaction very easily though it is a primary halide.

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

The products formed in the following reaction sequence are :

[Figure: benzene ring with $\mathrm{NO_2}$ group (nitrobenzene)]

$$\mathrm{C_6H_5NO_2} \xrightarrow{\text{(i) } \mathrm{Br_2}, \text{ AcOH; (ii) Sn, HCl; (iii) } \mathrm{NaNO_2}, \text{ HCl, 273 K; (iv) } \mathrm{C_2H_5OH}} \mathrm{A} + \mathrm{B}$$

Which of the following acids is a vitamin ?

The compounds which give positive Fehling's test are :

(A) [Figure: benzaldehyde, $\mathrm{C_6H_5-CHO}$] (B) [Figure: acetophenone, $\mathrm{C_6H_5-CO-CH_3}$] (C) $\mathrm{HOCH_2-CO-(CHOH)_3-CH_2-OH}$ (D) $\mathrm{CH_3-CHO}$ (E) [Figure: phenylacetaldehyde, $\mathrm{C_6H_5-CH_2-CHO}$]

Choose the correct answer from the options given below :

Some $\mathrm{CO_2}$ gas was kept in a sealed container at a pressure of 1 atm and at 273 K. This entire amount of $\mathrm{CO_2}$ gas was later passed through an aqueous solution of $\mathrm{Ca(OH)_2}$. The excess unreacted $\mathrm{Ca(OH)_2}$ was later neutralized with 0.1 M of 40 mL HCl. If the volume of the sealed container of $\mathrm{CO_2}$ was x, then x is ________ $\mathrm{cm^3}$ (nearest integer).

[Given : The entire amount of $\mathrm{CO_2(g)}$ reacted with exactly half the initial amount of $\mathrm{Ca(OH)_2}$ present in the aqueous solution.]

The number of molecules/ions that show linear geometry among the following is ________.

$\mathrm{SO_2}$, $\mathrm{BeCl_2}$, $\mathrm{CO_2}$, $\mathrm{N_3^-}$, $\mathrm{NO_2}$, $\mathrm{F_2O}$, $\mathrm{XeF_2}$, $\mathrm{NO_2^+}$, $\mathrm{I_3^-}$, $\mathrm{O_3}$

$$\mathrm{A} \rightarrow \mathrm{B}$$

The molecule A changes into its isomeric form B by following a first order kinetics at a temperature of 1000 K. If the energy barrier with respect to reactant energy for such isomeric transformation is $191.48\ \mathrm{kJ\ mol^{-1}}$ and the frequency factor is $10^{20}$, the time required for $50\%$ molecules of A to become B is ________ picoseconds (nearest integer). $[\mathrm{R} = 8.314\ \mathrm{J\ K^{-1}\ mol^{-1}}]$

In Carius method for estimation of halogens, 180 mg of an organic compound produced 143.5 mg of AgCl. The percentage composition of chlorine in the compound is ________ %.

(Given : molar mass in $\mathrm{g\ mol^{-1}}$ of Ag : 108, Cl : 35.5)

Consider the following sequence of reactions :

[Figure: benzene ring with $\mathrm{NO_2}$ group (nitrobenzene)]

$$\mathrm{C_6H_5NO_2} \xrightarrow{\text{(i) Sn + HCl; (ii) } \mathrm{NaNO_2}, \text{ HCl, } 0°\mathrm{C}\text{; (iii) } \mathrm{Cu_2Cl_2}\text{; (iv) Na, Ether}} \mathrm{A}\ (\text{Product})$$

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

Let $A=\{1,2,3,\ldots,10\}$ and $B=\left\{\frac{m}{n}: m, n \in A, m<n \text{ and } \gcd(m, n)=1\right\}$. Then $n(B)$ is equal to :

The number of non-empty equivalence relations on the set $\{1,2,3\}$ is :

The product of all solutions of the equation $e^{5(\log_e x)^2+3}=x^8, x>0$, is :

