JEE Main 2026 — 28 January, Evening Shift

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

Statement I : $25^{13}+20^{13}+8^{13}+3^{13}$ is divisible by 7 .

Statement II : The integral part of $(7+4 \sqrt{3})^{25}$ is an odd number.

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

The sum of the coefficients of $x^{499}$ and $x^{500}$ in $(1+x)^{1000}+x(1+x)^{999}+x^{2}(1+x)^{998}+\ldots \ldots+x^{1000}$ is

Let A be the focus of the parabola $\mathrm{y}^{2}=8 \mathrm{x}$. Let the line $\mathrm{y}=\mathrm{mx}+\mathrm{c}$ intersect the parabola at two distinct points B and C . If the centroid of the triangle ABC is $\left(\frac{7}{3}, \frac{4}{3}\right)$, then $(\mathrm{BC})^{2}$ is equal to :

The probability distribution of a random variable X is given below :

If $\mathrm{E}(\mathrm{X})=\frac{263}{15}$, then $\mathrm{P}(\mathrm{X}<20)$ is equal to :

5. Let $f(x)=\lim _{\theta \rightarrow 0}\left(\frac{\cos \pi x-x^{\left(\frac{2}{\theta}\right)} \sin (x-1)}{1+x^{\left(\frac{2}{\theta}\right)}(x-1)}\right), x \in R$. Consider the following two statements :

(I) $\mathrm{f}(\mathrm{x})$ is discontinous at $\mathrm{x}=1$.

(II) $\mathrm{f}(\mathrm{x})$ is continous at $\mathrm{x}=-1$. Then :

Considering the principal values of inverse trigonometric functions, the value of the expression $\tan \left(2 \sin ^{-1}\left(\frac{2}{\sqrt{13}}\right)-2 \cos ^{-1}\left(\frac{3}{\sqrt{10}}\right)\right)$ is equal to :

Let the arithmetic mean of $\frac{1}{\mathrm{a}}$ and $\frac{1}{\mathrm{~b}}$ be $\frac{5}{16}, \mathrm{a}>2$. If $\alpha$ is such that $\mathrm{a}, 4, \alpha, \mathrm{~b}$ are in A.P., then the equation $\alpha x^{2}-a x+2(\alpha-2 b)=0$ has :

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

Statement I : The function $\mathrm{f}: \mathbf{R} \rightarrow \mathbf{R}$ defined by $\mathrm{f}(\mathrm{x})=\frac{\mathrm{X}}{1+|\mathrm{x}|}$ is one-one.

Statement II : The function $\mathrm{f}: \mathbf{R} \rightarrow \mathbf{R}$ defined by $f(x)=\frac{x^{2}+4 x-30}{x^{2}-8 x+18}$ is many-one.

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

An ellipse has its center at $(1,-2)$, one focus at $(3,-2)$ and one vertex at (5,-2). Then the length of its latus rectum is :

Let the ellipse $\mathrm{E}: \frac{\mathrm{x}^{2}}{144}+\frac{\mathrm{y}^{2}}{169}=1$ and the hyperbola $\mathrm{H}: \frac{\mathrm{x}^{2}}{16}-\frac{\mathrm{y}^{2}}{\lambda^{2}}=-1$ have the same foci. If e and L respectively denote the eccentricity and the length of the latus rectum of H , then the value of $24(\mathrm{e}+\mathrm{L})$ is :

Let $P_{1}: y=4 x^{2}$ and $P_{2}: y=x^{2}+27$ be two parabolas. If the area of the bounded region enclosed between $P_{1}$ and $P_{2}$ is six times the area of the bounded region enclosed between the line $\mathrm{y}=\alpha \mathrm{x}, \alpha>0$ and $\mathrm{P}_{1}$, then $\alpha$ is equal to :

Let the circle $x^{2}+y^{2}=4$ intersect $x$-axis at the points $\mathrm{A}(\mathrm{a}, 0), \mathrm{a}>0$ and $\mathrm{B}(\mathrm{b}, 0)$. Let $\mathrm{P}(2 \cos \alpha$, $2 \sin \alpha), 0<\alpha<\frac{\pi}{2}$ and $\mathrm{Q}(2 \cos \beta, 2 \sin \beta)$ be two points such that $(\alpha-\beta)=\frac{\pi}{2}$. Then the point of intersection of AQ and BP lies on :

