JEE Main 2025 — 24 January, Evening Shift

The equation of the chord, of the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$, whose mid-point is $(3,1)$ is :

The function $f:(-\infty, \infty) \rightarrow(-\infty, 1)$, defined by $f(x)=\frac{2^{x}-2^{-x}}{2^{x}+2^{-x}}$ is :

If $\alpha>\beta>\gamma>0$, then the expression $\cot ^{-1}\left\{\beta+\frac{\left(1+\beta^{2}\right)}{(\alpha-\beta)}\right\}+\cot ^{-1}\left\{\gamma+\frac{\left(1+\gamma^{2}\right)}{(\beta-\gamma)}\right\}+\cot ^{-1}\left\{\alpha+\frac{\left(1+\alpha^{2}\right)}{(\gamma-\alpha)}\right\}$ is equal to:

Let $\mathrm{f}:(0, \infty) \rightarrow \mathbf{R}$ be a function which is differentiable at all points of its domain and satisfies the condition

$x^{2} f^{\prime}(x)=2 x f(x)+3$, with $f(1)=4$. Then $2 f(2)$ is equal to:

Let $A=\left\{x \in(0, \pi)-\left\{\frac{\pi}{2}\right\}: \log _{(2 / \pi)}|\sin x|+\log _{(2 / \pi)}|\cos x|=2\right\}$

and $B=\{x \geq 0: \sqrt{x}(\sqrt{x}-4)-3|\sqrt{x}-2|+6=0\}$. Then $\mathrm{n}(\mathrm{A} \cup \mathrm{B})$ is equal to:

Let the position vectors of three vertices of a triangle be $4 \overrightarrow{\mathrm{p}}+\overrightarrow{\mathrm{q}}-3 \overrightarrow{\mathrm{r}},-5 \overrightarrow{\mathrm{p}}+\overrightarrow{\mathrm{q}}+2 \overrightarrow{\mathrm{r}}$ and $2 \overrightarrow{\mathrm{p}}-\overrightarrow{\mathrm{q}}+2 \overrightarrow{\mathrm{r}}$.

If the position vectors of the orthocenter and the circumcenter of the triangle are $\frac{\overrightarrow{\mathrm{p}}+\overrightarrow{\mathrm{q}}+\overrightarrow{\mathrm{r}}}{4}$ and $\alpha \overrightarrow{\mathrm{p}}+\beta \overrightarrow{\mathrm{q}}+\gamma \overrightarrow{\mathrm{r}}$

respectively, then $\alpha+2 \beta+5 \gamma$ is equal to:

In an arithmetic progression, if $S_{40}=1030$ and $S_{12}=57$, then $S_{30}-S_{10}$ is equal to:

If $7=5+\frac{1}{7}(5+\alpha)+\frac{1}{7^{2}}(5+2 \alpha)+\frac{1}{7^{3}}(5+3 \alpha)+$ $\_\_\_\_\_$ $\infty$, then the value of $\alpha$ is :

If the system of equations

$x+2 y-3 z=2$

$2 x+\lambda y+5 z=5$

$14 x+3 y+\mu z=33$

has infinitely many solutions, then $\lambda+\mu$ is equal to:

Let $(2,3)$ be the largest open interval in which the function $f(x)=2 \log _{e}(x-2)-x^{2}+a x+1$

is strictly increasing and (b, c) be the largest open interval, in which the function

$g(x)=(x-1)^{3}(x+2-a)^{2}$ is strictly decreasing. Then $100(a+b-c)$ is equal to:

Suppose $A$ and $B$ are the coefficients of $30^{\text {th }}$ and $12^{\text {th }}$ terms respectively in the binomial expansion of $(1+x)^{2 n-1}$. If $2 A=5 B$, then $n$ is equal to:

Let $\overrightarrow{\mathrm{a}}=3 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=\overrightarrow{\mathrm{a}} \times(\hat{\mathrm{i}}-2 \hat{\mathrm{k}})$ and $\overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{b}} \times \hat{\mathrm{k}}$. Then the projection of $\vec{c}-2 \hat{j}$ on $\vec{a}$ is:

