JEE Main 2026 — 22 January, Morning Shift

Let $\overrightarrow{\mathrm{AB}}=2 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}-5 \mathrm{k}$ and $\overrightarrow{\mathrm{AD}}=\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\lambda \mathrm{k}, \lambda \in \mathbb{R}$. Let the projection of the vector $\vec{v}=\hat{i}+\hat{j}+\hat{k}$ on the diagonal $\overrightarrow{\mathrm{AC}}$ of the parallelogram ABCD be of length one unit. If $\alpha, \beta$, where $\alpha>\beta$, be the roots of the equation $\lambda^{2} \mathrm{x}^{2}- 6 \lambda \mathrm{x}+5=0$, then $2 \alpha-\beta$ is equal to

Let the relation R on the set $\mathrm{M}=\{1,2,3, \ldots \ldots .16\}$ be given by $\mathrm{R}=\{(\mathrm{x}, \mathrm{y}): 4 \mathrm{y}=5 \mathrm{x}-3, \mathrm{x}, \mathrm{y} \in \mathrm{M}\}$. Then the minimum number of elements required to be added in R , in order to make the relation symmetric, is equal to

Let the line $x=-1$ divide the area of the region $\left\{(x, y): 1+x^{2} \leq y \leq 3-x\right\}$ in the ratio $m: n, g c d (\mathrm{m}, \mathrm{n})=1$. Then $\mathrm{m}+\mathrm{n}$ is equal to

Two distinct numbers a and b are selected at random from $1,2,3, \ldots \ldots, 50$. The probability, that their product ab is divisible by 3 , is

Let $f(x)=x^{2025}-x^{2000}, x \in[0,1]$ and the minimum value of the function $f(x)$ in the interval $[0,1]$ be $(80)^{80}(\mathrm{n})^{-81}$. Then n is equal to :

Let $\mathrm{P}(\alpha, \beta, \gamma)$ be the point on the line $\frac{x-1}{2}=\frac{y+1}{-3}=z$ at a distance $4 \sqrt{14}$ from the point $(1,-1,0)$ and nearer to the origin. Then the shortest distance, between the lines $\frac{x-\alpha}{1}=\frac{y-\beta}{2}=\frac{z-\gamma}{3}$ and $\frac{x+5}{2}=\frac{y-10}{1}=\frac{z-3}{1}$ , is equal to

If a random variable x has the probability distribution

then $\mathrm{P}(3<\mathrm{x} \leq 6)$ is equal to

The number of distinct real solutions of the equation $\mathrm{x}|\mathrm{x}+4|+3|\mathrm{x}+2|+10=0$ is

Let $f:[1, \infty) \rightarrow \mathbb{R}$ be a differentiable function, If $6 \int_{1}^{x} f(t) d t=3 x f(x)+x^{3}-4$ for all $\mathrm{x} \geq 1$, then the value of $f(2)-f(1)$ is

If the line $\alpha x+2 y=1$, where $\alpha \in \mathbb{R}$, does not meet the hyperbola $x^{2}-9 y^{2}=9$, then a possible value of $\alpha$ is :

If the image of the point $\mathrm{P}(1,2, a)$ in the line $\frac{\mathrm{x}-6}{3}=\frac{\mathrm{y}-7}{2}=\frac{7-\mathrm{z}}{2}$ is $\mathrm{Q}(5, \mathrm{~b}, \mathrm{c})$, then $\mathrm{a}^{2}+\mathrm{b}^{2}+ \mathrm{c}^{2}$ is equal to

Let the set of all values of $r$, for which the circles $(x+1)^{2}+(y+4)^{2}=r^{2}$ and $x^{2}+y^{2}-4 x-2 y-4=0$ intersect at two distinct points be the interval ( $\alpha$, $\beta$ ). Then $\alpha \beta$ is equal to

If $\mathrm{A}=\left[\begin{array}{ll}2 & 3\\ 3 & 5\end{array}\right]$, then the determinant of the matrix ( $\mathrm{A}^{2025}-3 \mathrm{~A}^{2024}+\mathrm{A}^{2023}$ ) is

The value of $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\left(\frac{1}{[x]+4}\right) d x$, where $[\bullet]$ denotes the greatest integer function, is

The coefficient of $x^{48}$ in $(1+x)+2(1+x)^{2}+3(1 +\mathrm{x})^{3}+\ldots .+100(1+\mathrm{x})^{100}$ is equal to :

