JEE Main 2025 — 24 January, Morning Shift

Let $\overrightarrow{\mathrm{a}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=3 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{c}}$ be three vectors such that $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b}$. If the vector

$\overrightarrow{\mathrm{c}}$ is perpendicular to $\vec{b}$ and $\overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}=5$, then $|\overrightarrow{\mathbf{c}}|$ is equal to

In $I(m, n)=\int_{0}^{1} x^{m-1}(1-x)^{n-1} d x, m, n>0$, then $\mathrm{I}(9,14)+\mathrm{I}(10,13)$ is

Let $\mathrm{f}: \mathbb{R}-\{0\} \rightarrow \mathbb{R}$ be a function such that $f(x)-6 f\left(\frac{1}{x}\right)=\frac{35}{3 x}-\frac{5}{2}$. If the

$\lim _{x \rightarrow 0}\left(\frac{1}{\alpha x}+f(x)\right)=\beta ;$ $\alpha, \beta \in \mathbb{R}$, then $\alpha+2 \beta$ is equal to

Let $\mathrm{S}_{\mathrm{n}}=\frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\ldots$ upto n terms. If the sum of the first six terms of an A.P. with first

term -p and common difference p is $\sqrt{2026 \mathrm{~S}_{2025}}$, then the absolute difference between $20^{\text {th }}$ and $15^{\text {th }}$

terms of the A.P. is

Let $f(x)=\frac{2^{x+2}+16}{2^{2 x+1}+2^{x+4}+32}$. Then the value of $8\left(\mathrm{f}\left(\frac{1}{15}\right)+\mathrm{f}\left(\frac{2}{15}\right)+\ldots+\mathrm{f}\left(\frac{59}{15}\right)\right)$ is equal to

If $\alpha$ and $\beta$ are the roots of the equation $2 z^{2}-3 z-2 i=0$, where $i=\sqrt{-1}$, then 16.$\operatorname{Re}\left(\frac{\alpha^{19}+\beta^{19}+\alpha^{11}+\beta^{11}}{\alpha^{15}+\beta^{15}}\right) \cdot \operatorname{Im}\left(\frac{\alpha^{19}+\beta^{19}+\alpha^{11}+\beta^{11}}{\alpha^{15}+\beta^{15}}\right)$ is equal to

$\lim _{x \rightarrow 0} \operatorname{cosec} x\left(\sqrt{2 \cos ^{2} x+3 \cos x}-\sqrt{\cos ^{2} x+\sin x+4}\right)$ is

Let in a $\triangle A B C$, the length of the side $A C$ be 6 , the vertex $B$ be $(1,2,3)$ and the vertices $A, C$ lie on the line $\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}$. Then the area (in sq. units) of $\triangle A B C$ is

Let $\mathrm{y}=\mathrm{y}(\mathrm{x})$ be the solution of the differential equation $\left(x y-5 x^{2} \sqrt{1+x^{2}}\right) d x+\left(1+x^{2}\right) d y=0$, $y(0)=0$.

Then $y(\sqrt{3})$ is equal to

Let the product of the focal distances of the point $\left(\sqrt{3}, \frac{1}{2}\right)$ on the ellipse $\frac{\mathrm{x}^{2}}{\mathrm{a}^{2}}+\frac{\mathrm{y}^{2}}{\mathrm{~b}^{2}}=1,(\mathrm{a}>\mathrm{b})$, be $\frac{7}{4}$. Then the absolute difference of the eccentricities of two such ellipses is

A and B alternately throw a pair of dice. A wins if he throws a sum of 5 before B throws a sum of 8, and B wins if he throws a sum of 8 before A throws a sum of 5. The probability, that A wins if A makes the first throw, is

Consider the region $R=\left\{(x, y): x \leq y \leq 9-\frac{11}{3} x^{2}, x \geq 0\right\}$. The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in $R$, is :

The area of the region $\left\{(\mathrm{x}, \mathrm{y}): \mathrm{x}^{2}+4 \mathrm{x}+2 \leq \mathrm{y} \leq|\mathrm{x}+2|\right\}$ is equal to