For some $a, b$, let

$\lim _{x \rightarrow 0} f(x)=\lambda+\mu a+v b$. Then $(\lambda+\mu+v)^{2}$ is equal to:

Group $A$ consists of 7 boys and 3 girls, while group $B$ consists of 6 boys and 5 girls. The number of ways, 4 boys and 4 girls can be invited for a picnic if 5 of them must be from group A and the remaining 3 from group $B$, is equal to:

The area of the region enclosed by the curves $y=e^{x}, y=\left|e^{x}-1\right|$ and $y$-axis is :

The number of real solution(s) of the equation $x^{2}+3 x+2=\min \{|x-3|,|x+2|\}$ is :

Let $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]$ be a square matrix of order 2 with entries either 0 or 1 . Let E be the event that A is an invertible matrix. Then the probability $\mathrm{P}(\mathrm{E})$ is :

If the equation of the parabola with vertex $V\left(\frac{3}{2}, 3\right)$ and the directrix $x+2 y=0$ is

$\alpha x^{2}+\beta y^{2}-\gamma x y-30 x-60 y+225=0$, then $\alpha+\beta+\gamma$ is equal to:

Number of functions $f : \{1, 2, ..., 100\} \rightarrow \{0, 1\},$ that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to $\_\_\_\_\_$.

Let P be the image of the point Q(7, -2, 5) in the line $L: \frac{x-1}{2} = \frac{y+1}{3} = \frac{z}{4}$ and R(5, p, q) be a point on L. Then the square of the area of $\triangle$ PQR is$\_\_\_\_\_$.

Let $y=y(x)$ be the solution of the differential equation $2 \cos x \frac{d y}{d x}=\sin 2 x-4 y \sin x, x \in\left(0, \frac{\pi}{2}\right)$.

If $y\left(\frac{\pi}{3}\right)=0$, then $y^{\prime}\left(\frac{\pi}{4}\right)+y\left(\frac{\pi}{4}\right)$ is equal to $\_\_\_\_\_$ .

Let $H_{1}: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ and $H_{2}:-\frac{x^{2}}{A^{2}}+\frac{y^{2}}{B^{2}}=1$ be two hyperbolas having length of latus rectums $15 \sqrt{2}$ and $12 \sqrt{5}$ respectively. Let their eccentricities be $e_{1}=\sqrt{\frac{5}{2}}$ and $e_{2}$ respectively. If the product of the lengths of their transverse axes is $100 \sqrt{10}$, then $25 \mathrm{e}_{2}^{2}$ is equal to $\_\_\_\_\_$

If $\int \frac{2 x^{2}+5 x+9}{\sqrt{x^{2}+x+1}} d x=x \sqrt{x^{2}+x+1}+\alpha \sqrt{x^{2}+x+1}+$ $\beta \log _{e}\left|x+\frac{1}{2}+\sqrt{x^{2}+x+1}\right|+C$,

where $C$ is the constant of integration, then $\alpha+2 \beta$ is equal to $\_\_\_\_\_$

Based on the following data:

the strongest reducing agent among the following is:

Given below are two statements:

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

For reaction

The correct order of set of reagents for the above conversion is:

For hydrogen atom, the orbital/s with lowest energy is/are :

(A) 4 s

(B) $3 p_{x}$

(C) $3 d_{x^{2}-y^{2}}$

(D) $3 \mathrm{~d}_{\mathrm{z}^{2}}$

(E) $4 p_{z}$

Choose the correct answer from the options given below :

In the given structure, number of sp and $\mathrm{sp}^{2}$ hybridized carbon atoms present respectively are :

Given below are two statements :

Statement(I) : Experimentally determined oxygen-oxygen bond lengths in the $\mathrm{O}_{3}$ are found to be same and the bond length is greater than that of a $\mathrm{O}=\mathrm{O}$ (double bond) but less than that of a single ( $\mathrm{O}-\mathrm{O}$ ) bond.

