JEE Main 2025 Jan 24 Shift 1 — Question Paper

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

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

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

If the system of equations

$2x - y + z = 4$

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

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

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

Let $S_n = \frac{1}{2}+\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+\ldots$ upto n terms. If the sum of the first six terms of an A.P. with first term $-p$ and common difference $p$ is $\sqrt{2026\, S_{2025}}$, then the absolute difference between $20^{\text{th}}$ and $15^{\text{th}}$ terms of the A.P. is

For some $n \neq 10$, let the coefficients of the 5th, 6th and 7th 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:

For a statistical data $x_1, x_2, \ldots, x_{10}$ of 10 values, a student obtained the mean as 5.5 and $\sum_{i=1}^{10} x_i^2 = 371$. 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

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

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

Let circle C be the image of $x^2 + y^2 - 2x + 4y - 4 = 0$ in the line $2x - 3y + 5 = 0$ 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

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

$\lim_{x \to 0} \operatorname{cosec} x\left(\sqrt{2\cos^2 x + 3\cos x} - \sqrt{\cos^2 x + \sin x + 4}\right)$ is:

Let $\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$, $\vec{b}=3\hat{i}+\hat{j}-\hat{k}$ and $\vec{c}$ be three vectors such that $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b}$. If the vector $\vec{c}$ is perpendicular to $\vec{b}$ and $\vec{a} \cdot \vec{c} = 5$, then $|\vec{c}|$ is equal to

Let in a $\triangle ABC$, the length of the side AC 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 $\triangle ABC$ is:

Let the line passing through the points $(-1, 2, 1)$ and parallel to the line $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z}{4}$ 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 $Q(4, -5, 1)$ is

Let $f: \mathbb{R}-\{0\} \to \mathbb{R}$ be a function such that $f(x)-6f\left(\frac{1}{x}\right)=\frac{35}{3x}-\frac{5}{2}$. If the $\lim_{x \to 0}\left(\frac{1}{\alpha x}+f(x)\right)=\beta$ ; $\alpha, \beta \in \mathbb{R}$, then $\alpha + 2\beta$ is equal to

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:

If $I(m, n)=\int_0^1 x^{m-1}(1-x)^{n-1}\,dx$, $m, n > 0$, then $I(9, 14) + I(10, 13)$ is

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

Let $y = y(x)$ be the solution of the differential equation $\left(xy - 5x^2\sqrt{1+x^2}\right)dx + (1+x^2)dy = 0$, $y(0) = 0$. Then $y(\sqrt{3})$ is equal to

Let $S = \{p_1, p_2, \ldots, p_{10}\}$ be the set of first ten prime numbers. Let $A = S \cup P$, where P is the set of all possible products of distinct elements of S. Then the number of all ordered pairs $(x, y)$, $x \in S$, $y \in A$, such that x divides y, is _____.

Let A be a $3 \times 3$ matrix such that $X^T A X = O$ for all nonzero $3 \times 1$ matrices $X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$. If $A\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 4 \\ -5 \end{bmatrix}$, $A\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix} = \begin{bmatrix} 0 \\ 4 \\ -8 \end{bmatrix}$, and $\det(\operatorname{adj}(2(A + I))) = 2^{\alpha}\, 3^{\beta}\, 5^{\gamma}$, $\alpha, \beta, \gamma \in \mathbb{N}$, then $\alpha^2 + \beta^2 + \gamma^2$ is _____.

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

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

Let f be a differentiable function such that $2(x+2)^2 f(x) - 3(x+2)^2 = 10\int_0^x (t+2)f(t)\,dt$, $x \geq 0$. Then $f(2)$ is equal to _____.

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

An object of mass 'm' is projected from origin in a vertical xy plane at an angle $45°$ with the x-axis with an initial velocity $v_0$. 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]

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 $v_0$. The coefficient of friction $\mu$ between the wheels of the car and the banked road is

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

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

A satellite is launched into a circular orbit of radius 'R' around the earth. A second satellite 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

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

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)

Choose the correct answer from the options given below:

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 $\frac{D}{d}$ is

An alternating current is given by $I = I_A \sin\omega t + I_B \cos\omega t$. The r.m.s current will be

A parallel plate capacitor was made with two rectangular plates, each of length $l = 3$ cm and breadth $b = 1$ 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. $l = 30$ cm, $b = 1$ cm, $d = 1~\mu\mathrm{m}$

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

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

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

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

Choose the correct answer from the options given below:

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

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 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 face of the lens is silvered 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

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

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 $f_2$ 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 $f_1$ and $f_2$ will be

An electron of mass 'm' with an initial velocity $\vec{v}=v_0\hat{i}\,(v_0>0)$ enters an electric field $\vec{E}=-E_o\hat{k}$. If the initial de Broglie wavelength is $\lambda_0$, the value after time t would be

During the transition of electron from state A to state C of a Bohr atom, the wavelength of emitted radiation is 2000 Å and it becomes 6000 Å 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.

