JEE Main 2025 · 02 Apr 2025 · Shift 1

Q1.A square Lamina OABC of length 10 cm is pivoted at 'O'. Forces act at Lamina as shown in figure. If Lamina remains stationary, then the magnitude of $F$ is:

Q2.Let $B_1$ be the magnitude of magnetic field at center of a circular coil of radius R carrying current I. Let ${\mathrm{B}}_2$ be the magnitude of magnetic field at an axial distance 'x' from the center. For …

Q3.Considering Bohr's atomic model for hydrogen atom:
(A) the energy of H atom in ground state is same as energy of ${\mathrm{He}}^{+}$ ion in its first excited state.
(B) the energy of H atom in ground state is same as that for ${\mathrm{Li}}^{++}$ ion in its second excited state. …

Q4.Moment of inertia of a rod of mass 'M' and length 'L' about an axis passing through its center and normal to its length is '$\alpha$'. Now the rod is cut into two equal parts and these parts are joined symmetrically to form a cross shape. Moment of inertia of cross about an axi …

Q5.A spherical surface separates two media of refractive indices 1 and 1.5 as shown in figure.
Distance of the image of an object 'O', is:
(C is the center of curvature of the spherical surface and $R$ is the radius of curvature)

Q6.Match List-I with List-II.

List-I
(A) Coefficient of viscosity
(B) Intensity of wave
(C) Pressure gradient
(D) Compressibility

List-II
(I) $[{\mathrm{ML}}^0{\mathrm{T}}^{-3}]$
(II) $[{\mathrm{ML}}^{-2}{\mathrm{T}}^{-2}]$
(III) $[{\mathrm{M}}^{-1}{\mathrm{LT}}^2]$
(IV) …

Q7.A small bob of mass 100 mg and charge $+10\mu \mathrm{C}$ is connected to an insulating string of length 1 m. It is brought near to an infinitely long nonconducting sheet of charge density '$\sigma$' as shown in figure. If string subtends an angle of $45^{\circ }$ with the …

Q8.A monochromatic light is incident on a metallic plate having work function $\varphi$. An electron, emitted normally to the plate from a point A with maximum kinetic energy, enters a constant magnetic field, perpendicular to the initial velocity of electron. The electron passes thr …

Q9.A vessel with square cross-section and height of 6 m is vertically partitioned. A small window of $100{\mathrm{cm}}^2$ with hinged door is fitted at a depth of 3 m in the partition wall. One part of the vessel is filled completely with water and the other side is filled with …

Q10.A steel wire of length 2 m and Young's modulus $2.0\times 10^{11}{\mathrm{Nm}}^{-2}$ is stretched by a force. If Poisson ratio and transverse strain for the wire are 0.2 and $10^{-3}$ respectively, then the elastic potential energy density of the wire is _______ …

Q11.If the measured angular separation between the second minimum to the left of the central maximum and the third minimum to the right of the central maximum is $30^{\circ }$ in a single slit diffraction pattern recorded using 628 nm light, then the width of the slit is _______ …

Q12.${\gamma }_{\mathrm{A}}$ is the specific heat ratio of monoatomic gas A having 3 translational degrees of freedom. ${\gamma }_{\mathrm{B}}$ is the specific heat ratio of polyatomic gas B having 3 translational, 3 rotational degrees of freedom and 1 vibrational mode. If …

Q13.A person travelling on a straight line moves with a uniform velocity ${\mathrm{v}}_1$ for a distance x and with a uniform velocity $v_2$ for the next $\frac{3}{2}x$ distance. The average velocity in this motion is $\frac{50}{7}\mathrm{m}/\mathrm{s}$. If …

Q1.Designate whether each of the following compounds is aromatic or not aromatic.
(a) (see structure)
(b) (see structure)
(c) (see structure)
(d) (see structure)
(e) (see structure)
(f) (see structure)
(g) (see structure)
(h) (see structure)

Q2.An optically active alkyl halide $\mathrm{C}{}_4\mathrm{H}{}_9\mathrm{Br}$ [A] reacts with hot KOH dissolved in ethanol and forms alkene [B] as major product which reacts with bromine to give dibromide [C]. The compound [C] is converted into a gas [D] upon reacting with alcoholic $\mathrm{NaNH}{}_2$. During hydrat …

Q3.The property/properties that show irregularity in first four elements of group-17 is/are :
(A) Covalent radius
(B) Electron affinity
(C) Ionic radius
(D) First ionization energy
Choose the correct answer from the options given below:

Q4.Which of the following graph correctly represents the plots of $K_H$ at 1 bar gases in water versus temperature?