Statement (II) : The strong lone pair-lone pair repulsion between oxygen atoms is solely responsible for the fact that the bond length in ozone is smaller than that of a double bond $(\mathrm{O}=\mathrm{O})$ but more than that of a single bond $(\mathrm{O}-\mathrm{O})$.

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

Find the compound 'A' from the following reaction sequences.

$\mathrm{A} \xrightarrow{\text { aqua-regia }} \mathrm{B} \xrightarrow[(2) \mathrm{AcOH}]{(1) \mathrm{KNO}_{2} / \mathrm{NH}_{4} \mathrm{OH}}$ yellow ppt

Match List-I with List-II.

Choose the correct answer from the options given below :

The elemental composition of a compound is $54.2\%$ C, $9.2\%$ H and $36.6\%$ O. If the molar mass of the compound is $132\,\text{g mol}^{-1}$, the molecular formula of the compound is $\_\_\_\_\_$.

(Given: Relative atomic masses of C : H : O = 12 : 1 : 16)

When Ethane-1,2-diamine is added progressively to an aqueous solution of Nickel (II) chloride, the sequence of colour change observed will be :

The conditions and consequence that favours the $\mathrm{t}_{2 \mathrm{~g}}{ }^{3}, \mathrm{e}_{\mathrm{g}}{ }^{1}$ configuration in a metal complex are :

Identify correct statement/s :

(A) $-\mathrm{OCH}_{3}$ and $-\mathrm{NHCOCH}_{3}$ are activating group

(B) -CN and -OH are meta directing group

(C) -CN and $-\mathrm{SO}_{3} \mathrm{H}$ are meta directing group

(D) Activating groups act as ortho - and para directing groups

(E) Halides are activating groups

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 ionization energy of Pb is greater than that of Sn.

Statement II : The first ionization energy of Ge is greater than that of Si.

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

$\mathrm{S}(\mathrm{g})+\frac{3}{2} \mathrm{O}_{2}(\mathrm{~g}) \rightarrow \mathrm{SO}_{3}(\mathrm{~g})+2 \mathrm{xkcal}$

$\mathrm{SO}_{2}(\mathrm{~g})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{~g}) \rightarrow \mathrm{SO}_{3}(\mathrm{~g})+\mathrm{y} \mathrm{kcal}$

The heat of formation of $\mathrm{SO}_{2}(\mathrm{~g})$ is given by :

Match List-I with List-II

Choose the correct answer from the options given below :

The structure of the major product formed in the following reaction is :

Match List-I with List-II.

Choose the correct answer from the options given below :

The successive 5 ionisation energies of an element are $800,2427,3658,25024$ and $32824\; \mathrm{kJ\,mol^{-1}}$, respectively. By using the above values predict the group in which the above element is present :

The observed molar mass and normal molar mass of the compound $\mathrm{MX_2}$ are $65.6$ and $164$ respectively.

Find the percent degree of ionisation of $\mathrm{MX_2}$. $\quad (\text{Nearest \;integer})$

The possible number of stereoisomers for 5 -phenylpent-4-en-2-ol is___.

Consider a complex reaction taking place in three steps with rate constants $\mathrm{k}_{1}, \mathrm{k}_{2}$ and $\mathrm{k}_{3}$ respectively. The overall rate constant $k$ is given by the expression $k=\sqrt{\frac{k_{1} k_{3}}{k_{2}}}$. If the activation energies of the three steps are 60,30 and $10 \mathrm{~kJ} \mathrm{~mol}^{-1}$ respectively, then the overall energy of activation in $\mathrm{kJ} \mathrm{mol}^{-1}$ is___ . (Nearest integer)

The hydrocarbon (X) with molar mass $80 \mathrm{~g} \mathrm{~mol}^{-1}$ and $90 \%$ carbon has ___degree of unsaturation.

In Carius method of estimation of halogen, 0.25 g of an organic compound gave 0.15 g of silver bromide $(\mathrm{AgBr})$. The percentage of Bromine in the organic compound is____ $\times 10^{-1} \%$ (Nearest integer).

