JEE Main 2026 — 2 April, Evening Shift

Dimensions of universal gravitational constant (G) in terms of Planck's constant $(h)$ , distance $0$ , mass $(M)$ and time $(T)$ are

A $0.5\mathrm{kg}$ mass is in contact against the inner wall of a cylindrical drum of radius $4\mathrm{m}$ rotating about its vertical axis. The minimum rotational speed of the drum to enable the mass to remain stuck to the wall (without falling) is $5\mathrm{rad / s}$ . The coefficient of friction between the drum's inner wall surface and mass is (Take $\mathrm{g} = 10\mathrm{m / s}^2$)

Two blocks of masses $2\mathrm{kg}$ and $1\mathrm{kg}$ respectively, are tied to the ends of a string which passes over a light frictionless pulley as shown in the figure below. The masses are held at rest at the same horizontal level and then released. The distance traversed by the centre of mass in $2\mathrm{s}$ is $\mathrm{m}$. (Take $\mathrm{g} = 10\mathrm{m / s}^2$)

A particle having charge $10^{-9}$ C moving in $x-y$ plane in fields of $0.4 \hat{j} \mathrm{N/C}$ and $4 \times 10^{-3} \hat{k} \mathrm{T}$ experiences a force of $(4\hat{i}+2\hat{j})\times 10^{-10} \mathrm{N}$. The velocity of the particle at that instant is $\mathrm{m/s}$.

In a screw gauge the zero of main scale reference line coincides with the fifth division of the circular scale when two studs are in contact. There are 100 divisions in circular scale and pitch of screw gauge is $0.1\mathrm{mm}$. When diameter of a sphere is measured, the reading of main scale is $5\mathrm{mm}$ and $50^{\mathrm{th}}$ division of circular scale coincides with the reference line of main scale. The diameter of sphere is mm.

The surface tension of a soap bubble is $0.03 \mathrm{N/m}$. The work done in increasing the diameter of bubble from $2\mathrm{cm}$ to $6\mathrm{cm}$ is $\alpha \pi \times 10^{-4} \mathrm{J}$. The value of $\alpha$ is ______. (Take $\pi=3.14$)

A mixture of carbon dioxide and oxygen has volume $8310 \mathrm{cm}^3$, temperature $300\mathrm{K}$, pressure $100\mathrm{kPa}$ and mass $13.2\mathrm{g}$. The number of moles of carbon dioxide and oxygen gases in the mixture respectively are ______. (Assume both gases ideal) [$\mathrm{R}=8.31\mathrm{J/mol.K}$]

If an air bubble of diameter $2\mathrm{mm}$ rises steadily through a liquid of density $2000\mathrm{kg/m}^3$ at a rate of $0.5\mathrm{cm/s}$, then the coefficient of viscosity of liquid is ______. (Take $\mathrm{g}=10\mathrm{m/s}^2$)

A spherical ball of mass $2\mathrm{kg}$ falls from a height of $10\mathrm{m}$ and is brought to rest after penetrating $10\mathrm{cm}$ into sand. The average force exerted by sand on the ball is N.

An electromagnetic wave travels in free space along the $x$-direction. At a particular point in space and time, $\vec{B}=2\times10^{-7}\hat{j}\mathrm{T}$ is associated with this wave. The value of corresponding electric field $\vec{E}$ at this point is V/m.

Two resistors of $200\Omega$ and $400\Omega$ are connected in series with a battery of $100\mathrm{V}$. A bulb rated at $200\mathrm{V},100\mathrm{W}$ is connected across the $400\Omega$ resistance. The potential drop across the bulb is V.