Choose the correct answer from the options given below:

The temperature of 1 mole of an ideal monoatomic gas is increased by $50°C$ 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 _____.

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

A current of 5A exists in a square loop of side $\frac{1}{\sqrt{2}}$ m. Then the magnitude of the magnetic field B at the centre of the square loop will be $p \times 10^{-6}$ T. where, value of p is _____. [Take $\mu_0 = 4\pi \times 10^{-7}~\mathrm{T\,m\,A^{-1}}$].

A square loop of sides $a = 1$ m is held normally in front of a point charge $q = 1\mathrm{C}$. The flux of the electric field through the shaded region is $\frac{5}{p}\times\frac{1}{\varepsilon_0}~\frac{\mathrm{Nm^2}}{\mathrm{C}}$, where the value of p is _____.

[Figure: a square of side $a$ divided into four quadrants; the left half of the top-left quadrant (triangular region) and the bottom two quadrants are shaded; the point charge q lies on the axis through the centre of the square]

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

A. $2p_z$ and $2p_x$

B. $2s$ and $2p_x$

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

D. $2s$ and $2p_z$

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

Choose the correct answer from the options given below:

Which of the following statement is true with respect to $\mathrm{H_2O}$, $\mathrm{NH_3}$ and $\mathrm{CH_4}$ ?

A. The central atoms of all the molecules are $sp^3$ hybridized.

B. The H–O–H, H–N–H and H–C–H angles in the above molecules are 104.5°, 107.5° and 109.5°, respectively.

C. The increasing order of dipole moment is $\mathrm{CH_4 < NH_3 < H_2O}$.

D. Both $\mathrm{H_2O}$ and $\mathrm{NH_3}$ are Lewis acids and $\mathrm{CH_4}$ is a Lewis base.

E. A solution of $\mathrm{NH_3}$ in $\mathrm{H_2O}$ is basic. In this solution $\mathrm{NH_3}$ and $\mathrm{H_2O}$ act as Lowry-Bronsted acid and base respectively.

Choose the correct answer from the options given below:

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.

Consider the given plots of vapour pressure (VP) vs temperature (T/K). Which amongst the following options is correct graphical representation showing $\Delta T_f$, depression in the freezing point of a solvent in a solution?

$\mathrm{K_{sp}}$ for $\mathrm{Cr(OH)_3}$ is $1.6 \times 10^{-30}$. What is the molar solubility of this salt in water?

For the given cell $$\mathrm{Fe^{2+}_{(aq)} + Ag^{+}_{(aq)} \to Fe^{3+}_{(aq)} + Ag_{(s)}}$$ The standard cell potential of the above reaction is Given: $\mathrm{Ag^+ + e^- \to Ag} \quad E^{\ominus} = x\ \mathrm{V}$

$\mathrm{Fe^{2+} + 2e^- \to Fe} \quad E^{\ominus} = y\ \mathrm{V}$

$\mathrm{Fe^{3+} + 3e^- \to Fe} \quad E^{\ominus} = z\ \mathrm{V}$

For a reaction, $\mathrm{N_2O_{5(g)} \to 2NO_{2(g)} + \frac{1}{2}O_{2(g)}}$ 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 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 configurations are arranged in same period.

D. The position of an element indicates the quantum numbers of the last filled orbital.

E. The number of elements present in a period is equal to the number of atomic orbitals present in the energy level that is being filled.

Choose the correct answer from the options given below:

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

Which of the following ions is the strongest oxidizing agent? (Atomic Number of Ce = 58, Eu = 63, Tb = 65, Lu = 71)

Preparation of potassium permanganate from $\mathrm{MnO_2}$ involves two step process in which the 1st step is a reaction with KOH and $\mathrm{KNO_3}$ to produce

One mole of the octahedral complex compound $\mathrm{Co(NH_3)_5Cl_3}$ 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 $\mathrm{AgCl_{(s)}}$. The structure of the complex is:

Given below are two statements I and II.