Let $z_1, z_2$ and $z_3$ be three complex numbers on the circle $|z|=1$ with $\arg(z_1)=\frac{-\pi}{4}$, $\arg(z_2)=0$ and $\arg(z_3)=\frac{\pi}{4}$. If $\left|z_1\bar{z}_2+z_2\bar{z}_3+z_3\bar{z}_1\right|^2=\alpha+\beta\sqrt{2}$, $\alpha, \beta \in \mathbb{Z}$, then the value of $\alpha^2+\beta^2$ is :

If $\sum_{r=1}^{n} T_r=\frac{(2n-1)(2n+1)(2n+3)(2n+5)}{64}$, then $\lim_{n \to \infty} \sum_{r=1}^{n}\left(\frac{1}{T_r}\right)$ is equal to :

Let $a_1, a_2, a_3, \ldots$ be a G.P. of increasing positive terms. If $a_1 a_5=28$ and $a_2+a_4=29$, then $a_6$ is equal to :

From all the English alphabets, five letters are chosen and are arranged in alphabetical order. The total number of ways, in which the middle letter is 'M', is :

A coin is tossed three times. Let X denote the number of times a tail follows a head. If $\mu$ and $\sigma^2$ denote the mean and variance of X, then the value of $64(\mu+\sigma^2)$ is :

Two balls are selected at random one by one without replacement from a bag containing 4 white and 6 black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is $\frac{m}{n}$, where $\gcd(m, n)=1$, then $m+n$ is equal to :

Let the triangle PQR be the image of the triangle with vertices $(1,3)$, $(3,1)$ and $(2,4)$ in the line $x+2y=2$. If the centroid of $\Delta$PQR is the point $(\alpha, \beta)$, then $15(\alpha-\beta)$ is equal to :

A circle C of radius 2 lies in the second quadrant and touches both the coordinate axes. Let r be the radius of a circle that has centre at the point $(2,5)$ and intersects the circle C at exactly two points. If the set of all possible values of r is the interval $(\alpha, \beta)$, then $3\beta-2\alpha$ is equal to :

Let the parabola $y=x^2+px-3$, meet the coordinate axes at the points P, Q and R. If the circle C with centre at $(-1,-1)$ passes through the points P, Q and R, then the area of $\Delta$PQR is :

Let the foci of a hyperbola be $(1,14)$ and $(1,-12)$. If it passes through the point $(1,6)$, then the length of its latus-rectum is :

Using the principal values of the inverse trigonometric functions, the sum of the maximum and the minimum values of $16\left((\sec^{-1}x)^2+(\operatorname{cosec}^{-1}x)^2\right)$ is :

Let $L_1: \frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}$ and $L_2: \frac{x-2}{3}=\frac{y-4}{4}=\frac{z-5}{5}$ be two lines. Then which of the following points lies on the line of the shortest distance between $L_1$ and $L_2$ ?

Let $f: \mathbb{R} \to \mathbb{R}$ be a twice differentiable function such that $f(x+y)=f(x)f(y)$ for all $x, y \in \mathbb{R}$. If $f'(0)=4a$ and $f$ satisfies $f''(x)-3af'(x)-f(x)=0$, $a>0$, then the area of the region $R=\{(x, y) \mid 0 \le y \le f(ax), 0 \le x \le 2\}$ is :

Let $f(x)$ be a real differentiable function such that $f(0)=1$ and $f(x+y)=f(x)f'(y)+f'(x)f(y)$ for all $x, y \in \mathbb{R}$. Then $\sum_{n=1}^{100} \log_e f(n)$ is equal to :

Let for $f(x)=7\tan^8 x+7\tan^6 x-3\tan^4 x-3\tan^2 x$, $I_1=\int_0^{\pi/4} f(x)\,dx$ and $I_2=\int_0^{\pi/4} x f(x)\,dx$. Then $7I_1+12I_2$ is equal to :

The area of the region, inside the circle $(x-2\sqrt{3})^2+y^2=12$ and outside the parabola $y^2=2\sqrt{3}\,x$ is :

Let $x=x(y)$ be the solution of the differential equation $y^2 dx+\left(x-\frac{1}{y}\right)dy=0$. If $x(1)=1$, then $x\left(\frac{1}{2}\right)$ is :

Let A be a square matrix of order 3 such that $\det(A)=-2$ and $\det(3\,\mathrm{adj}(-6\,\mathrm{adj}(3A)))=2^{m+n} \cdot 3^{mn}$, $m>n$. Then $4m+2n$ is equal to ________.