For a statistical data $x_{1}, x_{2}, \ldots, x_{10}$ of 10 values, a student obtained the mean as 5.5 and $\sum_{i=1}^{10} \mathrm{x}_{\mathrm{i}}^{2}=371$. He later found that he had noted two values in the data incorrectly as 4 and 5 , instead of the correct values 6 and 8 , respectively. The variance of the corrected data is

Let circle $C$ be the image of $x^{2}+y^{2}-2 x+4 y-4=0$ in the line $2 x-3 y+5=0$ and $A$ be the point on $C$ such that OA is parallel to $x$-axis and $A$ lies on the right hand side of the centre $O$ of $C$. If $B(\alpha, \beta)$, with $\beta<4$, lies on $C$ such that the length of the arc AB is $(1 / 6)^{\text {th }}$ of the perimeter of C , then $\beta-\sqrt{3} \alpha$ is

equal to

For some $n \neq 10,$ let the coefficients of the $5^{th}, 6^{th}$ and $7^{th}$ terms in the binomial expansion of $(1+x)^{n+4}$ be in A.P. Then the largest coefficient in the expansion of $(1+x)^{n+4}$ is:

The product of all the rational roots of the equation $(x^2 - 9x + 11)^2 - (x-4)(x-5) = 3,$ is equal to:

Let the line passing through the points $(-1,2,1)$ and parallel to the line $\frac{\mathrm{x}-1}{2}=\frac{\mathrm{y}+1}{3}=\frac{\mathrm{z}}{4}$

intersect the line $\frac{x+2}{3}=\frac{y-3}{2}=\frac{z-4}{1}$ at the point $P$. Then the distance of P from the point $\mathrm{Q}(4,-5,1)$ is :

Let the lines $3 x-4 y-\alpha=0,8 x-11 y-33=0$, and $2 x-3 y+\lambda=0$ be concurrent. If the image of the point $(1,2)$ in the line $2 x-3 y+\lambda=0$ is $\left(\frac{57}{13}, \frac{-40}{13}\right)$, then $|\alpha \lambda|$ is equal to :

If the system of equations

$2 \mathrm{x}-\mathrm{y}+\mathrm{z}=4$

$5 x+\lambda y+3 z=12$

$100 x-47 y+\mu z=212$,

has infinitely many solutions, then $\mu-2 \lambda$ is equal to

Let f be a differentiable function such that $2(x+2)^{2} f(x)-3(x+2)^{2}=10 \int_{0}^{x}(t+2) f(t) d t$,

$x \geq 0$. Then $f(2)$ is equal to $\_\_\_\_\_$ .

If for some $\alpha, \beta ; \alpha \leq \beta, \alpha+\beta=8$ and $\sec ^{2}\left(\tan ^{-1} \alpha\right)+\operatorname{cosec}^{2}\left(\cot ^{-1} \beta\right)=36$, then $\alpha^{2}+\beta$

is $\_\_\_\_\_$ .

The number of 3-digit numbers, that are divisible by 2 and 3 , but not divisible by 4 and 9 , is

Let be a $3 \times 3$ matrix such that $X^T A X = O$ for all nonzero $3 \times 1$ matrices $X = \begin{bmatrix} x \\ y \\ z \end{bmatrix} .$

Let $S=\left\{p_{1}, p_{2} \ldots \ldots, p_{10}\right\}$ be the set of first ten prime numbers. Let $A=S \cup P$, where $P$ is the set

of all possible products of distinct element of $S$. Then the number of all ordered pairs ( $x, y$ ), $x \in S$, $y \in A$,

such that $x$ divides $y$, is $\_\_\_\_\_$ .