Two metal plates (A,B) are kept horizontally with separation of $\left(\frac{12}{\pi}\right)\mathrm{cm}$, with plate A on the top. An atomizer jet sprays oil (density $1.5\mathrm{g/cm}^3$) droplets of radius $1\mathrm{mm}$ horizontally. All oil droplets carry a charge $5\mathrm{nC}$. The potentials $V_A$ and $V_B$ required on plates A and B respectively in order to ensure the droplets do not descend are (Neglect air resistance, $\mathrm{g}=10\mathrm{m/s}^2$)

Two point charges $8\mu\mathrm{C}$ and $-2\mu\mathrm{C}$ are located at $x=2\mathrm{cm}$ and $x=4\mathrm{cm}$ respectively on the $x$-axis. The ratio of electric flux due to these charges through two spheres of radii $3\mathrm{cm}$ and $5\mathrm{cm}$ with their centers at the origin is

One side of an equilateral prism is painted by a transparent material of refractive index $n_2$. The refractive index of prism is 1.6. The minimum value of $n_2$ required for total internal reflection from painted face is

The figure given below shows an LCR series circuit with two switches $S_1$ and $S_2$. When switch $S_1$ is closed keeping $S_2$ open, the phase difference $\phi$ between current and source voltage is $30^\circ$ and phase difference is $60^\circ$ when $S_2$ is closed keeping $S_1$ open. The value of $(3L_1 - L_2)$ is H.

The binding energy per nucleon of $^{209}_{83}\mathrm{Bi}$ is MeV. [Take $m(^{209}_{83}\mathrm{Bi})=208.980388\mathrm{u}$, $m_p=1.007825\mathrm{u}$, $m_n=1.008665\mathrm{u}$, $1\mathrm{u}=931\mathrm{MeV}/c^2$]

The equation of motion of a particle is given by $x = a\sin(50t + \pi/3)\mathrm{cm}$. The particle will come to rest at time $t_1$ and it will have zero acceleration at time $t_2$. The $t_1$ and $t_2$ respectively are

In a Young's double slit experiment, the intensity at some point on the screen is found to be $\frac{3}{4}$ times the maximum intensity. The path difference between the interfering waves at this point is $\frac{\lambda}{x}$ where $\lambda$ is wavelength. The value of $x$ is _____

Using Bohr's model, calculate the ratio of the magnetic fields generated due to the motion of the electrons in the $2^{\mathrm{nd}}$ and $4^{\mathrm{th}}$ orbits of hydrogen atom

5 moles of unknown gas is heated at constant volume from $10^\circ\mathrm{C}$ to $20^\circ\mathrm{C}$. The molar specific heat at constant pressure $c_p = 8\,\mathrm{cal/mol.^\circ C}$ and $R = 8.36\,\mathrm{J/mol.^\circ C}$. The change in internal energy of the gas is ______ calorie.

If sunlight is focused on a paper using convex lens, it starts burning the paper in shortest time when the lens is kept at $30\mathrm{cm}$ above the paper. If the radius of curvature of the lens is $60\mathrm{cm}$ then the refractive index of the lens material is $\frac{\alpha}{10}$. The value of $\alpha$ is ______.

Moment of inertia about an axis AB for a rod of mass $40\mathrm{kg}$ and length $3\mathrm{m}$ is same as that of a solid sphere of mass $10\mathrm{kg}$ and radius R about an axis parallel to AB axis with separation of $3\mathrm{m}$ as shown in figure. The value of R is given as $\sqrt{\frac{\alpha}{2}}$. The value of $\alpha$ is ______.

Let $\alpha ,\beta$ be the roots of the equation $\mathrm{x}^{2} - 3\mathrm{x} + \mathrm{r} = 0$ and $\frac{\alpha}{2}, 2\beta$ be the roots of the equation $\mathrm{x}^{2} + 3\mathrm{x} + \mathrm{r} = 0$. If the roots of the equation $\mathrm{x}^{2} + 6\mathrm{x} = \mathrm{m}$ are $2\alpha +\beta +2\mathrm{r}$ and $\alpha - 2\beta -\frac{\mathrm{r}}{2}$, then m is equal to :-