If $\sum_{r=0}^{5} \frac{{}^{11}C_{2r+1}}{2r+2}=\frac{m}{n}$, $\gcd(m, n)=1$, then $m-n$ is equal to ________.

Let $\vec{c}$ be the projection vector of $\vec{b}=\lambda\hat{i}+4\hat{k}$, $\lambda>0$, on the vector $\vec{a}=\hat{i}+2\hat{j}+2\hat{k}$. If $|\vec{a}+\vec{c}|=7$, then the area of the parallelogram formed by the vectors $\vec{b}$ and $\vec{c}$ is ________.

Let $L_1: \frac{x-1}{3}=\frac{y-1}{-1}=\frac{z+1}{0}$ and $L_2: \frac{x-2}{2}=\frac{y}{0}=\frac{z+4}{\alpha}$, $\alpha \in \mathbb{R}$, be two lines, which intersect at the point B. If P is the foot of perpendicular from the point A$(1,1,-1)$ on $L_2$, then the value of $26\,\alpha(PB)^2$ is ________.

Let the function, $f(x)=\begin{cases}-3ax^2-2, & x<1 \\ a^2+bx, & x \geq 1\end{cases}$ be differentiable for all $x \in \mathbb{R}$, where $a>1$, $b \in \mathbb{R}$. If the area of the region enclosed by $y=f(x)$ and the line $y=-20$ is $\alpha+\beta\sqrt{3}$, $\alpha, \beta \in \mathbb{Z}$, then the value of $\alpha+\beta$ is ________.

If B is magnetic field and $\mu_0$ is permeability of free space, then the dimensions of $(B/\mu_0)$ is

Given below are two statements :

Statement I : In a vernier callipers, one vernier scale division is always smaller than one main scale division.

Statement II : The vernier constant is given by one main scale division multiplied by the number of vernier scale divisions.

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

A bob of mass m is suspended at a point O by a light string of length $l$ and left to perform vertical motion (circular) as shown in figure. Initially, by applying horizontal velocity $v_0$ at the point 'A', the string becomes slack when, the bob reaches at the point 'D'. The ratio of the kinetic energy of the bob at the points B and C is ________.

[Figure: vertical circle with centre O; A at the bottom, D at the top; points B (lower right) and C (upper right) on the circle with OB and OC each making 60° with the horizontal through O; horizontal velocity $v_0$ applied at A]

A uniform circular disc of radius 'R' and mass 'M' is rotating about an axis perpendicular to its plane and passing through its centre. A small circular part of radius R/2 is removed from the original disc as shown in the figure. Find the moment of inertia of the remaining part of the original disc about the axis as given above.

[Figure: disc of radius R with a removed circular part of radius R/2 whose diameter spans from the centre of the disc to its rim]

A small point of mass m is placed at a distance 2R from the centre 'O' of a big uniform solid sphere of mass M and radius R. The gravitational force on 'm' due to M is $F_1$. A spherical part of radius R/3 is removed from the big sphere as shown in the figure and the gravitational force on m due to remaining part of M is found to be $F_2$. The value of ratio $F_1 : F_2$ is

[Figure: solid sphere of mass M with centre O; a small spherical cavity carved out between O and the surface along the line OP; point mass m at P with OP = 2R]

Two spherical bodies of the same material having radii $0.2\ \mathrm{m}$ and $0.8\ \mathrm{m}$ are kept in the same atmosphere. The temperature of the smaller body is $800\ \mathrm{K}$ and the temperature of the bigger body is $400\ \mathrm{K}$. If the energy radiated from the smaller body is E, the energy radiated from the bigger body is [neglect the effect of the temperature of the surroundings],