(Given : Molar mass of Ag is 108 and Br is $80 \mathrm{~g} \mathrm{~mol}^{-1}$ )

Young's double slit interference apparatus is immersed in a liquid of refractive index 1.44. It has slit separation of 1.5 mm . The slits are illuminated by a parallel beam of light whose wavelength in air is 690 nm . The fringe-width on a screen placed behind the plane of slits at a distance of 0.72 m , will be :

Arrange the following in the ascending order of wavelength ( $\lambda$ ) : (A) Microwaves $\left(\lambda_{1}\right)$ (B) Ultraviolet rays ( $\lambda_{2}$ ) (C) Infrared rays $\left(\lambda_{3}\right)$ (D) X-rays $\left(\lambda_{4}\right)$ Choose the most appropriate answer from the options

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason(R). Assertion (A) : A electron in a certain region of uniform magnetic field is moving with constant velocity in a straight line path.

Reason (A) : The magnetic field in that region is along the direction of velocity of the electron.

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

A solid sphere is rolling without slipping on a horizontal plane. The ratio of the linear kinetic energy of the centre of mass of the sphere and rotational kinetic energy is :

. A long straight wire of a circular cross-section with radius 'a' carries a steady current I. The current I is a uniformly distributed across this cross-section. The plot of magnitude of magnetic field B with distance r from the centre of the wire is given by (1)

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason(R). Assertion (A) : In an insulated container, a gas is adiabatically shrunk to half of its initial volume. The temperature of the gas decreases.

Reason (R): Free expansion of an ideal gas is an irreversible and an adiabatic process.

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

In the first configuration (1) as shown in the figure, four identical charges $\left(q_{0}\right)$ are kept at the corners $A, B, C$ and $D$ of square of side length ' $a$ '. In the second configuration (2), the same charges are shifted to mid points $\mathrm{G}, \mathrm{E}, \mathrm{H}$ and F , of the square, If $\mathrm{K}=\frac{1}{4 \pi \varepsilon_{0}}$, the difference between the potential energies of configuration (2) and (1) is given by :

The position vector of a moving body at any instant of time is given as $\overrightarrow{\mathrm{r}}=\left(5 \mathrm{t}^{2} \hat{\mathrm{i}}-5 \mathrm{t} \hat{\mathrm{j}}\right) \mathrm{m}$. The magnitude and direction of velocity at $\mathrm{t}=2 \mathrm{~s}$ is,

A solid sphere and a hollow sphere of the same mass and of same radius are rolled on an inclined plane. Let the time taken to reach the bottom by the solid sphere and the hollow sphere be $t_1$ and $t_2$, respectively, then

N equally spaced charges each of value q , are placed on a circle of radius R. The circle rotates about its axis with an angular velocity $\omega$ as shown in the figure. A bigger Amperian loop B encloses the whole circle where as a smaller Amperian loop A encloses a small segment. The difference between enclosed currents, $I_{A}-I_{B}$, for the given Amperian loops is images

In photoelectric effect, the stopping potential $\left(\mathrm{V}_{0}\right)$ $\mathrm{v} / \mathrm{s}$ frequency ( v ) curve is plotted.

(h is the Planck's constant and $\phi_{0}$ is work function of metal)

(A) $V_{0} v / s v$ is linear

(B) The slope of $\mathrm{V}_{0} \mathrm{v} / \mathrm{s} v$ curve $=\frac{\phi_{0}}{\mathrm{~h}}$

(C) h constant is related to the slope of $\mathrm{V}_{0} \mathrm{v} / \mathrm{s} \mathrm{v}$ line

(D) The value of electric charge of electron is not required to determine $h$ using the $V_{0} v / s v$ curve.

(E) The work function can be estimated without knowing the value of $h$.