If the chord joining the points $\mathrm{P}_{1}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)$ and $\mathrm{P}_{2}\left(\mathrm{x}_{2}\right.$, $y_{2}$ ) on the parabola $y^{2}=12 x$ subtends a right angle at the vertex of the parabola, then $\mathrm{x}_{1} \mathrm{x}_{2}-\mathrm{y}_{1} \mathrm{y}_{2}$ is equal to

The number of solutions of $\tan ^{-1} 4 x+\tan ^{-1} 6 x=\frac{\pi}{6}$, where $-\frac{1}{2\sqrt{6}} < x < \frac{1}{2\sqrt{6}}$ is equal to

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

If the sum of the first four terms of an A.P. is $6$ and the sum of its first six terms is $4$, then the sum of its first twelve terms is

Let $\alpha=\frac{-1+i \sqrt{3}}{2}$ and $\beta=\frac{-1-i \sqrt{3}}{2}, i=\sqrt{-1}$. If $(7-7 \alpha+9 \beta)^{20}+(9+7 \alpha-7 \beta)^{20}+(-7+9 \alpha+7 \beta)^{20} +(14+7 \alpha+7 \beta)^{20}=\mathrm{m}^{10}$, then m is $\_\_\_\_$ .

Let A be a $3 \times 3$ matrix such that $\mathrm{A}+\mathrm{A}^{\mathrm{T}}=\mathrm{O}$. If A $\mathrm{A}\left[\begin{array}{c}1 \\-1 \\0\end{array}\right]=\left[\begin{array}{l}3\\ 3\\ 2\end{array}\right], \mathrm{A}^{2}\left[\begin{array}{c}1 \\-1 \\0\end{array}\right]=\left[\begin{array}{c}-3 \\19 \\-24\end{array}\right]$ and $\operatorname{det}(\operatorname{adj}(2 \mathrm{adj} (\mathrm{A}+\mathrm{I})))=(2)^{\alpha} \cdot^{\beta} \cdot(11)^{\gamma}, \alpha, \beta, \gamma$ are non-negative integers, then $\alpha+\beta+\gamma$ is equal to $\_\_\_\_$

If $\int(\sin \mathrm{x})^{\frac{-11}{2}}(\cos \mathrm{x})^{\frac{-5}{2}} \mathrm{dx}=$ $-\frac{\mathrm{p}_{1}}{\mathrm{q}_{1}}(\cot \mathrm{x})^{\frac{9}{2}}-\frac{\mathrm{p}_{2}}{\mathrm{q}_{2}}(\cot \mathrm{x})^{\frac{5}{2}}-\frac{\mathrm{p}_{3}}{\mathrm{q}_{3}}(\cot \mathrm{x})^{\frac{1}{2}}+\frac{\mathrm{p}_{4}}{\mathrm{q}_{4}}(\cot \mathrm{x})^{\frac{-3}{2}}+\mathrm{C}$, where $p_{i}$ and $q_{i}$ are positive integers with $\operatorname{gcd}\left(p_{i}, q_{i}\right) =1$ for $\mathrm{i}=1,2,3,4$ and C is the constant of integration, then $\frac{15 p_{1} p_{2} p_{3} p_{4}}{q_{1} q_{2} q_{3} q_{4}}$ is equal to $\_\_\_\_$ .

If $\frac{\cos ^{2} 48^{\circ}-\sin ^{2} 12^{\circ}}{\sin ^{2} 24^{\circ}-\sin ^{2} 6^{\circ}}=\frac{\alpha+\beta \sqrt{5}}{2}$, where $\alpha, \beta \in \mathbb{N}$, then $\alpha+\beta$ is equal to $\_\_\_\_$

Let ABC be a triangle. Consider four points $\mathrm{p}_{1}, \mathrm{p}_{2}$, $p_{3}, p_{4}$ on the side $A B$, five points $p_{5}, p_{6}, p_{7}, p_{8}, p_{9}$ on the side BC and four points $\mathrm{p}_{10}, \mathrm{p}_{11}, \mathrm{p}_{12}, \mathrm{p}_{13}$ on the side AC . None of these points is a vertex of the triangle ABC . Then the total number of pentagons, that can be formed by taking all the vertices from the points $\mathrm{p}_{1}, \mathrm{p}_{2}, \ldots . \mathrm{p}_{13}$, is $\_\_\_\_$

A solid sphere of mass 5 kg and radius 10 cm is kept in contact with another solid sphere of mass 10 kg and radius 20 cm . The moment of inertia of this pair of spheres about the tangent passing through the point of contact is $\_\_\_\_$ $\mathrm{kg} . \mathrm{m}^{2}$.

