JEE Main 2025 — 7 April, Evening Shift

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

1 M aqueous solutions of each of $\mathrm{Cu}\left(\mathrm{NO}_{3}\right)_{2}$, $\mathrm{AgNO}_{3}, \mathrm{Hg}_{2}\left(\mathrm{NO}_{3}\right)_{2} ; \mathrm{Mg}\left(\mathrm{NO}_{3}\right)_{2}$ are electrolysed using inert electrodes.

Given : $\quad \mathrm{E}_{\mathrm{Ag}^{+} / \mathrm{Ag}}^{0}=0.80 \mathrm{~V}, \quad \mathrm{E}_{\mathrm{Hg}_{2}^{2+} / \mathrm{Hg}}^{0}=0.79 \mathrm{~V}$, $\mathrm{E}_{\mathrm{Cu}^{2+} / \mathrm{Cu}}^{0}=0.24 \mathrm{~V}$ and $\mathrm{E}_{\mathrm{Mg}^{2+} / \mathrm{Mg}}^{0}=-2.37 \mathrm{~V}$

Statement I: With increasing voltage, the sequence of deposition of metals on the cathode will be $\mathrm{Ag}, \mathrm{Hg}$ and

Cu .

Statement II: Magnesium will not be deposited at cathode instead oxygen gas will be evolved at the cathode.

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

Mixture of 1 g each of chlorobenzene, aniline and benzoic acid is dissolved in 50 mL ethyl acetate and placed in a separating funnel. $5 \mathrm{M} \mathrm{NaOH}(30 \mathrm{~mL})$ was added in the same funnel. The funnel was shaken vigorously and then kept aside. The ethyl acetate layer in the funnel contains

Match List-I with List-II.

Choose the correct answer from the options given below.

Match List-I with List-II.

LIST-I (Conversion)	LIST-II (Reagents, Conditions used)
P.	1. $\text{Warm,\; }\mathrm{H_{2}O}$
Q.	2. (a) $\mathrm{NaOH},\;368\;\mathrm{K};$ (b) $\mathrm{H_{3}O^{+}}$
R.	3. (a) $\mathrm{NaOH},\;443\;\mathrm{K};$ (b) $\mathrm{H_{3}O^{+}}$
S.	4. (a) $\mathrm{NaOH},\;623\;\mathrm{K},\;300\;\mathrm{atm};$ (b) $\mathrm{H_{3}O^{+}}$

Choose the correct answer from the options given below.

$\mathrm{A}(\mathrm{g}) \rightarrow \mathrm{B}(\mathrm{g})+\mathrm{C}(\mathrm{g})$ is a first order reaction.

The reaction was started with reactant A only. Which of the following expression is correct for rate constant k?

" $P$ " is an optically active compound with molecular formula $\mathrm{C}_{6} \mathrm{H}_{12} \mathrm{O}$. When ' P ' is treated with 2,4dinitrophenylhydrazine, it gives a positive test. However, in presence of Tollens reagent, "P" gives a negative test. Predict the structure of "P".

The number of unpaired electrons responsible for the paramagnetic nature of the following complex species are

respectively: $\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]^{3-},\left[\mathrm{FeF}_{6}\right]^{3-},\left[\mathrm{CoF}_{6}\right]^{3-},\left[\mathrm{Mn}(\mathrm{CN})_{6}\right]^{3-}$

The correct statements from the following are:

(A) $\mathrm{Tl}^{3+}$ is a powerful oxidising agent

(B) $\mathrm{Al}^{3+}$ does not get reduced easily

(C) Both $\mathrm{Al}^{3+}$ and $\mathrm{Tl}^{3+}$ are very stable in solution

(D) $\mathrm{Tl}^{+}$is more stable than $\mathrm{Tl}^{3+}$

(E) $\mathrm{Al}^{3+}$ and $\mathrm{Tl}^{+}$are highly stable

Choose the correct answer from the options given below:

Liquid $A$ and $B$ form an ideal solution. The vapour pressures of pure liquids $A$ and $B$ are 350 and 750 mm Hg respectively at the same temperature. If $X_{A}$ and $X_{B}$ are the mole fraction of $A$ and $B$ in solution while $y_{A}$ and $y_{B}$ are the mole fraction of $A$ and $B$ in vapour phase then,

The correct statement amongst the following is

In $\mathrm{SO}_{2}, \mathrm{NO}_{2}^{-}$and $\mathrm{N}_{3}^{-}$the hybridization at the central atom are respectively

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

Statement I: On hydrolysis, oligo peptides give rise to fewer number of $\alpha$-amino acids while proteins give rise to a large number of $\beta$-amino acids.