Choose the correct answer from the options given below :

A photograph of a landscape is captured by a drone camera at a height of 18 km . The size of the camera film is $2 \mathrm{~cm} \times 2 \mathrm{~cm}$ and the area of the landscape photographed is $400 \mathrm{~km}^{2}$. The focal length of the lens in the drone camera is :

The output of the circuit is low (zero) for :

(A) $\mathrm{X}=0, \mathrm{Y}=0$

(B) $\mathrm{X}=0, \mathrm{Y}=1$

(C) $\mathrm{X}=1, \mathrm{Y}=0$

(D) $\mathrm{X}=1, \mathrm{Y}=1$

Choose the correct answer from the options given below :

The temperature of a body in air falls from $40^{\circ} \mathrm{C}$ to $24^{\circ} \mathrm{C}$ in 4 minutes. The temperature of the air is $16^{\circ} \mathrm{C}$. The temperature of the body in the next 4 minutes will be :

The energy E and momentum p of a moving body of mass $m$ are related by some equation. Given that c represents the speed of light, identify the correct equation.

A small uncharged conducting sphere is placed in contact with an identical sphere but having $4 \times 10^{-8}$ C charge and then removed to a distance such that the force of repulsion between them is $9 \times 10^{-3} \mathrm{~N}$. The distance between them is (Take $\frac{1}{4 \pi \varepsilon_{0}}$ as $9 \times 10^{9}$ in SI units)

In a Young's double slit experiment, three polarizers are kept as shown in the figure. The transmission axes of $P_{1}$ and $P_{2}$ are orthogonal to each other. The polarizer $P_{3}$ covers both the slits with its transmission axis at $45^{\circ}$ to those of $\mathrm{P}_{1}$ and $P_{2}$. An unpolarized light of wavelength $\lambda$ and intensity $\mathrm{I}_{0}$ is incident on $\mathrm{P}_{1}$ and $\mathrm{P}_{2}$. The intensity at a point after $P_{3}$ where the path difference between the light waves from $\mathrm{s}_{1}$ and $\mathrm{s}_{2}$ is $\frac{\lambda}{3}$, is

A tightly wound long solenoid carries a current of 1.5 A. An electron is executing

uniform circular motion inside the solenoid with a time period of 75 ns . The number

of turns per metre in the solenoid is $\qquad$ .[0pt]

[Take mass of electron $m_{e}=9 \times 10^{-31} \mathrm{~kg}$, charge of electron $\left|\mathrm{q}_{\mathrm{e}}\right|=1.6 \times 10^{-19} \mathrm{C}$,

$\left.\mu_{0}=4 \pi \times 10^{-7} \frac{\mathrm{~N}}{\mathrm{~A}^{2}}, 1 \mathrm{~ns}=10^{-9} \mathrm{~s}\right]$

A string of length $L$ is fixed at one end and carries a mass of M at the other end. The mass makes $\left(\frac{3}{\pi}\right)$ rotations per second about the vertical axis passing through end of the string as shown. The tension in the string is $\qquad$ ML.A string of length $L$ is fixed at one end and carries a mass of M at the other end. The mass makes $\left(\frac{3}{\pi}\right)$ rotations per second about the vertical axis passing through end of the string as shown. The tension in the string is $\qquad$ ML.. images

The ratio of the power of a light source $S_{1}$ to that the light source $S_{2}$ is $2 . S_{1}$ is emitting $2 \times 10^{15}$ photons per second at 600 nm . If the wavelength of the source $S_{2}$ is 300 nm , then the number of photons per second emitted by $S_{2}$ is $\qquad$ $\times 10^{14}$.

The increase in pressure required to decrease the volume of a water sample by $0.2 \%$ is $\mathrm{P} \times 10^{5} \mathrm{Nm}^{-2}$. Bulk modulus of water is $2.15 \times 10^{9} \mathrm{Nm}^{-2}$. The value of P is $\qquad$ —.

Acceleration due to gravity on the surface of earth is ' g '. If the diameter of earth is reduced to one third of its original value and mass remains unchanged, then the acceleration due to gravity on the surface of the earth is $\qquad$ g.