Let [] denote the greatest integer function. Then $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{12(3+[x])}{3+[\sin x]+[\cos x]}\right) d x$ is equal to:

Let $\mathrm{y}=\mathrm{y}(\mathrm{x})$ be the solution of the differential equation $x \frac{d y}{d x}-y=x^{2} \cot x, x \in(0, \pi)$. Ify $\left(\frac{\pi}{2}\right)=\frac{\pi}{2}$, then $6 y\left(\frac{\pi}{6}\right)-8 y\left(\frac{\pi}{4}\right)$ is equal to :

The sum of all the elements in the range of $f(x)=\operatorname{Sgn}(\sin x)+\operatorname{Sgn}(\cos x)+\operatorname{Sgn}(\tan x)+\operatorname{Sgn}(\cot x)$, $\mathrm{x} \neq \frac{\mathrm{n} \pi}{2}, \mathrm{n} \in \mathbf{Z}$, where $\operatorname{Sgn}(t)=\left\{\begin{array}{lll}1, & \text { if } & t>0 \\ -1 & \text { if } & t<0\end{array}\right.$, is

Let $\mathrm{Q}(\mathrm{a}, \mathrm{b}, \mathrm{c})$ be the image of the point $\mathrm{P}(3,2,1)$ in the line $\frac{x-1}{1}=\frac{y}{2}=\frac{z-1}{1}$. Then the distance of $Q$ from the line $\frac{x-9}{3}=\frac{y-9}{2}=\frac{z-5}{-2}$ is

Let P be a point in the plane of the vector $\overrightarrow{\mathrm{AB}}=3 \hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{AC}}=\hat{\mathrm{i}}-\hat{\mathrm{j}}+3 \hat{\mathrm{k}}$ such that P is equidistant from the lines AB and AC . If $|\overrightarrow{\mathrm{AP}}|=\frac{\sqrt{5}}{2}$, then the area of the triangle ABP is :

Let $A=\{z \in \mathbb{C}:|z-2| \leq 4\}$ and $B=\{z \in \mathbb{C}:|z-2|+|z+2|=5\}$. Then the max $\left\{\left|z_{1}-z_{2}\right|: z_{1} \in A\right.$ and $\left.z_{2} \in B\right\}$ is

Let $\mathrm{f}(\mathrm{x})=\int \frac{\mathrm{dx}}{\mathrm{x}^{\left(\frac{2}{3}\right)}+2 \mathrm{x}^{\left(\frac{1}{2}\right)}}$ be such that $\mathrm{f}(0)=-26+24 \log _{\mathrm{e}}(2)$. If $\mathrm{f}(1)=\mathrm{a}+\mathrm{b} \log _{\mathrm{e}}$, where $\mathrm{a}, \mathrm{b} \in \mathbf{Z}$, then $\mathrm{a}+\mathrm{b}$ is equal to:

$\frac{6}{3^{26}}+\frac{10.1}{3^{25}}+\frac{10.2}{3^{24}}+\frac{10.2^{2}}{3^{23}}+\ldots+\frac{10.2^{24}}{3}$ is equal to

$\sum_{\mathrm{r}=1}^{25}\left(\frac{\mathrm{r}}{\mathrm{r}^{4}+\mathrm{r}^{2}+1}\right)=\frac{\mathrm{p}}{\mathrm{q}}$, where p and q are positive integers such that $\operatorname{gcd}(p, q)=1$, then $p+q$ is equal to $\_\_\_\_$ .