Statement II: Natural proteins are denatured by acids which convert the water soluble form of fibrous proteins to their water insoluble form.

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

$X$ ' is the number of acidic oxides among $\mathrm{VO}_{2}, \mathrm{~V}_{2} \mathrm{O}_{3}$, $\mathrm{CrO}_{3}, \mathrm{~V}_{2} \mathrm{O}_{5}$ and $\mathrm{Mn}_{2} \mathrm{O}_{7}$. The primary valency

of cobalt in $\left[\mathrm{Co}\left(\mathrm{H}_{2} \mathrm{NCH}_{2} \mathrm{CH}_{2} \mathrm{NH}_{2}\right)_{3}\right]_{2}\left(\mathrm{SO}_{4}\right)_{3}$ is Y . The value of $X+Y$ is $\_\_\_\_\_$ .

Choose the incorrect trend in the atomic radii $(r)$ of the elements.

The extra stability of half-filled subshell is due to :

(A) Symmetrical distribution of electrons

(B) Smaller coulombic repulsion energy

(C) The presence of electrons with the same spin in non-degenerate orbitals

(D) Larger exchange energy

(E) Relatively smaller shielding of electrons by one another

Identify the correct statements:

The hydration energies of $\mathrm{K}^{+}$and $\mathrm{Cl}^{-}$are $-x$ and $-\mathrm{y} \mathrm{kJ} / \mathrm{mol}$ respectively. If lattice energy of KCl is $-z \mathrm{~kJ} / \mathrm{mol}$, then the heat of solution of KCl is:

Match List-I with List-II.

Choose the correct answer from the options given below:

Butane reacts with oxygen to produce carbon dioxide and water following the equation given below.

$\mathrm{C}_{4} \mathrm{H}_{10}(\mathrm{~g})+\frac{13}{2} \mathrm{O}_{2}(\mathrm{~g}) \rightarrow 4 \mathrm{CO}_{2}(\mathrm{~g})+5 \mathrm{H}_{2} \mathrm{O}(\mathrm{I})$

If 174.0 kg of butane is mixed with 320.0 kg of $\mathrm{O}_{2}$, the volume of water formed in liters is $\_\_\_\_\_$ .

(Nearest integer)

[Given: (a) Molar mass of C, H, O are 12, 1, 16 g $\mathrm{mol}^{-1}$ respectively, (b) Density of water $=1 \mathrm{~g} \mathrm{~mL}^{-1}$ ]

One litre buffer solution was prepared by adding 0.10 mol each of $\mathrm{NH}_{3}$ and $\mathrm{NH}_{4} \mathrm{Cl}$ in deionised water. The change in pH on addition of 0.05 mol of HCl to the above solution is $\_\_\_\_\_$ $\times 10^{-2}$. (Nearest integer) Given: $\mathrm{pK}_{\mathrm{b}}$ of $\mathrm{NH}_{3}=4.745$ and $\log _{10} 3=0.477$

The number of paramagnetic metal complex species among

$\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}\right]^{3+}, \quad\left[\mathrm{Co}\left(\mathrm{C}_{2} \mathrm{O}_{4}\right)_{3}\right]^{3-}$, $\left[\mathrm{MnCl}_{6}\right]^{3-},\left[\mathrm{Mn}(\mathrm{CN})_{6}\right]^{3-},\left[\mathrm{CoF}_{6}\right)^{3-},\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]^{3-}$ and $\left[\mathrm{FeF}_{6}\right]^{3-}$ with same number of unpaired electrons is

In Dumas' method 292 mg of an organic compound released 50 mL of nitrogen gas $\left(\mathrm{N}_{2}\right)$ at 300 K temperature and 715 mm Hg pressure. The percentage composition of ' N ' in the organic compound is $\_\_\_\_\_$ % (Nearest integer) (Aqueous tension at $300 \mathrm{~K}=15 \mathrm{~mm} \mathrm{Hg}$ )

The number of solutions of the equation $\cos 2 \theta \cos \frac{\theta}{2}+\cos \frac{5 \theta}{2}=2 \cos ^{3} \frac{5 \theta}{2}$ in $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ is:

If the orthocenter of the triangle formed by the lines $y$ $=x+1, y=4 x-8$ and $y=m x+c$ is at

$(3,-1)$, then $m$ $-c$ is:

If the equation of the line passing through the point $\left(0,-\frac{1}{2}, 0\right)$ and perpendicular to the lines $\vec{r}=\lambda(\hat{i}+a \hat{j}+b \hat{k}) \quad$ and $\quad \vec{r}=(\hat{i}-\hat{j}-6 \hat{k})+\mu$ $(-b \hat{i}+a \hat{j}+5 \hat{k})$ is $\frac{x-1}{-2}=\frac{y+4}{d}=\frac{z-c}{-4}$, then $a+b$ $+c+d$ is equal to:

Let $\vec{a}$ and $\vec{b}$ be the vectors of the same magnitude such that $\frac{|\vec{a}+\vec{b}|+|\vec{a}-\vec{b}|}{|\vec{a}+\vec{b}|-|\vec{a}-\vec{b}|}=\sqrt{2}+1$. Then $\frac{|\vec{a}+\vec{b}|^{2}}{|\vec{a}|^{2}}$ is:

If the area of the region $\left\{(x, y): 1+x^{2} \leq y \leq \min \{x\right.$ $+7,11-3 x\}\}$ is $A$, then $3 A$ is equal to

Let a random variable $X$ take values $0,1,2,3$ with $P(X=0)=P(X=1)=p, P(X=2)=$ $P(X=3)=q$ and $E\left(X^{2}\right)=2 E(X)$. Then the value of $8 p-1$ is :

Let $p$ be the number of all triangles that can be formed by joining the vertices of a regular polygon $P$ of $n$ sides and $q$ be the number of all quadrilaterals that can be formed by joining the vertices of $P$. If $p+q=126$, then the eccentricity of the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{n}=1$ is :

If the range of the function $f(x)=\frac{5-x}{x^{2}-3 x+2}, x \neq$ 1 , 2 , is $(-\infty, \alpha] \cup[\beta, \infty)$, then $\alpha^{2}+\beta^{2}$ is equal to :

If the locus of $z \in C$, such that $\operatorname{Re}\left(\frac{z-1}{2 z+i}\right)+\operatorname{Re}\left(\frac{\bar{z}-1}{2 \bar{z}-i}\right)=2$, is a circle of radius $r$ and center $(a, b)$,

then $\frac{15 a b}{r^{2}}$ is equal to :

Let $a_{n}$ be the $n^{\text {th }}$ term of an A.P. If $S_{n}=a_{1}+a_{2}+a_{3}$ $+\ldots+a_{n}=700, a_{6}=7$ and $S_{7}=7$,

then $a_{n}$ is equal to :

A bag contains 19 unbiased coins and one coin with head on both sides. One coin drawn at random is tossed and head turns up. If the probability that the drawn coin was unbiased, is $\frac{m}{n}, \operatorname{gcd}(m, n)=1$, then $n^{2}-m^{2}$ is equal to :

Let $A=\{(\alpha, \beta) \in \mathbb{R} \times \mathbb{R}:|\alpha-1| \leq 4$ and $|\beta-5| \leq 6\}$ and

$B=\left\{(\alpha, \beta) \in \mathbb{R} \times \mathbb{R}: 16(\alpha-2)^{2}+9(\beta-6)^{2} \leq 144\right\}$. Then

If the sum of the second, fourth and sixth terms of a G.P. of positive terms is 21 and the sum of its eighth, tenth and twelfth terms is 15309, then the sum of its first nine terms is:

Let the length of a latus rectum of an ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ be 10 . If its eccentricity is the minimum value of

the function $f(t)=t^{2}+t+\frac{11}{12}, t \in \mathbb{R}$, then $a^{2}+b^{2}$ is equal to

Let $y=y(x)$ be the solution of the differential equation

$\left(x^{2}+1\right) y^{\prime}-2 x y=\left(x^{4}+2 x^{2}+1\right) \cos x$, $y(0)=1$. Then $\int_{-3}^{3} y(x) d x$ is :

Consider the lines $L_{1}: x-1=y-2=z$ and $L_{2}: x-$ $2=y=z-1$. Let the feet of the perpendiculars from the point $P(5,1,-3)$ on the lines $L_{1}$ and $L_{2}$ be $Q$ and $R$ respectively. If the area of the triangle $P Q R$ is $A$, then $4 A^{2}$ is equal to