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

Statement II: In Dumas method, the organic compound is heated with concentrated $\mathrm{H_2SO_4}$ to form ammonium sulphate.

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

Which one of the carbocations from the following is most stable?

Following are the four molecules "P", "Q", "R" and "S". Which one among the four molecules will react with $\mathrm{H{-}Br_{(aq)}}$ at the fastest rate?

[Figure: P = 3,6-dihydro-2H-pyran (six-membered ring with O, C=C not adjacent to O); Q = 3,4-dihydro-2H-pyran (six-membered ring with O, C=C adjacent to O); R = 1-methylcyclohexene; S = 1,2-dimethylcyclohexene]

Given below are two statements:

Statement I: The conversion proceeds well in the less polar medium. $$\mathrm{CH_3-CH_2-CH_2-CH_2-Cl \xrightarrow{HO^-} CH_3-CH_2-CH_2-CH_2-OH + Cl^{(-)}}$$

Statement II: The conversion proceeds well in the more polar medium. $$\mathrm{CH_3-CH_2-CH_2-CH_2-Cl \xrightarrow{R_3N:} CH_3-CH_2-CH_2-CH_2-\overset{(+)}{N}R_3\ Cl^{(-)}}$$

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

Aman has been asked to synthesise the molecule [Figure: 1-(cyclopent-1-en-1-yl)ethan-1-one — a cyclopentene ring bearing a $\mathrm{-C(=O)-CH_3}$ group on a double-bond carbon] (x). He thought of preparing this molecule by aldol condensation. He found some cyclic alkenes in his laboratory. He planned ozonolysis of an alkene to obtain 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 arrangements with respect to their reactivity in nucleophilic addition reaction is correct?

The product (A) formed in the following reaction sequence is $$\mathrm{CH_3-C\equiv CH \xrightarrow[\text{iii) } H_2/Ni]{\text{i) } Hg^{2+},\ H_2SO_4\ \ \text{ii) } HCN} (A)}$$

The carbohydrate "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:

Consider the following reaction occurring in the blast furnace: $$\mathrm{Fe_3O_{4(s)} + 4CO_{(g)} \to 3Fe_{(l)} + 4CO_{2(g)}}$$ '$x$' kg of iron is produced when $2.32 \times 10^3$ kg $\mathrm{Fe_3O_4}$ and $2.8 \times 10^2$ kg CO are brought together in the furnace. The value of '$x$' is ______. (nearest integer) {Given: molar mass of $\mathrm{Fe_3O_4}$ = 232 g mol$^{-1}$ molar mass of CO = 28 g mol$^{-1}$ molar mass of Fe = 56 g mol$^{-1}$}

Standard entropies of $\mathrm{X_2}$, $\mathrm{Y_2}$ and $\mathrm{XY_5}$ are 70, 50 and 110 J K$^{-1}$ mol$^{-1}$ respectively. The temperature in Kelvin at which the reaction $$\frac{1}{2}\mathrm{X_2} + \frac{5}{2}\mathrm{Y_2} \rightleftharpoons \mathrm{XY_5} \qquad \Delta H^{\ominus} = -35\ \mathrm{kJ\ mol^{-1}}$$ will be at equilibrium is ______ (nearest integer).

37.8 g $\mathrm{N_2O_5}$ was taken in a 1 L reaction vessel and allowed to undergo the following reaction at 500 K $$\mathrm{2N_2O_{5(g)} \rightleftharpoons 2N_2O_{4(g)} + O_{2(g)}}$$ The total pressure at equilibrium was found to be 18.65 bar. Then, $K_p$ = ______ $\times 10^{-2}$ [nearest integer]

Assume $\mathrm{N_2O_5}$ to behave ideally under these conditions.

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

X g of benzoic acid on reaction with aq $\mathrm{NaHCO_3}$ released $\mathrm{CO_2}$ that occupied 11.2 L volume at STP. X is ______.

Among the following cations, the number of cations which will give characteristic precipitate in their identification tests with $\mathrm{K_4[Fe(CN)_6]}$ is ______. $\mathrm{Cu^{2+},\ Fe^{3+},\ Ba^{2+},\ Ca^{2+},\ NH_4^{+},\ Mg^{2+},\ Zn^{2+}}$