Three persons enter in a lift at the ground floor. The lift will go upto $10^{\text {th }}$ floor. The number of ways, in which the three persons can exit the lift at three different floors, if the lift does not stop at first, second and third floors, is equal to $\_\_\_\_$

Let f be a differentiable function satisfying $\mathrm{f}(\mathrm{x})=1-2 \mathrm{x}+\int_{0}^{\mathrm{x}} \mathrm{e}^{(\mathrm{x}-\mathrm{t})} \mathrm{f}(\mathrm{t}) \mathrm{dt}, \mathrm{x} \in \mathbf{R}$ and let $g(\mathrm{x})=\int_{0}^{\mathrm{x}}(\mathrm{f}(\mathrm{t})+2)^{15}(\mathrm{t}-4)^{6}(\mathrm{t}+12)^{17} \mathrm{dt}, \mathrm{x} \in \mathbf{R}$. If $p$ and $q$ are respectively the points of local minima and local maxima of g , then the value of $|p+q|$ is equal to $\_\_\_\_$ .

If the distance of the point $\mathrm{P}(43, \alpha, \beta), \beta<0$, from the line $\overrightarrow{\mathrm{r}}=4 \hat{\mathrm{i}}-\hat{\mathrm{k}}+\mu(2 \hat{\mathrm{i}}+3 \hat{\mathrm{k}}), \mu \in \mathbf{R}$ along a line with direction ratios $3,-1,0$ is $13 \sqrt{10}$, then $\alpha^{2}+\beta^{2}$ is equal to $\_\_\_\_$ .

Let $A=\left[\begin{array}{ll}3 & -4\\ 1 & -1\end{array}\right]$ and $B$ be two matrices such that $A^{100}=100 B+I$. Then the sum of all the elements of $B^{100}$ is $\_\_\_\_$ .

A nucleus has mass number $\alpha$ and radius $R_{\alpha}$. Another nucleus has mass number $\beta$ and radius $\mathrm{R}_{\beta}$. If $\beta=8 \alpha$ then $R_{\alpha} / R_{\beta}$ is :

A plane electromagnetic wave is moving in free space with velocity $\mathrm{c}=3 \times 10^{8} \mathrm{~m} / \mathrm{s}$ and its electric field is given as $\overrightarrow{\mathrm{E}}=54 \sin (\mathrm{kz}-\omega \mathrm{t}) \hat{\mathrm{j}} \mathrm{V} / \mathrm{m}$, where $\hat{j}$ is the unit vector along y-axis. The magnetic field vector $\overrightarrow{\mathrm{B}}$ of the wave is :

A biconvex lens is formed by using two thin planoconvex lenses, as shown in the figure. The refractive index and radius of curved surfaces are also mentioned in figure. When an object is placed on the left side of lens at a distance of 30 cm from the biconvex lens, the magnification of the image will be :

The mean free path of a molecule of diameter $5 \times 10^{-10} \mathrm{~m}$ at the temperature $41^{\circ} \mathrm{C}$ and pressure $1.38 \times 10^{5} \mathrm{~Pa}$, is given as m. (Given $\mathrm{k}_{\mathrm{B}}=1.38 \times 10^{-23} \mathrm{~J} / \mathrm{K}$ ).

Two p-n junction diodes $\mathrm{D}_{1}$ and $\mathrm{D}_{2}$ are connected as shown in figure. A and B are input signals and C is the output. The given circuit will function as a $\_\_\_\_$ .

A wheatstone bridge is initially at room temperature and all arms of the bridge have same value of resistances ( $\mathrm{R}_{1}=\mathrm{R}_{2}=\mathrm{R}_{3}=\mathrm{R}_{4}$ ). When $\mathrm{R}_{3}$ resistance is heated to some temperature, its resistance value has gone up by $10 \%$. The potential difference $\left(\mathrm{V}_{\mathrm{a}}-\mathrm{V}_{\mathrm{b}}\right)$ (after $\mathrm{R}_{3}$ is heated) is $\_\_\_\_$ V.

In an experiment, a set of reading are obtained $-1.24 \mathrm{~mm}, 1.25 \mathrm{~mm}, 1.23 \mathrm{~mm}, 1.21 \mathrm{~mm}$. The expected least count of the instrument used in recording these readings is $\_\_\_\_$ mm.

A particle starts moving from time $\mathrm{t}=0$ and its coordinate is given as $\mathrm{x}(\mathrm{t})=4 \mathrm{t}^{3}-3 \mathrm{t}$.