The number of real roots of the equation $x|x-2|+$ $3|x-3|+1=0$ is

Let the system of equations

$x+5 y-z=1$

$4 x+3 y-3 z=7$

$24 x+y+\lambda z=m$

$\lambda, \mu \in \mathbf{R}$, have infinitely many solutions. Then the number of the solutions of this system, if $x, y, z$ are integers and satisfy $7 \leq x+y+z \leq 77$, is

Let $f: R \rightarrow R$ be a polynomial function of degree four having extreme values at $x=4$ and $x=5$. If $\lim _{x \rightarrow 0} \frac{f(x)}{x^{2}}=5$, then $f(2)$ is equal to

For $t>-1$, let $\alpha_{t}$ and $\beta_{t}$ be the roots of the equation $\left((t+2)^{\frac{1}{7}}-1\right) x^{2}+\left((t+2)^{\frac{1}{6}}-1\right) x+$

$\left((t+2)^{\frac{1}{21}}-1\right)=0$. If $\lim _{t \rightarrow-1^{+}} \alpha_{t}=a$ and $\lim _{t \rightarrow-1^{+}} \beta_{t}=b$, then $72(a+b)^{2}$ is equal

to $\_\_\_\_$ .

If $\int\left(\frac{1}{x}+\frac{1}{x^{3}}\right)\left(23 \sqrt{3 x^{-24}+x^{-26}}\right) d x$ $=-\frac{\alpha}{3(\alpha+1)}\left(3 x^{\beta}+x^{\gamma}\right)^{\frac{\alpha+1}{\alpha}}+C, x>0,$

$(\alpha, \beta, \gamma \in Z)$, where $C$ is the constant of integration, then $\alpha+\beta+$ $\gamma$ is equal to $\_\_\_\_\_$ -.

The sum of the series $2 \times 1 \times{ }^{20} C_{4}-3 \times 2 \times{ }^{20} C_{5}+4 \times 3 \times{ }^{20} C_{6}-5 \times 4 \times$ ${ }^{20} \mathrm{C}_{7}+\ldots . .$ $+18 \times 17 \times{ }^{20} \mathrm{C}_{20}$, is equal to $\_\_\_\_$ .

Let the lengths of the transverse and conjugate axes of a hyperbola in standard form $2 a$ and $2 b$, respectively, and one focus and the corresponding directrix of this hyperbola be $(-5,0)$ and $5 x+9=0$, respectively. If the product of the focal distances of a point $(\alpha, 2 \sqrt{5})$ on the hyperbola is $p$, then $4 p$ is equal to $\qquad$ .

If the function $f(x)=\frac{\tan (\tan x)-\sin (\sin x)}{\tan x-\sin x}$ is continuous at $x=0$, then $f(0)$ is equal to $\_\_\_\_$ .

Consider the following logic circuit. The output is $Y=0$ when :

The unit of $\sqrt{\frac{2 I}{\epsilon_{0} c}}$ is : ( $I=$ intensity of an electromagnetic wave, $c$ : speed of light)

A dipole with two electric charges of $2 \mu \mathrm{C}$ magnitude each, with separation distance $0.5 \mu \mathrm{~m}$, is placed between the plates of a capacitor such that its axis is parallel to an electric field established between the plates when a potential difference of 5 V is applied. Separation between the plates is 0.5 mm . If the dipole is rotated by $30^{\circ}$ from the axis, it tends to realign in the direction due to a torque. The value of torque is :

Match List-I with List-II. Choose the correct answer from the options given below:

A transparent block $A$ having refractive index $\mu=1.25$ is surrounded by another medium of refractive index $\mu=1.0$ as shown in figure. A light ray is incident on the flat face of the block with incident angle $\theta$ as shown in figure. What is the maximum value of $\theta$ for which light suffers total internal reflection at the top surface of the block?

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

Assertion (A) : The outer body of an air craft is made of metal which protects persons sitting inside from lightning-strikes.

Reason ( R ): The electric field inside the cavity enclosed by a conductor is zero.