For the given cell

$\mathrm{Fe}^{2+}(\mathrm{eq})+\mathrm{Ag}^{+}(\mathrm{aq}) \rightarrow \mathrm{Fe}^{3+}(\mathrm{aq})+\mathrm{Ag}(\mathrm{s})$

The standard cell potential of the above reaction is Given :

$\mathrm{Ag}^{+}+\mathrm{e}^{-} \rightarrow \mathrm{Ag}$

$\mathrm{E}^{0}=\mathrm{xV}$

$\mathrm{Fe}^{2+}+2 \mathrm{e}^{-} \rightarrow \mathrm{Fe}$

$E^{0}=y V$

$\mathrm{Fe}^{3+}+3 \mathrm{e}^{-} \rightarrow \mathrm{Fe}$

$\mathrm{E}^{0}=\mathrm{zV}$

Following are the four molecules "P", "Q", "R" and "S".

Which one among the four molecules will react with $\mathrm{H}-\mathrm{Br}(\mathrm{aq})$ at the fastest rate ?

One mole of the octahedral complex compound $\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{5} \mathrm{Cl}_{3}$ gives 3 moles of ions on dissolution in water. One mole of the same complex reacts with excess of $\mathrm{AgNO}_{3}$ solution to yield two moles of $\mathrm{AgCl}_{(\mathrm{s})}$. The structure of the complex is :

Which of the following linear combination of atomic orbitals will lead to formation of molecular orbitals in homonuclear diatomic molecules [internuclear axis in z-direction] ?

A. $2 p_{z}$ and $2 p_{x}$

B. 2 s and $2 \mathrm{p}_{\mathrm{x}}$

C. $3 d_{x y}$ and $3 d_{x^{2}-y^{2}}$

D. 2 s and $2 \mathrm{p}_{\mathrm{z}}$

E. $2 p_{z}$ and $3 d_{x^{2}-y^{2}}$

Which of the following ions is the strongest oxidizing agent?

(Atomic Number of $\mathrm{Ce}=58, \mathrm{Eu}=63, \mathrm{~Tb}=65, \mathrm{Lu}=71$ ]

Ksp for $\mathrm{Cr}(\mathrm{OH})_{3}$ is $1.6 \times 10^{-30}$. What is the molar solubility of this salt in water?

Let us consider an endothermic reaction which is non-spontaneous at the freezing point of water. However, the reaction is spontaneous at boiling point of water. Choose the correct option.

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

Statement I : Dumas method is used for estimation of "Nitrogen" in an organic compound.

Statement II : Dumas method involves the formation of ammonium sulphate by heating the organic compound with conc $\mathrm{H}_{2} \mathrm{SO}_{4}$.

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

Which of the following statements are NOT true about the periodic table?

A. The properties of elements are function of atomic weights.

B. The properties of elements are function of atomic numbers.

C. Elements having similar outer electronic configuration are arranged in same period.

D. An element's location reflects the quantum numbers of the last filled orbital.

E. The number of elements in a period is same as the number of atomic orbitals available in energy level that is being filled.

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

The carbohydrates "Ribose" present in DNA, is A.

A pentose sugar

B. present in pyranose from

C. in "D" configuration

D. a reducing sugar, when free

E. in $\alpha$-anomeric form

Choose the correct answer from the options given below:

Preparation of potassium permanganate from $\mathrm{MnO}_{2}$ involves two step process in which the $1^{\text {st }}$ step is a reaction with KOH and $\mathrm{KNO}_{3}$ to produce

The large difference between the melting and boiling points of oxygen and sulphur may be explained on the basis of:

For a reaction, $\mathrm{N}_{2} \mathrm{O}_{5(\mathrm{~g})} \rightarrow 2 \mathrm{NO}_{2(\mathrm{~g})}+\frac{1}{2} \mathrm{O}_{2(\mathrm{~g})}$ in a constant volume container, no

products were present initially. The final pressure of the system when $50 \%$ of reaction gets

completed is

Which of the following arrangements with respect to their reactivity in nucleophilic addition reaction is correct?

Aman has been asked to synthesise the molecule

He thought of preparing the molecule using an aldol condensation reaction. He found a few cyclic alkenes in his laboratory. He thought of performing ozonolysis reaction on alkene to produce a dicarbonyl compound followed by aldol reaction to prepare "x". Predict the suitable alkene that can lead to the formation of "x".