A. The particle returns to its original position (origin) 0.866 units later

B. The particle is 1 unit away from origin at its turning point.

C. Acceleration of the particle is non-negative.

D. The particle is 0.5 units away from origin at its turning point.

E. Particle never turns back as acceleration is non-negative.

Choose the correct answer from the options given below:

The speed of a longitudinal wave in a metallic bar is $400 \mathrm{~m} / \mathrm{s}$. If the density and Young's modulus of the bar material are increased by $0.5 \%$ and $1 \%$ respectively then the speed of the wave is changed approximately to $\_\_\_\_$ $\mathrm{m} / \mathrm{s}$.

Identify the correct statements :

A. Effective capacitance of a series combination of capacitors is always smaller than the smallest capacitance of the capacitor in the combination.

B. When a dielectric medium is placed between the charged plates of a capacitor, displacement of charges cannot occur due to insulation property of dielectric.

C. Increasing of area of capacitor plate or decreasing of thickness of dielectric is an alternate method to increase the capacitance.

D. For a point charge, concentric spherical shells cantered at the location of the charge are equipotential surfaces.

Choose the correct answer from the options given below.

Number of photons of equal energy emitted per second by a 6 mW laser source operating at 663 nm is $\_\_\_\_$ . (Given : $\mathrm{h}=6.63 \times 10^{-34}$ J.s and $\mathrm{c}=3 \times 10^{8} \mathrm{~m} / \mathrm{s}$ )

When the position vector $\vec{r}=x \hat{i}+y \hat{j}+z \hat{k}$ changes sign as $-\overrightarrow{\mathrm{r}}$, which one of the following vector will not flip under sign change ?

Which one of the following is not a measurable quantity?

A long cylindrical conductor with large cross section carries an electric current distributed uniformly over its cross-section. Magnetic field due to this current is :

A. maximum at either ends of the conductor and minimum at the midpoint

B. maximum at the axis of the conductor

C. minimum at the surface of the conductor D. minimum at the axis of the conductor

E. same at all points in the cross-section of the conductor

Choose the correct answer from the options given below :

A small block of mass $m$ slides down from the top of a frictionless inclined surface, while the inclined plane is moving towards left with constant acceleration $\mathrm{a}_{0}$. The angle between the inclined plane and ground is $\theta$ and its base length is L . Assuming that initially the small block is at the top of the inclined plane, the time it takes to reach the lowest point of the inclined plane is $\_\_\_\_$ .

Identify the correct statements : A. Electrostatic field lines form closed loops. B. The electric field lines point radially outward when charge is greater than zero. C. The Gauss-Law is valid only for inverse-square force. D. The workdone in moving a charged particle in a static electric field around a closed path is zero. E. The motion of a particle under Coulomb's force must take place in a plane. Choose the correct answer from the options given below:

As shown in the figure, a spring is kept in a stretched position with some extension by holding the masses 1 kg and 0.2 kg with a separation more than spring natural length and are released. Assuming the horizontal surface to be frictionless, the angular frequency (in SI unit) of the system is :

For a transparent prism, if the angle of minimum deviation is equal to its refracting angle, the refractive index $n$ of the prism satisfies.

The time period of a simple harmonic oscillator is $\mathrm{T}=2 \pi \sqrt{\frac{\mathrm{k}}{\mathrm{m}}}$. The measured value of mass (m) of the object is 10 g with an accuracy of 10 mg , and time for 50 oscillations of the spring is found to be 60 s using a watch of 2 s resolution. Percentage error in determination of spring constant(k) is $\_\_\_\_$ %.

Match List-I with List-II.

Choose the correct answer from the options given below:

Two tuning forks A and B are sounded together giving rise to 8 beats in 2 s . When fork A is loaded with wax, the beat frequency is reduced to 4 beats in 2 s . If the original frequency of tuning fork B is 380 Hz , then the original frequency of tuning fork A is $\_\_\_\_$ Hz.

A thermodynamic system is taken through the cyclic process ABC as shown in the figure. The total work done by the system during the cycle ABC is $\_\_\_\_$ J.