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

The dimension of $\sqrt{\frac{\mu_{0}}{\varepsilon_{0}}}$ is equal to that of : ( $\mu_{0}=$ Vacuum permeability and $\varepsilon_{0}=$ Vacuum permittivity)

The helium and argon are put in the flask at the same room temperature ( 300 K ). The ratio of average kinetic energies (per molecule) of helium and argon is (Give : Molar mass of helium $=4 \mathrm{~g} / \mathrm{mol}$, Molar mass of argon $=40 \mathrm{~g} / \mathrm{mol}$ )

A capillary tube of radius 0.1 mm is partly dipped in water (surface tension $70 \mathrm{dyn} / \mathrm{cm}$ and glass water contact angle $\approx 0^{\circ}$ ) with $30^{\circ}$ inclined with the vertical. The length of water risen in the capillary is $\qquad$ cm . (Take $g=9.8 \mathrm{~m} / \mathrm{s}^{2}$ )

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

Assertion (A) : Refractive index of glass is higher than that of air.

Reason (R): Optical density of a medium is directly proportionate to its mass density which results in a proportionate refractive index.

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

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

Assertion (A): The radius vector from the Sun to a planet sweeps out equal areas in equal intervals of time and thus areal velocity of planet is constant.

Reason (R): For a central force field the angular momentum is a constant.

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

Match List-I with List-II. Choose the correct answer from the options given below.

A photoemissive substance is illuminated with a radiation of wavelength $\lambda_{i}$ so that it releases electrons with de-Broglie wavelength $\lambda_{e}$. The longest wavelength of radiation that can emit photoelectron is $\lambda_{0}$. Expression for de-Broglie wavelength is given by: ( $m$ : mass of the electron, $h$ : Planck's constant and $c$ : speed of light)

An object with mass 500 g moves along $x$-axis with speed $v=4 \sqrt{x} \mathrm{~m} / \mathrm{s}$. The force acting on the object is:

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

Assertion (A) : The density of the copper $\left({ }_{29}^{64} \mathrm{Cu}\right)$ nucleus is greater than that of the carbon $\left({ }_{6}^{12} \mathrm{C}\right)$ nucleus.

Reason (R) : The nucleus of mass number $A$ has a radius proportional to $A^{1 / 3}$.

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

Which one of the following forces cannot be expressed in terms of potential energy?

A mirror is used to produce an image with magnification of $\frac{1}{4}$ If the distance between object and its image is 40 cm , then the focal length of the mirror is $\qquad$ .

The equation of a wave travelling on a string is $y=\sin$ $[20 \pi x+10 \pi t]$, where $x$ and $t$ are distance

and time in SI units. The minimum distance between two points having the same oscillating speed is :

The electric field in a region is given by $\vec{E}=(2 \hat{i}+4 \hat{j}+6 \hat{k}) \times 10^{3} \mathrm{~N} / \mathrm{C}$.

The flux of the field through a rectangular surface parallel to $x-z$ plane is

$6.0 \mathrm{Nm}^{2} \mathrm{C}^{-1}$. The area of the surface is $\qquad$ $\mathrm{cm}^{2}$.

An inductor of reactance $100 \Omega$, a capacitor of reactance $50 \Omega$, and a resistor of resistance

$50 \Omega$ are connected in series with an AC source of $10 \mathrm{~V}, 50 \mathrm{~Hz}$.

Average power dissipated by the circuit is $\qquad$ W.

$M$ and $R$ be the mass and radius of a disc. A small disc of radius $\frac{R}{3}$ is removed from the bigger disc as shown in figure. The moment of inertia of remaining part of bigger disc about an axis $A B$ passing through the centre $O$ and perpendicular to the plane of disc is $\frac{4}{x} M R^{2}$. The value of $x$ is $\qquad$ .

Two cylindrical rods $A$ and $B$ made of different materials, are joined in a straight line. The ratios of lengths, radii and thermal conductivites of these rods are: $\frac{L_{A}}{L_{B}}=\frac{1}{2}, \frac{r_{A}}{r_{B}}=2$ and $\frac{K_{A}}{K_{B}}=\frac{1}{2}$. The free ends of rods $A$ and $B$ are maintained at $400 \mathrm{~K}, 200 \mathrm{~K}$, respectively. The temperature of rods interface is $\qquad$ K, when equilibrium is established.

A parallel plate capacitor has charge $5 \times 10^{-6} \mathrm{C}$.

A dielectric slab is inserted between the plates and almost fills the space between the plates. If the induced charge on one face of the slab is $4 \times 10^{-6} \mathrm{C}$ then the dielectric constant of the slab is $\qquad$ .