Let the circle $\mathrm{C}_{1}:\left|\mathrm{z}\right| = \mathrm{r}$ and $\mathrm{C}_{2}:\left|\mathrm{z} - 3 - 4\mathrm{i}\right| = 5$, $\mathrm{z}\in \mathbb{C}$, be such that $\mathrm{C}_{2}$ lies within $\mathrm{C}_{1}$. If $\mathrm{z}_{1}$ moves on $\mathrm{C}_{1}$, $\mathrm{z}_{2}$ moves on $\mathrm{C}_{2}$ and $\min \left|\mathrm{z}_{1} - \mathrm{z}_{2}\right| = 2$, then $\max \left|\mathrm{z}_{1} - \mathrm{z}_{2}\right|$ is equal to :-

If the system of equations $\mathrm{x} + 5\mathrm{y} + 6\mathrm{z} = 4$, $2\mathrm{x} + 3\mathrm{y} + 4\mathrm{z} = 7$, $\mathrm{x} + 6\mathrm{y} + \mathrm{az} = \mathrm{b}$ has infinitely many solutions, then the point (a, b) lies on the line

Let $\mathbf{a}_1,\mathbf{a}_2,\mathbf{a}_3,\ldots$ be an A.P. and $\mathbf{g}_1 = \mathbf{a}_1,\mathbf{g}_2,\mathbf{g}_3,\ldots$ be an increasing G.P. If $\mathbf{a}_1 = \mathbf{a}_2 + \mathbf{g}_2 = 1$ and $\mathbf{a}_3 + \mathbf{g}_3 = 4$ then $\mathbf{a}_{10} + \mathbf{g}_5$ is equal to:

The sum $\frac{1^3}{1} +\frac{1^3 + 2^3}{1 + 3} +\frac{1^3 + 2^3 + 3^3}{1 + 3 + 5} +\ldots$ up to 8 terms, is:

If for $3\leq \mathrm{r}\leq 30$ $\mathrm{^3C_{30 - r}} + 3(\mathrm{^3C_{31 - r}}) + 3(\mathrm{^3C_{32 - r}}) + (\mathrm{^3C_{33 - r}}) = ^{m}C_r$ then m equals :

Let $\mathbf{p}_{\mathrm{n}}$ denote the total number of triangles formed by joining the vertices of an n-side regular polygon. If $\mathbf{p}_{\mathrm{n} + 1} - \mathbf{p}_{\mathrm{n}} = 66$ then the sum of all distinct prime divisors of n is:

A man throws a fair coin repeatedly. He gets 10 points for each head he throws and 5 points for each tail he throws. If the probability that he gets exactly 30 points is $\frac{\mathrm{m}}{\mathrm{n}}$, $\mathrm{gcd}(\mathrm{m},\mathrm{n}) = 1$, then $\mathrm{m} + \mathrm{n}$ is equal to :

The mean and variance of n observations are 8 and 16, respectively. If the sum of the first (n - 1) observations is 48 and the sum of squares of the first (n - 1) observations is 496, then the value of n is:

Let a circle pass through the origin and its centre be the point of intersection of two mutually perpendicular lines $\mathbf{x} + (\mathbf{k} - 1)\mathbf{y} + 3 = 0$ and $2\mathbf{x}+ \mathbf{k}^2\mathbf{y} - 4 = 0$. If the line $\mathbf{x} - \mathbf{y} + 2 = 0$ intersects the circle at the points A and B, then (AB)^2 is equal to :

Let O be the origin, and P and Q be two points on the rectangular hyperbola $\mathbf{x}\mathbf{y} = 12$ such that the mid point of the line segment PQ is $\left(\frac{1}{2}, -\frac{1}{2}\right)$. Then the area of the triangle OPQ equals:

Let the parabola $\mathbf{y} = \mathbf{x}^{2} + \mathbf{px} + \mathbf{q}$ passing through the point $(1, -1)$ be such that the distance between its vertex and the x-axis is minimum. Then the value of $\mathbf{p}^{2} + \mathbf{q}^{2}$ is:

Let $\mathrm{P} = \{\theta \in [0,4\pi ]:\tan^{2}\theta \neq 1\}$ and $\mathrm{S} = \{\mathrm{a}\in \mathrm{Z}:2(\cos^{8}\theta -\sin^{8}\theta)\sec 2\theta = \mathrm{a}^{2},\theta \in \mathrm{P}\}$. Then $\mathrm{n}(\mathrm{S})$ is:

Let the vectors $\vec{\mathrm{a}} = -\hat{\mathrm{i}} +\hat{\mathrm{j}} +3\hat{\mathrm{k}}$ and $\vec{\mathrm{b}} = \hat{\mathrm{i}} +3\hat{\mathrm{j}} +\hat{\mathrm{k}}$. For some $\lambda ,\mu \in \mathbb{R}$ let $\vec{\mathrm{c}} = \lambda \vec{\mathrm{a}} +\mu \vec{\mathrm{b}}$. If $\vec{\mathrm{c}}.\left(3\hat{\mathrm{i}} - 6\hat{\mathrm{j}} +2\hat{\mathrm{k}}\right) = 10$ and $\vec{\mathrm{c}}.\left(\hat{\mathrm{i}} +\hat{\mathrm{j}} +\hat{\mathrm{k}}\right) = -2$, then $|\vec{\mathrm{c}} |^{2}$ is equal to:

Let the point A be the foot of perpendicular drawn from the point P(a, b, 0) on the line $\frac{x - 1}{2} = \frac{y - 2}{1} = \frac{z - \alpha}{3}$. If the midpoint of the line segment PA is $\left(0, \frac{3}{4}, -\frac{1}{4}\right)$, then the value of $\mathrm{a}^2 + \mathrm{b}^2 + \mathrm{\alpha}^2$ is equal to :

Two adjacent sides of a parallelogram PQRS are given by $\overrightarrow{\mathrm{PQ}} = \mathbf{j} + \mathbf{k}$ and $\overrightarrow{\mathrm{PS}} = \mathbf{i} - \mathbf{j}$. If the side PS is rotated about the point P by an acute angle $\alpha$ in the plane of the parallelogram so that it becomes perpendicular to the side PQ, then $\sin^2\left(\frac{5\alpha}{2}\right) - \sin^2\left(\frac{\alpha}{2}\right)$ is equal to :

The value of $\int_{0}^{20\pi}(\sin^{4}x + \cos^{4}x)\mathrm{d}x$ is equal to:

Let $\mathrm{f}(\mathbf{x})$ be a polynomial of degree 5, and have extrema at $\mathbf{x} = 1$ and $\mathbf{x} = -1$. If $\lim_{x\to 0}\left(\frac{\mathrm{f}(\mathbf{x})}{\mathbf{x}^3}\right) = -5$ then $\mathrm{f}(2) - \mathrm{f}(-2)$ is equal to :

Let $\mathrm{f}(\mathbf{x}) = \int \left(\frac{16\mathbf{x} + 24}{\mathbf{x}^2 + 2\mathbf{x} - 15}\right)\mathrm{d}\mathbf{x}$. If $\mathrm{f}(4) = 14\log_e(3)$ and $\mathrm{f}(7) = \log_e(2^{\alpha}.3^{\beta})$, $\alpha ,\beta \in \mathbf{N}$, then $\alpha +\beta$ is equal to:

Let $\mathbf{x} = \mathbf{x}(\mathbf{y})$ be the solution of the differential equation $2\mathbf{y}^2 \frac{\mathrm{d}\mathbf{x}}{\mathrm{d}\mathbf{y}} - 2\mathbf{x}\mathbf{y} + \mathbf{x}^2 = 0$, $\mathbf{y} > 1$, $\mathbf{x}(\mathbf{e}) = \mathbf{e}$. Then $\mathbf{x}(\mathbf{e}^2)$ is equal to:

Let $\mathrm{A} = \{2,3,4,5,6\}$. Let R be a relation on the set $\mathrm{A}\times \mathrm{A}$ given by $(\mathbf{x},\mathbf{y})\mathbf{R}(\mathbf{z},\mathbf{w})$ if and only if $\mathbf{x}$ divides $\mathbf{z}$ and $\mathbf{y}\leq \mathbf{w}$. Then the number of elements in R is

Consider the matrices $\mathrm{A} = \left[ \begin{array}{cc}2 & -2\\ 4 & -2 \end{array} \right]$ and $\mathrm{B} = \left[ \begin{array}{cc}3 & 9\\ 1 & 3 \end{array} \right]$. If matrices $\mathbf{P}$ and $\mathbf{Q}$ are such that $\mathbf{PA} = \mathbf{B}$ and $\mathbf{AQ} = \mathbf{B}$, then the absolute value of the sum of the diagonal elements of $2(\mathbf{P} + \mathbf{Q})$ is

Let A be the point (3, 0) and circles with variable diameter AB touch the circle $\mathbf{x}^{2} + \mathbf{y}^{2} = 36$ internally. Let the curve C be the locus of the point B. If the eccentricity of C is e, then $72\mathrm{e}^{2}$ is equal to.

If the area of the region bounded by $16x^{2} - 9y^{2} = 144$ and $8x - 3y = 24$ is A, then $3(\mathrm{A} + 6\log_{e}(3))$ is equal to

The number of points in the interval [2, 4], at which the function $\mathrm{f(x)} = \left[\mathrm{x}^{2} - \mathrm{x} - \frac{1}{2}\right]$, where [·] denotes the greatest integer function, is discontinuous, is

The ratio of mass percentage $(\mathrm{w} / \mathrm{w})$ of $\mathrm{C}: \mathrm{H}$ in a hydrocarbon is $12: 1$. It has two carbon atoms. The weight (in g ) of $\mathrm{CO}_{2}(\mathrm{~g})$ formed when 3.38 g of this hydrocarbon is completely burnt in oxygen is :(Given : Molar mass in $\mathrm{g} \mathrm{mol}^{-1} \mathrm{C}: 12, \mathrm{H}: 1, \mathrm{O}: 16$ )

The first and second ionization constants of a weak dibasic acid $\mathrm{H}_{2} \mathrm{~A}$ are $8.1 \times 10^{-8}$ and $1.0 \times 10^{-13}$ respectively. 0.1 mol of $\mathrm{H}_{2} \mathrm{~A}$ was dissolved in 1L of 0.1 M HCl solution. The concentration of $\mathrm{HA}^{-}$ in the resultant solution is :

$\mathrm{SF}_{4}$ is isostructural with : A. $\mathrm{BrF}_{4}^{\ominus}$ B. $\mathrm{CH}_{4}$ C. $\mathrm{IF}_{4}^{\oplus}$ D. $\mathrm{XeF}_{4}$ E. $\mathrm{XeO}_{2} \mathrm{~F}_{2}$

Choose the correct answer from the options given below:

Gas 'A' undergoes change from state ' X ' to state ' Y '. In this process, the heat absorbed and work done by the gas is 10 J and 18 J respectively. Now gas is brought back to state ' X ' by another process during which 6 J of heat is evolved. In the reverse process of ' Y ' to ‘ X ’.

Solution A is prepared by dissolving 1 g of a protein (molar mass $=50000 \mathrm{~g} \mathrm{~mol}^{-1}$ ) in 0.5 L of water at 300 K . Its osmotic pressure is x bar. Solution B is made by dissolving 2 g of same protein in 1 L of water at 300 K . Osmotic pressure of solution $B$ is y bar. Entire solution of $A$ is mixed with entire solution of $B$ at same temperature. The osmotic pressure of resultant solution is $z$ bar. $x, y$ and z respectively are :( $\mathrm{R}=0.083 \mathrm{~L}$ bar $\mathrm{mol}^{-1} \mathrm{~K}^{-1}$ )