An inductor stores 16 J of magnetic field energy and dissipates 32 W of thermal energy due to its resistance when an a.c. current of 2 A (rms) and frequency 50 Hz flows through it. The ratio of inductive reactance to its resistance is $\_\_\_\_$ . ( $\pi=3.14$ )

A beam of light consisting of wavelengths 650 nm and 550 nm illuminates the Young's double slits with separation of 2 mm such that the interference fringes are formed on a screen, placed at a distance of 1.2 m from the slits. The least distance of a point from the central maximum, where the bright fringes due to both the wavelengths coincide, is $\_\_\_\_$ $\times 10^{-5} \mathrm{~m}$.

A fly wheel having mass 3 kg and radius 5 m is free to rotate about a horizontal axis. A string having negligible mass is wound around the wheel and the loose end of the string is connected to a 3 kg mass. The mass is kept at rest initially and released. Kinetic energy of the wheel when the mass descends by 3 m is $\_\_\_\_$ J. $\left(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}\right)$

Identify the correct statements :

The presence of $-\mathrm{NO}_{2}$ group in benzene ring

A. activates the ring towards electrophilic substitutions.

B. deactivates the ring towards electrophilic substitutions.

C. activates the ring towards nucleophilic substitutions.

D. deactivates the ring towards nucleophilic substitutions.

Consider the elements $\mathrm{N}, \mathrm{P}, \mathrm{O}, \mathrm{S}, \mathrm{Cl}$ and F . The number of valence electrons present in the elements with most and least metallic character from the above list is respectively.

Observe the following equilibrium in a 1 L flask.

At $\mathrm{T}(\mathrm{K})$, the equilibrium concentrations of A and B are 0.5 M and 0.375 M respectively. 0.1 moles of A is added into the flask and heated to $\mathrm{T}(\mathrm{K})$ to establish the equilibrium again. The new equilibrium concentrations (in M ) of A and B are respectively.

The plot of $\log _{10} \mathrm{~K}$ vs $\frac{1}{\mathrm{~T}}$ gives a straight line. The intercept and slope respectively are (where K is equilibrium constant).

The reactions which produce alcohol as the product area :

A. $\mathrm{CH}_{4}+\mathrm{O}_{2} \xrightarrow[\Delta]{\mathrm{Mo}_{2} \mathrm{O}_{3}}$

B. $2 \mathrm{CH}_{3} \mathrm{CH}_{3}+3 \mathrm{O}_{2} \xrightarrow[\Delta]{\left(\mathrm{CH}_{3} \mathrm{COO}\right)_{2} \mathrm{Mn}}$

C. $\left(\mathrm{CH}_{3}\right)_{3} \mathrm{CH} \xrightarrow{\mathrm{KMnO}_{4}}$

D. $2 \mathrm{CH}_{4}+\mathrm{O}_{2} \xrightarrow{\mathrm{Cu} / 523 \mathrm{~K} / 100 \mathrm{~atm} .}$

E. $\mathrm{CH}_{3}-\mathrm{CH}=\mathrm{CH}-\mathrm{CH}_{3} \xrightarrow{\mathrm{KMnO}_{4} / \mathrm{H}^{+}}$

Choose the correct answer from the options given below :

Consider the following statements about manganate and permanganate ions. Identify the correct statements :

A. The geometry of both manganate and permanganate ions is tetrahedral.

B. The oxidation states of Mn in manganate and permanganate are +7 and +6 , respectively.

C. Oxidation of Mn (II) salt by peroxodisulphate gives manganate ion as the final product.

D. Manganate ion is paramagnetic and permanganate ions is diamagnetic.

E. Acidified permanganate ion reduces oxalate, nitrite and iodide ions.

Choose the correct answer from the options given below:

The wavelength of photon A is $400\,\mathrm{nm}$. The frequency of photon B is $10^{16}\,\mathrm{s^{-1}}$. The wave number of photon C is $10^{4}\,\mathrm{cm^{-1}}$.

The correct order of energy of these photons is:

Match List-I with List-II according to shape.