$7.9 \mathrm{MeV} \alpha$-particle scatters from a target material of atomic number 79. From the given data the estimated diameter of nuclei of the target material is (approximately) $\_\_\_\_$ m. $\left[\frac{1}{4 \pi \epsilon_{\mathrm{o}}}=9 \times 10^{9} \mathrm{Nm}^{2} / \mathrm{C}^{2}\right.$ and electron charge $\left.=1.6 \times 10^{-19} \mathrm{C}\right]$

Six point charges are kept $60^{\circ}$ apart from each other on the circumference of a circle of radius R as shown in figure. The net electric field at the centre of the circle is $\_\_\_\_$ . ( $\epsilon_{\mathrm{o}}$ is permittivity of free space)

XPQY is a vertical smooth long loop having a total resistance R where PX is parallel to QY and separation between them is $l$. A constant magnetic field B perpendicular to the plane of the loop exists in the entire space. A rod CD of length L ( $\mathrm{L}>l$ ) and mass $m$ is made to slide down from rest under the gravity as shown in figure. The terminal speed acquired by the rod is $\_\_\_\_$ $\mathrm{m} / \mathrm{s}$. ( $\mathrm{g}=$ acceleration due to gravity)

The escape velocity from a spherical planet A is 10 $\mathrm{km} / \mathrm{s}$. The escape velocity from another planet B whose density and radius are $10 \%$ of those of planet A, is $\_\_\_\_$ $\mathrm{m} / \mathrm{s}$.

A meter bridge with two resistances $\mathrm{R}_{1}$ and $\mathrm{R}_{2}$ as shown in figure was balanced (null point) at 40 cm from the point P . The null point changed to 50 cm from the point P , when $16 \Omega$ resistance is connected in parallel to $R_{2}$. The values of resistances $R_{1}$ and $\mathrm{R}_{2}$ are $\_\_\_\_$ .

A projectile is thrown upward at an angle $60^{\circ}$ with the horizontal. The speed of the projectile is $20 \mathrm{~m} / \mathrm{s}$ when its direction of motion is $45^{\circ}$ with the horizontal. The initial speed of the projectile is $\_\_\_\_$ $\mathrm{m} / \mathrm{s}$.

Given below are two statements:

Statement I: Pressure of fluid is exerted only on a solid surface in contact as the fluid-pressure does not exist everywhere in a still fluid.

Statement II: Excess potential energy of the molecules on the surface of a liquid, when compared to interior, results in surface tension.

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

Net gravitational force at the centre of a square is found to be $F_{1}$ when four particles having mass $M, 2 M, 3 M$ and $4 M$ are placed at the four corners of the square as shown in figure and it is $F_{2}$ when the positions of $3 M$ and $4 M$ are interchanged. The ratio $\frac{F_{1}}{F_{2}}$ is $\frac{\alpha}{\sqrt{5}}$. The value of $\alpha$ is $\_\_\_\_$ .

The minimum frequency of photon required to break a particle of mass 15.348 amu into $4 \alpha$ particles is $\_\_\_\_$ kHz .[0pt] [mass of He nucleus = 4.002 amu, $1 \mathrm{amu}=1.66 \times 10^{-27} \mathrm{~kg}, \mathrm{~h}=6.6 \times 10^{-34} \mathrm{~J} . \mathrm{s}$ and $\mathrm{c}= 3 \times 10^{8} \mathrm{~m} / \mathrm{s}$ ]

A cylindrical tube $A B$ of length $l$, closed at both ends contains an ideal gas of 1 mol having molecular weight $M$. The tube is rotated in a horizontal plane with constant angular velocity $\omega$ about an axis perpendicular to $A B$ and passing through the edge at end A , as shown in the figure. If $P_{A}$ and $P_{B}$ are the pressures at $A$ and $B$ respectively, then (Consider the temperature is same at all points in the tube)

Rods x and y of equal dimensions but of different materials are joined as shown in figure. Temperatures of end points $A$ and $F$ are maintained at $100^{\circ} \mathrm{C}$ and $40^{\circ} \mathrm{C}$ respectively. Given the thermal conductivity of rod x is three times of that of rod y , the temperature at junction points $B$ and $E$ are (close to) :

A thin convex lens of focal length 5 cm and a thin concave lens of focal length 4 cm are combined together (without any gap) and this combination has magnification $\mathrm{m}_{1}$ when an object is placed 10 cm before the convex lens. Keeping the positions of convex lens and object undisturbed a gap of 1 cm is introduced between the lenses by moving the concave lens away, which lead to a change in magnification of total lens system to $\mathrm{m}_{2}$. The value of $\left|\frac{m_{1}}{m_{2}}\right|$ is $\_\_\_\_$ .