Q5.According to Bohr's model of hydrogen atom, which of the following statement is incorrect?

Q6.Two vessels A and B are connected via stopcock. The vessel A is filled with a gas at a certain pressure. The entire assembly is immersed in water and is allowed to come to thermal equilibrium with water. After opening the stopcock the gas from vessel A expands into vessel B and n …

Q7.A solution is made by mixing one mole of volatile liquid A with 3 moles of volatile liquid B. The vapour pressure of pure A is 200 mm Hg and that of the solution is 500 mm Hg. The vapour pressure of pure B and the least volatile component of the solution, respectively, are :

Q8.$\mathrm{CaCO}{}_3(\mathrm{s})+2\mathrm{HCl}(\mathrm{aq})\to \mathrm{CaCl}{}_2(\mathrm{aq})+\mathrm{CO}{}_2(\mathrm{g})+\mathrm{H}{}_2\mathrm{O}(\mathrm{l})$
Consider the above reaction, what mass of $\mathrm{CaCl}{}_2$ will be formed if 250 mL of 0.76 M HCl reacts with 1000 g of $\mathrm{CaCO}{}_3$?
(Given : Molar mass of Ca, C, O, H and Cl are 40, 12, 16, 1 and 35.5 g mol${}^{-1}$, respectiv …

Q9.If equal volumes of $\mathrm{AB}{}_2$ and XY (both are salts) aqueous solutions are mixed, which of the following combination will give a precipitate of $\mathrm{AY}{}_2$ at 300 K?
(Given $K_{sp}$ (at 300 K) for $\mathrm{AY}{}_2=5.2\times 10^{-7}$)

Q10.Among $\mathrm{SO}{}_2$, $\mathrm{NF}{}_3$, $\mathrm{NH}{}_3$, $\mathrm{XeF}{}_2$, $\mathrm{ClF}{}_3$ and $\mathrm{SF}{}_4$, the hybridization of the molecule with non-zero dipole moment and highest number of lone-pairs of electrons on the central atom is

Q11.Given below are two statements :
Statement (I) : Vanillin (see structure) will react with NaOH and also with Tollen's reagent.
Statement (II) : (see structure) will undergo self aldol condensation very easily.
In the light of the above statements, choose the most appropriate answ …

Q12.Identify the correct statement among the following:

Q13.The correct order of basic nature on aqueous solution for the bases ${\mathrm{NH}}_3,{\mathrm{H}}_2\mathrm{N}-{\mathrm{NH}}_2,{\mathrm{CH}}_3{\mathrm{CH}}_2{\mathrm{NH}}_2$, ${\left({\mathrm{CH}}_3{\mathrm{CH}}_2\right)}_2\mathrm{NH}$ and …

Q14.Given below are two statements :

Statement (I) : The metallic radius of Al is less than that of Ga.
Statement (II) : The ionic radius of ${\mathrm{Al}}^{3+}$ is less than that of ${\mathrm{Ga}}^{3+}$.
In the light of the above statements, choose the most appropriate answer from the …

Q15.Given below are two statements :

Statement (I) : In octahedral complexes, when ${\Delta }_{\mathrm{o}}<\mathrm{P}$ high spin complexes are formed. When ${\Delta }_{\mathrm{o}}>\mathrm{P}$ low spin complexes are formed.
Statement (II) : In tetrahedral complexes because of …

Q16.Choose the correct tests with respective observations.
(A) ${\mathrm{CuSO}}_4$ (acidified with acetic acid) + ${\mathrm{K}}_4\left[\mathrm{Fe}(\mathrm{CN}{)}_6\right]\to$ Chocolate brown precipitate.
(B) …

Q17.On complete combustion 1.0 g of an organic compound (X) gave 1.46 g of ${\mathrm{CO}}_2$ and 0.567 g of ${\mathrm{H}}_2\mathrm{O}$. The empirical formula mass of compound (X) is ______ g.
(Given molar mass in …

Q18.Consider the following compound (X)

The most stable and least stable carbon radicals, respectively, produced by homolytic cleavage of corresponding $\mathrm{C}-\mathrm{H}$ bond are :

Q19.Consider the following molecules :

The correct order of rate of hydrolysis is :

Q20.A molecule with the formula ${\mathrm{AX}}_4\mathrm{Y}$ has all it's elements from p-block. Element A is rarest, monoatomic, non-radioactive from its group and has the lowest ionization enthalpy value among A, X and Y . Elements X and Y have first and second highest electronegat …