Choose the correct answer from the options given below :

The correct order of acidic strength of the major products formed in the given reactions, is :

A. $\mathrm{PhNH}_{2} \xrightarrow[\substack{\text { (2) } \mathrm{CuCN}\; \text { (3) } \mathrm{H}_{3} \mathrm{O}^{+} / \Delta}]{\text { (1) } \mathrm{NaNO}_{2}+\mathrm{HCl}\left(<5^{\circ} \mathrm{C}\right)}[\mathrm{A}]$

B. $\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{CHO} \xrightarrow[\Delta]{\left[\mathrm{Ag}\left(\mathrm{NH}_{3}\right)_{2}\right]^{+}, \mathrm{OH}^{-}}[\mathrm{B}]$

C. $\mathrm{CH}_{4}+\mathrm{O}_{2} \xrightarrow[\text { (ii) } \mathrm{Na}_{2} \mathrm{Cr}_{2} \mathrm{O}_{7} / \mathrm{H}^{+}]{\text {(i) } \mathrm{Mo}_{2} \mathrm{O}_{3}}[\mathrm{C}]$

D. $\mathrm{PhCH}_{2} \mathrm{MgBr}+\mathrm{CO}_{2} \xrightarrow[\mathrm{H}_{3} \mathrm{O}^{+}]{\text {Dryether }}[\mathrm{D}]$

Choose the correct answer from the options given below :

Total number of alkali insoluble solid sulphonamides obtained by reaction of given amines with Hinsberg's reagent is Aniline, N-Methylaniline, Methanamine, $\mathrm{N}, \mathrm{N}$-Dimethylmethanamine,

N-Methyl methanamine, Phenylmethanamine, N-propylaniline, N-phenylaniline, N, N-Dimethylaniline, Allyl amine, Isopropyl amine

Consider the following reactions.

$\mathrm{Na}_{2} \mathrm{~B}_{4} \mathrm{O}_{7} \xrightarrow{\Delta} 2 \mathrm{X}+\mathrm{Y}$

$\mathrm{CuSO}_{4}+\mathrm{Y} \xrightarrow{\text { Non-Luminous\; flame }} \mathrm{Z}+\mathrm{SO}_{3}$

$2 \mathrm{Z}+2 \mathrm{X}+$ Carbon $\xrightarrow{\text { Luminous\; flame }} 2 \mathrm{Q}+\mathrm{Na}_{2} \mathrm{~B}_{4} \mathrm{O}_{7}+\mathrm{CO}$

The oxidation states of Cu in Z and Q , respectively are :

For the reaction, $\mathrm{CaCO_3 + 2HCl \rightarrow CaCl_2 + H_2O + CO_2}$. If $90\,\mathrm{g}$ of $\mathrm{CaCO_3}$ is added to $300\,\mathrm{mL}$ of $\mathrm{HCl}$ which contains $38.55\%$ $\mathrm{HCl}$ by mass and has density $1.13\,\mathrm{g\,mL^{-1}}$, then which of the following option is correct?

(Given molar masses: $\mathrm{H}=1$, $\mathrm{Cl}=35.5$, $\mathrm{Ca}=40$, $\mathrm{O}=16\,\mathrm{g\,mol^{-1}}$)

The correct increasing order of spin-only magnetic moment values of the complex ions $\left[\mathrm{MnBr}_{4}\right]^{2-}(\mathrm{A}),\left[\mathrm{Cu}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+}(\mathrm{B}),\left[\mathrm{Ni}\left(\mathrm{CN}_{4}\right)\right]^{2-} (\mathrm{C})$ and $\left[\mathrm{Ni}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+}(\mathrm{D})$ is:

A student has been given 0.314 g of an organic compound and asked to estimate Sulphur. During the experiment, the student has obtained 0.4813 g of barium sulphate. The percentage of sulphur present in the compound is $\_\_\_\_$。 (Given Molar mass in $\mathrm{g} \mathrm{mol}^{-1} \mathrm{~S}: 32, \mathrm{BaSO}_{4}: 233$ )