Which of the following statement is true with respect to $\mathrm{H}_{2} \mathrm{O}, \mathrm{NH}_{3}$ and $\mathrm{CH}_{4}$ ?

A. The central atoms of all the molecules are $\mathrm{sp}^{3}$ hybridized.

B. The $\mathrm{H}-\mathrm{O}-\mathrm{H}, \mathrm{H}-\mathrm{N}-\mathrm{H}$ and $\mathrm{H}-\mathrm{C}-\mathrm{H}$ angles in the above molecules are $104.5^{\circ}, 107.5^{\circ}$ and $109.5^{\circ}$ respectively.

C. The increasing order of dipole moment is $\mathrm{CH}_{4}<\mathrm{NH}_{3}<\mathrm{H}_{2} \mathrm{O}$.

D. Both $\mathrm{H}_{2} \mathrm{O}$ and $\mathrm{NH}_{3}$ are Lewis acids and $\mathrm{CH}_{4}$ is a Lewis base

E. A solution of $\mathrm{NH}_{3}$ in $\mathrm{H}_{2} \mathrm{O}$ is basic. In this solution $\mathrm{NH}_{3}$ and $\mathrm{H}_{2} \mathrm{O}$ act as Lowry-Bronsted acid and base respectively.

Choose the correct answer from the options given below :

$\quad 37.8 \mathrm{~g} \mathrm{~N}_{2} \mathrm{O}_{5}$ was taken in a 1 L reaction vessel and allowed to undergo the following reaction

at 500 K $2 \mathrm{~N}_{2} \mathrm{O}_{5(\mathrm{~g})} \rightarrow 2 \mathrm{~N}_{2} \mathrm{O}_{4(\mathrm{~g})}+\mathrm{O}_{2(\mathrm{~g})}$

The total pressure at equilibrium was found to be 18.65 bar.

Then, $\mathrm{Kp}=$___$\times 10^{-2}$ [nearest integer] Assume $\mathrm{N}_{2} \mathrm{O}_{5}$ to behave ideally under these

conditions

Given : $\mathrm{R}=0.082$ bar $\mathrm{L} \mathrm{mol}^{-1} \mathrm{~K}^{-1}$

Standard entropies of $X_{2}, Y_{2}$ and $X Y_{5}$ are 70, 50 and $110 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}$ respectively. The

temperature in Kelvin at which the reaction

$\frac{1}{2} \mathrm{X}_{2}+\frac{5}{2} \mathrm{Y}_{2} \rightarrow \mathrm{XY}_{5} \Delta \mathrm{H}^{-}=-35 \mathrm{~kJ} \mathrm{~mol}^{-1}$

Will be at equilibrium is____(Nearest integer)

Xg of benzoic acid on reaction with aq. $\mathrm{NaHCO}_{3}$ release $\mathrm{CO}_{2}$ that occupied 11.2 L volume

at STP. X is $\qquad$ g.

Among the following cations, the number of cations which will give characteristic precipitate in

their identification tests with $\mathrm{K}_{4}\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]$ is :

$\mathrm{Cu}^{2+}, \mathrm{Fe}^{3+}, \mathrm{Ba}^{2+}, \mathrm{Ca}^{2+}, \mathrm{NH}_{4}^{+}, \mathrm{Mg}^{2+}, \mathrm{Zn}^{2+}$

Consider the following reaction occurring in the blast furnace.

$\mathrm{Fe_3O_4(s) + 4CO(g) \xrightarrow{} 3Fe(l) + 4CO_2(g)}$

$x$ kg of iron is produced when $2.32\times10^3\,\mathrm{kg}$ $\mathrm{Fe_3O_4}$ and $2.8\times10^2\,\mathrm{kg}$ $\mathrm{CO}$ are brought together in the furnace.

The value of $x$ is $\_\_\_\_\_\_$ (nearest integer).