An amount of ice of mass $10^{-3}$ kg and temperature $-10°\mathrm{C}$ is transformed to vapour of temperature $110°\mathrm{C}$ by applying heat. The total amount of work required for this conversion is,

(Take, specific heat of ice $= 2100\ \mathrm{Jkg^{-1}K^{-1}}$, specific heat of water $= 4180\ \mathrm{Jkg^{-1}K^{-1}}$, specific heat of steam $= 1920\ \mathrm{Jkg^{-1}K^{-1}}$, Latent heat of ice $= 3.35 \times 10^5\ \mathrm{Jkg^{-1}}$ and Latent heat of steam $= 2.25 \times 10^6\ \mathrm{Jkg^{-1}}$)

A closed organ and an open organ tube are filled by two different gases having same bulk modulus but different densities $\rho_1$ and $\rho_2$, respectively. The frequency of $9^{th}$ harmonic of closed tube is identical with $4^{th}$ harmonic of open tube. If the length of the closed tube is 10 cm and the density ratio of the gases is $\rho_1 : \rho_2 = 1 : 16$, then the length of the open tube is :

A parallel-plate capacitor of capacitance $40\ \mu\mathrm{F}$ is connected to a 100 V power supply. Now the intermediate space between the plates is filled with a dielectric material of dielectric constant $K = 2$. Due to the introduction of dielectric material, the extra charge and the change in the electrostatic energy in the capacitor, respectively, are

An electron is made to enter symmetrically between two parallel and equally but oppositely charged metal plates, each of 10 cm length. The electron emerges out of the electric field region with a horizontal component of velocity $10^6$ m/s. If the magnitude of the electric field between the plates is 9.1 V/cm, then the vertical component of velocity of electron is

(mass of electron $= 9.1 \times 10^{-31}$ kg and charge of electron $= 1.6 \times 10^{-19}$ C)

Which of the following resistivity ($\rho$) v/s temperature (T) curves is most suitable to be used in wire bound standard resistors ?

A line charge of length $\frac{'a'}{2}$ is kept at the center of an edge BC of a cube ABCDEFGH having edge length 'a' as shown in the figure. If the density of line charge is $\lambda$ C per unit length, then the total electric flux through all the faces of the cube will be ________. (Take, $\epsilon_0$ as the free space permittivity)

[Figure: cube ABCDEFGH of edge a with a short line charge centred on the bottom edge BC]

Given below are two statements :

Statement I : The equivalent emf of two non-ideal batteries connected in parallel is smaller than the emf of either of the two batteries.

Statement II : The equivalent internal resistance of two non-ideal batteries connected in parallel is smaller than the internal resistance of either of the two batteries.

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

Sliding contact of a potentiometer is in the middle of the potentiometer wire having resistance $R_p = 1\,\Omega$ as shown in the figure. An external resistance of $R_e = 2\,\Omega$ is connected via the sliding contact. The electric current in the circuit is :

[Figure: a 0.9 V cell connected across a potentiometer wire of resistance $R_p = 1\,\Omega$; an external resistance $R_e = 2\,\Omega$ is connected between the sliding contact (at the middle of the wire) and one end of the wire]

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

Assertion (A) : If Young's double slit experiment is performed in an optically denser medium than air, then the consecutive fringes come closer.

Reason (R) : The speed of light reduces in an optically denser medium than air while its frequency does not change.

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

Given is a thin convex lens of glass (refractive index $\mu$) and each side having radius of curvature R. One side is polished for complete reflection. At what distance from the lens, an object be placed on the optic axis so that the image gets formed on the object itself ?