Two positively charged particles $\mathrm{m}_{1}$ and $\mathrm{m}_{2}$ have been accelerated across the same potential difference of 200 keV as shown below. [Given mass of $\mathrm{m}_{1}=1$ amu and $\mathrm{m}_{2}=4$ amu]. The de Broglie wavelength of $\mathrm{m}_{1}$ will be $x$ times of $\mathrm{m}_{2}$. The value of $x$ is $\_\_\_\_$ . (nearest integer)

$\mathrm{A} \rightarrow \mathrm{B}$ (first reaction)

$\mathrm{C} \rightarrow \mathrm{D}$ (second reaction)

Consider the above two first-order reactions. The rate constant for first reaction at 500 K is double of the same at 300 K . At $500 \mathrm{~K}, 50 \%$ of the reaction becomes complete in 2 hour. The activation energy of the second reaction is half of that of first reaction. If the rate constant at 500 K of the second reaction becomes double of the rate constant of first reaction at the same temperature; then rate constant for the second reaction at 300 K is $\_\_\_\_$ $\times 10^{-1}$ hour $^{-1}$ (nearest integer).

For strong electrolyte $\Lambda_{\mathrm{m}}$ increases slowly with dilution and can be represented by the equation $\Lambda_{\mathrm{m}}=\Lambda^{\circ}{ }_{\mathrm{m}}-\mathrm{Ac}^{1 / 2}$ Molar conductivity values of the solutions of strong electrolyte AB at $18^{\circ} \mathrm{C}$ are given below :

The value of constant A based on the above data [ in $\mathrm{S} \mathrm{cm}^{2} \mathrm{~mol}^{-1} /(\mathrm{mol} / \mathrm{L})^{1 / 2}$ ] unit is $\_\_\_\_$ .

A volume of x mL of $5 \mathrm{M} \mathrm{NaHCO}_{3}$ solution was mixed with 10 mL of $2 \mathrm{M} \mathrm{H}_{2} \mathrm{CO}_{3}$ solution to make an electrolytic buffer. If the same buffer was used in the following electrochemical cell to record a cell potential of 235.3 mV , then the value of $\mathrm{x}=$ $\_\_\_\_$ mL (nearest integer). $\mathrm{Sn}(\mathrm{s})\left|\mathrm{Sn}(\mathrm{OH})_{6}^{2-}(0.5 \mathrm{M})\right| \mathrm{HSnO}_{2}^{-}(0.05 \mathrm{M}) \mid \mathrm{OH}^{-} \left|\mathrm{Bi}_{2} \mathrm{O}_{3}(\mathrm{~s})\right| \operatorname{Bi}(\mathrm{s})$ Consider upto one place of decimal for intermediate calculations $\left[\begin{array}{ll}\text { Given : } & \mathrm{E}_{\mathrm{HSnO}_{2}^{-} \mid \mathrm{Sn}(\mathrm{OH})_{6}^{2-}}^{\mathrm{o}}=-0.9 \mathrm{~V} \\& \mathrm{E}_{\mathrm{Bi}_{2} \mathrm{O}_{3} \mid \mathrm{Bi}}^{\mathrm{o}}=-0.44 \mathrm{~V} \\& \mathrm{pKa}_{\left(\mathrm{H}_{2} \mathrm{CO}_{3}\right)}=6.11 \\& \frac{2.303 \mathrm{RT}}{\mathrm{F}}=0.059 \mathrm{~V} \\& \text { Anti } \log (1.29)=19.5\end{array}\right]$

The number of isoelectronic species among $\mathrm{Sc}^{3+}$, $\mathrm{Cr}^{2+}, \mathrm{Mn}^{3+}, \mathrm{Co}^{3+}$ and $\mathrm{Fe}^{3+}$ is ' n '. If ' n ' moles of AgCl is formed during the reaction of complex with formula $\mathrm{CoCl}_{3}(\mathrm{en})_{2} \mathrm{NH}_{3}$ with excess of $\mathrm{AgNO}_{3}$ solution, then the number of electrons present in the $t_{2 g}$ orbital of the complex is $\_\_\_\_$。