Given: Molar mass of $\mathrm{Fe_3O_4} = 232\,\mathrm{g\,mol^{-1}}$ Molar mass of $\mathrm{CO} = 28\,\mathrm{g\,mol^{-1}}$ Molar mass of $\mathrm{Fe} = 56\,\mathrm{g\,mol^{-1}}$

Consider a parallel plate capacitor of area A (of each plate) and separation 'd' between the plates. If E is the electric field and $\varepsilon_{0}$ is the permittivity of free space between the plates, then potential energy stored in the capacitor is :-

What is the relative decrease in focal length of a lens for an increase in optical power by 0.1 D from 2.5 D? ['D' stands for dioptre]

An air bubble of radius 0.1 cm lies at a depth of 20 cm below the free surface of a liquid of density $1000 \mathrm{~kg} / \mathrm{m}^{3}$. If the pressure inside the bubble is $2100 \mathrm{~N} / \mathrm{m}^{2}$ greater than the atmospheric pressure, then the surface tension of the liquid in SI unit is (use $g=10 \mathrm{~m} / \mathrm{s}^{2}$ )

For an experimental expression $\mathrm{y}=\frac{32.3 \times 1125}{27.4}$, where all the digits are significant. Then to report the value of $y$ we should write :-

During the transition of electron from state A to state C of a Bohr atom, the wavelength of emitted radiation is $2000 A$ and it becomes $6000 A$ when the electron jumps from state B to state C. Then the wavelength of the radiation emitted during the transition of electrons from state A to state B is :-

Consider the following statements : A. The junction area of solar cell is made very narrow compared to a photo diode. B. Solar cells are not connected with any external bias. C. LED is made of lightly doped p-n junction. D. Increase of forward current results in continuous increase of LED light intensity. E. LEDs have to be connected in forward bias for emission of light.

The amount of work done to break a big water drop of radius ' R ' into 27 small drops of equal

radius is 10 J . The work done required to break the same big drop into 64 small drops of equal

radius will be :-

An object of mass ' $m$ ' is projected from origin in a vertical xy plane at an angle $45^{\circ}$ with the x -axis with an initial velocity $\mathrm{v}_{0}$. The magnitude and direction of the angular momentum of the object with respect to origin, when it reaches at the maximum height, will be [ g is acceleration due to gravity]

The Young's double slit interference experiment is performed using light consisting of 480 nm and 600 nm wavelengths to form interference patterns. The least number of the bright fringes of 480 nm light that are required for the first coincidence with the bright fringes formed by 600 nm light is :-

A car of mass ' $m$ ' moves on a banked road having radius ' r ' and banking angle $\theta$. To avoid slipping from banked road, the maximum permissible speed of the car is $\mathrm{v}_{0}$. The coefficient of friction $\mu$ between the wheels of the car and the banked road is :-

A uniform solid cylinder of mass ' m ' and radius ' r ' rolls along an inclined rough plane of inclination $45^{\circ}$. If it starts to roll from rest from the top of the plane then the linear acceleration of the cylinder axis will be:-

A thin plano convex lens made of glass of refractive index 1.5 is immersed in a liquid of refractive index 1.2 . When the plane side of the lens is silver coated for complete reflection, the lens immersed in the liquid behaves like a concave mirror of focal length 0.2 m . The radius of curvature of the curved surface of the lens is :-

A particle is executing simple harmonic motion with time period 2 s and amplitude 1 cm . If D and d are the total distance and displacement covered by the particle in 12.5 s , then D/d is :-

A satellite is launched into a circular orbit of radius ' R ' around the earth. A second statellite is launched into an orbit of radius 1.03 R . The time period of revolution of the second satellite is larger than the first one approximately by :-

A plano-convex lens having radius of curvature of first surface 2 cm exhibits focal length of $f_{1}$ in air. Another plano-convex lens with first surface radius of curvature 3 cm has focal length of $\mathrm{f}_{2}$ when it is immersed in a liquid of refractive index 1.2. If both the lenses are made of same glass of refractive index 1.5, the ratio of $\mathrm{f}_{1}$ and $\mathrm{f}_{2}$ will be :-

An alternating current is given by

$\mathrm{I}=\mathrm{I}_{\mathrm{A}} \sin \omega \mathrm{t}+\mathrm{I}_{\mathrm{B}} \cos \omega \mathrm{t}$.