At $25^{\circ} \mathrm{C}, 20.0 \mathrm{~mL}$ of 0.2 M weak monoprotic acid HX is titrated against 0.2 M NaOH . The pH of the solution (a) at the start of the titration (when NaOH has not been added) and (b) when 10 mL of NaOH is added respectively, are : Given : $\mathrm{K}_{\mathrm{a}}=5 \times 10^{-4}, \mathrm{pK}_{\mathrm{a}}=3.3, \alpha \ll 1$

The correct set that contains all kinds (basic, acidic, amphoteric and neutral) of oxides is:

Given below are two statements :

Statement-I : The second ionization enthalpy of $\mathrm{B}, \mathrm{Al}$ and Ga is in the order of $\mathrm{B}>\mathrm{Al}>\mathrm{Ga}$.

Statement-II : The correct order in terms of first ionization enthalpy is $\mathrm{Si}<\mathrm{Ge}<\mathrm{Pb}<\mathrm{Sn}$.

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

Which of the following complexes will show coordination isomerism?

(A) $\left[\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}\right]\left[\mathrm{Ag}(\mathrm{CN})_{2}\right]$

(B) $\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}\right]\left[\mathrm{Cr}(\mathrm{CN})_{6}\right]$

(C) $\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}\right]\left[\mathrm{Co}(\mathrm{CN})_{6}\right]$

(D) $\left[\mathrm{Fe}\left(\mathrm{NH}_{3}\right)_{6}\right]\left[\mathrm{Co}(\mathrm{CN})_{6}\right]$

(E) $\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{6}\right]\left[\mathrm{Fe}(\mathrm{CN})_{6}\right]$

Choose the correct answer from the options given below :

Complete combustion of X g of an organic compound gave 0.25 g of $\mathrm{CO}_{2}$ and 0.12 g of $\mathrm{H}_{2} \mathrm{O}$. If the % of carbon is $25 \%$ and of hydrogen is $4.89 \%$, then $\mathrm{X}=$ $\_\_\_\_$ $\times 10^{-3} \mathrm{~g}$. (Nearest integer) (Molar mass of $\mathrm{C}, \mathrm{H}$ and O are 12, 1 and $16 \mathrm{~g} \mathrm{~mol}^{-1}$ respectively.)

Given below are two statements :

Statement I: In

the carbocation is stabilised by +R effect of $-\mathrm{OCH}_{3}$ group.

Statement II : In

the carbanion is stabilised by -R effect of $-\mathrm{NO}_{2}$ group.

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

Correct statements regarding alkyl halides ( $\mathrm{R}-\mathrm{X}$ ) among the following are : (A) Alcohol being less polar solvent as compared to water, alcoholic KOH favours elimination reaction with $\mathrm{R}-\mathrm{X}$. (B) Order of reactivity towards $\mathrm{S}_{\mathrm{N}} 1$ mechanism is $\mathrm{C}_{6} \mathrm{H}_{5}-\mathrm{CH}_{2}-\mathrm{Cl}>\mathrm{C}_{6} \mathrm{H}_{5}-\mathrm{CHCl}-\mathrm{C}_{6} \mathrm{H}_{5}$. (C) Non substituted aryl halides exhibit properties similar to alkyl halides. (D) Vinyl chloride is an example of haloalkene and allyl chloride is an example of haloalkyne. (E) $\mathrm{R}-\mathrm{Cl}$ can be prepared by reacting $\mathrm{R}-\mathrm{OH}$ with $\mathrm{SOCl}_{2}$ but $\mathrm{Ar}-\mathrm{Cl}$ cannot be prepared by reacting $\mathrm{Ar}-\mathrm{OH}$ with $\mathrm{SOCl}_{2}$. Choose the correct answer from the options given below:

A molecule ( X ) with following structure under mild acidic condition is hydrolysed to produce (Y) and (Z). Identify the correct statements about (Y) and (Z).