Q21.A transition metal (M) among Mn, Cr, Co and Fe has the highest standard electrode potential ( ${\mathrm{M}}^{3+}/{\mathrm{M}}^{2+}$ ). It forms a metal complex of the type ${\left[\mathrm{M}(\mathrm{CN}{)}_6\right]}^{4-}$. The number of electrons present in the …

Q22.Consider the following electrochemical cell at standard condition. …

Q23.$0.1\mathrm{mol}$ of the following given antiviral compound $(\mathrm{P})$ will weigh ______ $\times 10^{-1}\mathrm{g}$

(Given : molar mass in $\mathrm{g}{\mathrm{mol}}^{-1}\mathrm{H}:1,\mathrm{C}:12,\mathrm{N}:14$, O : 16, F : 19, I : 127)

Q24.Consider the following equilibrium,
$\mathrm{CO}(\mathrm{g})+2{\mathrm{H}}_2(\mathrm{g})\rightleftharpoons {\mathrm{CH}}_3\mathrm{OH}(\mathrm{g})$
0.1 mol of CO along with a catalyst is present in a $2{\mathrm{dm}}^3$ flask maintained at 500 K . Hydrogen is introduced into t …

Q25.For the reaction $\mathrm{A}\to$ products.

The concentration of A at 10 minutes is $\_\_\_\_$ $\times 10^{-3}\mathrm{mol}{\mathrm{L}}^{-1}$ (nearest integer).
The reaction was started with $2.5\mathrm{mol}{\mathrm{L}}^{-1}$ of A.

Q1.The largest $n\in \mathbb{N}$ such that $3^n$ divides 50! is:

Q2.Let one focus of the hyperbola $H:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ be at $(\sqrt{10},0)$ and the corresponding directrix be $x=\frac{9}{\sqrt{10}}$. If $e$ and $\ell$ respectively are the eccentricity and the length of the latus rectum of $H$, then $9(e^2+\ell )$ is …

Q3.The number of sequences of ten terms, whose terms are either 0 or 1 or 2, that contain exactly five 1s and exactly three 2s, is equal to

Q4.Let $f:\mathbb{R}\to \mathbb{R}$ be a twice differentiable function such that $(\sin x\cos y)(f(2x+2y)-f(2x-2y))=(\cos x\sin y)(f(2x+2y)+f(2x-2y))$, for all $x,y\in \mathbb{R}$.
If $f^{\prime }(0)=\frac{1}{2}$, then the value of $24f^{\prime \prime }\left(\frac{5\pi }{3}\right)$ is:

Q5.Let $A=\left[\begin{array}{cc}\alpha & -1\\ 6& \beta \end{array}\right]$, $\alpha >0$, such that $\det (A)=0$ and $\alpha +\beta =1$. If $I$ denotes $2\times 2$ identity matrix, then the matrix $(I+A{)}^8$ is:

Q6.The term independent of $x$ in the expansion of ${\left(\frac{(x+1)}{(x^{2/3}+1-x^{1/3})}-\frac{(x-1)}{(x-x^{1/2})}\right)}^{10}$, $x>1$ is:

Q7.If $\theta \in [-2\pi ,2\pi ]$, then the number of solutions of $2\sqrt{2}{\cos }^2\theta +(2-\sqrt{6})\cos \theta -\sqrt{3}=0$, is equal to:

Q8.Let $a_1,a_2,a_3,\dots$ be in an A.P. such that ${\sum }_{k=1}^{12}{a}_{2k-1}=-\frac{72}{5}{a}_1$, $a_1\ne 0$. If ${\sum }_{k=1}^n{a}_k=0$, then $n$ is:

Q9.If the function $f(x)=2x^3-9a{x}^2+12a^{2}x+1$, where $a>0$, attains its local maximum and local minimum values at $p$ and $q$, respectively, such that $p^2=q$, then $f(3)$ is equal to:

Q10.Let $z$ be a complex number such that $|z|=1$. If $\frac{2+k^{2}z}{k+\overline{z}}=kz$, $k\in \mathbb{R}$, then the maximum distance of $k+i{k}^2$ from the circle $|z-(1+2i)|=1$ is:

Q11.If $\vec{a}$ is nonzero vector such that its projections on the vectors $2\hat{i}-\hat{j}+2\hat{k}$, $\hat{i}+2\hat{j}-2\hat{k}$ and $\hat{k}$ are equal, then a unit vector along $\vec{a}$ is:

Q12.Let $A$ be the set of all functions $f:\mathbb{Z}\to \mathbb{Z}$ and $R$ be a relation on $A$ such that $R=\{(f,g):f(0)=g(1)$ and $f(1)=g(0)\}$. Then $R$ is:

Q13.For $\alpha ,\beta ,\gamma ,\in \mathbf{R}$, if ${\lim }_{x\to 0}\frac{x^2\sin \alpha x+(\gamma -1)e^{x^2}}{\sin 2x-\beta x}=3$, then $\beta +\gamma -\alpha$ is equal to:

Q14.If the system of linear equations
$3x+y+\beta z=3$
$2\mathrm{x}+\alpha \mathrm{y}-\mathrm{z}=-3$
$\mathrm{x}+2\mathrm{y}+\mathrm{z}=4$
has infinitely many solutions, then the value of $22\beta -9\alpha$ is :

Q15.Let $P_n={\alpha }^n+{\beta }^n,n\in \mathbf{N}$. If $P_{10}=123,P_9=76$, ${\mathrm{P}}_8=47$ and ${\mathrm{P}}_1=1$, then the quadratic equation having roots $\frac{1}{\alpha }$ and $\frac{1}{\beta }$ is :

Q16.If $S$ and $S^{\prime }$ are the foci of the ellipse $\frac{x^2}{18}+\frac{y^2}{9}=1$ and P be a point on the ellipse, then $\min \left(\mathrm{SP}.{\mathrm{S}}^{\prime }\mathrm{P}\right)+\max \left({\mathrm{SP}}^{\prime }{\mathrm{S}}^{\prime }\mathrm{P}\right)$ is equal …

Q17.Let the vertices $Q$ and $R$ of the triangle $PQR$ lie on the line $\frac{\mathrm{x}+3}{5}=\frac{\mathrm{y}-1}{2}=\frac{\mathrm{z}+4}{3},\mathrm{QR}=5$ and the coordinates of the point P be $(0,2,3)$. If the area of the triangle $PQR$ is $\frac{m}{n}$ then :

Q18.Let ABCD be a tetrahedron such that the edges $\mathrm{AB},\mathrm{AC}$ and AD are mutually perpendicular. Let the areas of the triangles $\mathrm{ABC},\mathrm{ACD}$ and ADB be 5, 6 and 7 square units respectively. Then the area (in square units) of the …

Q19.Let $\mathrm{a}\in \mathbf{R}$ and A be a matrix of order $3\times 3$ such that $\det (\mathrm{A})=-4$ and $\mathrm{A}+\mathrm{I}=\left[\begin{array}{lll}1& \mathrm{a}& 1\\ 2& 1& 0\\ \mathrm{a}& 1& 2\end{array}\right]$, where I is the identity matri …

Q20.Let the focal chord PQ of the parabola ${\mathrm{y}}^2=4\mathrm{x}$ make an angle of $60^{\circ }$ with the positive x -axis, where P lies in the first quadrant. If the circle, whose one diameter is PS, S being the focus of the parabola, touches the y -axis at the point …

Q21.Let [ • ] denote the greatest integer function. If ${\int }_0^{{\mathrm{e}}^3}\left[\frac{1}{{\mathrm{e}}^{\mathrm{x}-1}}\right]\mathrm{dx}=\alpha -{\log }_{\mathrm{e}}2$, then ${\alpha }^3$ is equal to ______.

Q22.Let $f:\mathbf{R}\to \mathbf{R}$ be a thrice differentiable odd function satisfying $f^{\prime }(\mathrm{x})\ge 0,f^{\prime \prime }(\mathrm{x})=f(\mathrm{x}),f(0)=0,f^{\prime }(0)=3$. Then $9f\left({\log }_{\mathrm{e}}3\right)$ is equal to ______.

Q23.If the area of the region
$\left\{(\mathrm{x},\mathrm{y}):\left|4-{\mathrm{x}}^2\right|\le \mathrm{y}\le {\mathrm{x}}^2,\mathrm{y}\le 4,\mathrm{x}\ge 0\right\}$
is …

Q24.Three distinct numbers are selected randomly from the set $\{1,2,3,\dots \dots ,40\}$. If the probability, that the selected numbers are in an increasing G.P. is $\frac{m}{n}$, $\gcd (\mathrm{m},\mathrm{n})=1$, then $\mathrm{m}+\mathrm{n}$ is equal to ___ …

Q25.The absolute difference between the squares of the radii of the two circles passing through the point $(-9,4)$ and touching the lines $x+y=3$ and $x-y=3$, is equal to ______.