The r.m.s. current will be :-

An electron of mass ' $m$ ' with an initial velocity $\overrightarrow{\mathrm{v}}=\mathrm{v}_{0} \hat{\mathrm{i}}\left(\mathrm{v}_{0}>0\right) \quad$ enters an electric field $\overrightarrow{\mathrm{E}}=-\mathrm{E}_{0} \hat{\mathrm{k}}$. If the initial de Broglie wavelength is $\lambda_{0}$, the value after time $t$ would be :-

A parallel plate capacitor was made with two rectangular plates, each with a length of

$l=3 \mathrm{~cm}$ and breath of $\mathrm{b}=1 \mathrm{~cm}$. The distance

between the plates is $3 \mu \mathrm{~m}$. Out of the following, which are the ways

to increase the capacitance by a factor of 10 ?

A. $l=30 \mathrm{~cm}, \mathrm{~b}=1 \mathrm{~cm}, \mathrm{~d}=1 \mu \mathrm{~m}$

B. $l=3 \mathrm{~cm}, \mathrm{~b}=1 \mathrm{~cm}, \mathrm{~d}=30 \mu \mathrm{~m}$

C. $l=6 \mathrm{~cm}, \mathrm{~b}=5 \mathrm{~cm}, \mathrm{~d}=3 \mu \mathrm{~m}$

D. $l=1 \mathrm{~cm}, \mathrm{~b}=1 \mathrm{~cm}, \mathrm{~d}=10 \mu \mathrm{~m}$

E. $l=5 \mathrm{~cm}, \mathrm{~b}=2 \mathrm{~cm}, \mathrm{~d}=1 \mu \mathrm{~m}$

A force $F=\alpha+\beta x^{2}$ acts on an object in the x -direction. The work done by the force is 5 J when the object is displaced by 1 m . If the constant $\alpha=1 \mathrm{~N}$ then $\beta$ will be

An ideal gas goes from an initial state to final state. During the process, the pressure of gas increases linearly with temperature.

A. The work done by gas during the process is zero.

B. The heat added to gas is different from change in its internal energy.

C. The volume of the gas is increased.

D. The internal energy of the gas is increased.

E. The process is isochoric (constant volume process)

A square loop of sides $\mathrm{a}=1 \mathrm{~m}$ is held normally in front of a point charge $\mathrm{q}=1 \mathrm{C}$. The flux of the electric field through the shaded region is $\frac{5}{\mathrm{p}} \times \frac{1}{\varepsilon_{0}} \frac{\mathrm{Nm}^{2}}{\mathrm{C}}$, where the value of p is $\qquad$

images

The least count of a screw guage is 0.01 mm . If the pitch is increased by $75 \%$ and number of divisions on the circular scale is reduced by $50 \%$, the new least count will be $\qquad$ $\times 10^{-3} \mathrm{~mm}$.

A wire of resistance $9 \Omega$ is bent to form an equilateral triangle. Then the equivalent resistance across any two vertices will be $\qquad$ ohm.

A current of 5 A exists in a square loop of side $\frac{1}{\sqrt{2}} \mathrm{~m}$.

Then the magnitude of the magnetic field $B$ at the centre of the square loop will be

$\mathrm{p} \times 10^{-6} \mathrm{~T}$. where, value of p is $\qquad$ .[0pt]

[Take $\mu_{0}=4 \pi \times 10^{-7} \mathrm{~T} \mathrm{~mA}^{-1}$ ].

The temperature of 1 mole of an ideal monoatomic gas is increased by $50^{\circ} \mathrm{C}$ at constant pressure. The total heat added and change in internal energy are $E_{1}$ and $E_{2}$, respectively. If $\frac{E_{1}}{E_{2}}=\frac{x}{9}$ then the value of $x$ is $\qquad$ .