(X) (A) Both (Y) and (Z) have same molar mass (B) (Y) and (Z) can be distinguished from each other by $\mathrm{NaHCO}_{3}$ (C) (Y) and (Z) react with HCN with same rates (D) (Y) and (Z) undergo addition reaction with 2, 4-DNP Choose the correct answer from the options given below :

Identify the correct pair having amino acid (A) and the hormone (B) that is iodinated derivative of the amino acid (A). (T and Y represent one letter code for amino acids)

Among $\mathrm{Fe}^{2+}, \mathrm{Fe}^{3+}, \mathrm{Cr}^{2+}$ and $\mathrm{Zn}^{2+}$, the ion that shows positive borax bead test and with highest ionisation enthalpy is:

The surface of sodium metal is irradiated with radiation of wavelength x nm . The kinetic energy of ejected electrons is $2.8 \times 10^{-20} \mathrm{~J}$. The work function of sodium is 2.3 eV . The value of x is $\_\_\_\_$ $\times 10^{2} \mathrm{~nm}$. (Nearest integer) (Given : $\mathrm{h}=6.6 \times 10^{-34} \mathrm{~J} \mathrm{~s} ; 1 \mathrm{eV}=1.60 \times 10^{-19} \mathrm{~J}$; $\mathrm{c}=3.0 \times 10^{8} \mathrm{~ms}^{-1}$ )

Consider the following gas phase reaction being carried out in a closed vessel at $25^{\circ} \mathrm{C}$. $2 \mathrm{~A}(\mathrm{~g}) \rightarrow 4 \mathrm{~B}(\mathrm{~g})+\mathrm{C}(\mathrm{~g})$

The pressure of $\mathrm{C}(\mathrm{g})$ at 30 minutes time interval would be $\_\_\_\_$ mm Hg. (Nearest integer).

Consider the following two half-cell reactions along with the standard reduction potential given :

The fuel cell was set up using the above two reactions such that the cell operates under the standard condition of 1 bar pressure and 298 K temperature. The fuel cell works with $80 \%$ efficiency. If the work derived from the cell using 1 mol of $\mathrm{CH}_{3} \mathrm{OH}$ is used to compress an ideal gas isothermally against a constant pressure of 1 kPa , then the change in the volume of the gas, $\Delta \mathrm{V}=$ $\_\_\_\_$ $\mathrm{m}^{3}$. (Nearest integer) Given : $\mathrm{F}=96500 \mathrm{C} \mathrm{mol}^{-1}$

Number of paramagnetic ions among the following d- and f-block metal ions is $\_\_\_\_$ . $\mathrm{Mn}^{2+}, \mathrm{Cu}^{2+}, \mathrm{Zn}^{2+}, \mathrm{Yb}^{2+}, \mathrm{Sc}^{3+}, \mathrm{La}^{3+}, \mathrm{Gd}^{3+}, \mathrm{Lu}^{3+}, \mathrm{Ti}^{4+}$, $\mathrm{Ce}^{4+}$ (Atomic number of $\mathrm{Mn}=25, \mathrm{Cu}=29, \mathrm{Zn}=30$, $\mathrm{Yb}=70, \mathrm{Sc}=21, \mathrm{La}=57, \mathrm{Gd}=64, \mathrm{Lu}=71$, $\mathrm{Ti}=22, \mathrm{Ce}=58$ )

Consider the following reactions sequence When the product $(\mathrm{P})$ is subjected to Carius analysis using $\mathrm{AgNO}_{3}, 1.0 \mathrm{~g}$ of the product $(\mathrm{P})$ will produce $\_\_\_\_$ g of the precipitate of AgBr . (Nearest interger) (Given : molar mass in $\mathrm{g} \mathrm{mol}^{-1} \mathrm{C}: 12, \mathrm{H}: 1$, $\mathrm{O}: 16, \mathrm{~N}: 14, \mathrm{Br}: 80, \mathrm{Ag}: 108)$