In the diagram given below, there are three lenses formed. Considering negligible thickness of each of them as compared to $|R_1|$ and $|R_2|$, i.e., the radii of curvature for upper and lower surfaces of the glass lens, the power of the combination is

[Figure: a container holding three layers — water ($\mu = \frac{4}{3}$) on top, a thin curved glass lens ($\mu = \frac{3}{2}$) in the middle, and water ($\mu = \frac{4}{3}$) below; the upper and lower curved surfaces of the glass lens have radii of curvature $|R_1|$ and $|R_2|$]

The work functions of cesium (Cs) and lithium (Li) metals are 1.9 eV and 2.5 eV, respectively. If we incident a light of wavelength 550 nm on these two metal surfaces, then photo-electric effect is possible for the case of

In the lowest state of a hydrogen atom, the orbital radius of an electron is $5.3 \times 10^{-11}$ m, while in the third excited state the orbital radius of the same electron is $8.48 \times 10^{-10}$ m. The ratio of the de-Broglie wavelengths of the electron in the excited state with that in the lowest state is :

Which of the following circuits represents a forward biased diode ?

[Figure: five circuits — (A) diode with p-side terminal at −10 V and n-side terminal at 0 V; (B) diode with p-side at −10 V and n-side at −15 V; (C) diode with p-side at 4 V and n-side at 2 V; (D) diode with n-side terminal at −5 V and p-side terminal at −10 V; (E) diode with n-side connected to ground and p-side terminal at 2 V]

Choose the correct answer from the options given below :

The position vectors of two 1 kg particles, (A) and (B), are given by $\vec{r}_A = \left(\alpha_1 t^2 \hat{i} + \alpha_2 t \hat{j} + \alpha_3 t \hat{k}\right)$ m and $\vec{r}_B = \left(\beta_1 t \hat{i} + \beta_2 t^2 \hat{j} + \beta_3 t \hat{k}\right)$ m, respectively; $(\alpha_1 = 1\ \mathrm{m/s^2}$, $\alpha_2 = 3n\ \mathrm{m/s}$, $\alpha_3 = 2\ \mathrm{m/s}$, $\beta_1 = 2\ \mathrm{m/s}$, $\beta_2 = -1\ \mathrm{m/s^2}$, $\beta_3 = 4p\ \mathrm{m/s})$, where t is time, n and p are constants. At $t = 1$ s, $\left|\vec{V}_A\right| = \left|\vec{V}_B\right|$ and velocities $\vec{V}_A$ and $\vec{V}_B$ of the particles are orthogonal to each other. At $t = 1$ s, the magnitude of angular momentum of particle (A) with respect to the position of particle (B) is $\sqrt{L}\ \mathrm{kg\,m^2 s^{-1}}$. The value of L is ________.

A particle is projected at an angle of 30° from horizontal at a speed of 60 m/s. The height traversed by the particle in the first second is $h_0$ and height traversed in the last second, before it reaches the maximum height, is $h$. The ratio $h_0/h$ is ________. [Note: the final line of this stem is corrupted in the source PDF after "maximum h"; the closing wording is reconstructed]

Two soap bubbles of radius 2 cm and 4 cm, respectively, are in contact with each other. The radius of curvature of the common surface, in cm, is ________.

Three conductors of same length having thermal conductivity $k_1$, $k_2$ and $k_3$ are connected as shown in figure.

[Figure: conductors 1 ($k_1$) and 2 ($k_2$) side by side in parallel between the 100°C end and a common junction at temperature $\theta$°C; conductor 3 ($k_3$) connects the junction to the 0°C end]

Area of cross sections of $1^{st}$ and $2^{nd}$ conductor are same and for $3^{rd}$ conductor it is double of the $1^{st}$ conductor. The temperatures are given in the figure. In steady state condition, the value of $\theta$ is ________ °C.

(Given : $k_1 = 60\ \mathrm{Js^{-1}m^{-1}K^{-1}}$, $k_2 = 120\ \mathrm{Js^{-1}m^{-1}K^{-1}}$, $k_3 = 135\ \mathrm{Js^{-1}m^{-1}K^{-1}}$)

The driver sitting inside a parked car is watching vehicles approaching from behind with the help of his side view mirror, which is a convex mirror with radius of curvature $R = 2$ m. Another car approaches him from behind with a uniform speed of 90 km/hr. When the car is at a distance of 24 m from him, the magnitude of the acceleration of the image of the car in the side view mirror is 'a'. The value of 100a is ________ $\mathrm{m/s^2}$.