Consider an equilateral prism (refractive index $\sqrt{2}$ ). A ray of light is incident on its one surface at a certain angle $i$. If the emergent ray is found to graze along the other surface then the angle of refraction at the incident surface is close to $\_\_\_\_$ .

The volume of an ideal gas increases 8 times and temperature becomes $(1 / 4)^{\text {th }}$ of initial temperature during a reversible change. If there is no exchange of heat in this process ( $\Delta \mathrm{Q}=0$ ) then identify the gas from the following options (Assuming the gases given in the options are ideal gases) :

Electric field in a region is given by $\vec{E}=A x \hat{i}+B y \hat{j}$, where $A=10 \mathrm{~V} / \mathrm{m}^{2}$ and $B=5 \mathrm{~V} / \mathrm{m}^{2}$. If the electric potential at a point $(10,20)$ is 500 V , then the electric potential at origin is $\_\_\_\_$ V.

A simple pendulum has a bob with mass $m$ and charge q . The pendulum string has negligible mass. When a uniform and horizontal electric field $\vec{E}$ is applied, the tension in the string changes. The final tension in the string, when pendulum attains an equilibrium position is $\_\_\_\_$ . (g : acceleration due to gravity)

Match the LIST-I with LIST-II

Choose the correct answer from the options given below:

Three identical coils $\mathrm{C}_{1}, \mathrm{C}_{2}$ and $\mathrm{C}_{3}$ are closely placed such that they share a common axis. $\mathrm{C}_{2}$ is exactly midway. $\mathrm{C}_{1}$ carries current $I$ in anti-clockwise direction while $\mathrm{C}_{3}$ carries current $I$ in clockwise direction. An induced current flows through $\mathrm{C}_{2}$ will be in clockwise direction when

Two loudspeakers ( $L_{1}$ and $L_{2}$ ) are placed with a separation of 10 m , as shown in figure. Both speakers are fed with an audio input signal of same frequency with constant volume. A voice recorder, initially at point $A$, at equidistance to both loud speakers, is moved by 25 m along the line $A B$ while monitoring the audio signal. The measured signal was found to undergo 10 cycles of minima and maxima during the movement. The frequency of the input signal is $\_\_\_\_$ Hz (Speed of sound in air is $324 \mathrm{~m} / \mathrm{s}$ and $\sqrt{5}=2.23$ )

The electric field of a plane electromagnetic wave, travelling in an unknown non-magnetic medium is given by, $\mathrm{E}_{\mathrm{y}}=20 \sin \left(3 \times 10^{6} \mathrm{x}-4.5 \times 10^{14} \mathrm{t}\right) \mathrm{V} / \mathrm{m}$ (where $\mathrm{x}, \mathrm{t}$ and other values have S.I. units). The dielectric constant of the medium is (speed of light in free space is $3 \times 10^{8} \mathrm{~m} / \mathrm{s}$ )

A parallel beam of light travelling in air (refractive index 1.0) is incident on a convex spherical glass surface of radius of curvature 50 cm . Refractive index of glass is 1.5 . The rays converge to a point at a distance x cm from the centre of the curvature of the spherical surface. The value of x is $\_\_\_\_$ cm .

A circular disc has radius $R_{1}$ and thickness $T_{1}$. Another circular disc made of the same material has radius $R_{2}$ and thickness $T_{2}$. If the moment of inertia of both discs are same and $\frac{R_{1}}{R_{2}}=2$ then $\frac{T_{1}}{T_{2}}=\frac{1}{\alpha}$. The value of $\alpha$ is $\_\_\_\_$ .

Inductance of a coil with $10^{4}$ turns is 10 mH and it is connected to a dc source of 10 V with internal resistance of $10 \Omega$. The energy density in the inductor when the current reaches $\left(\frac{1}{\mathrm{e}}\right)$ of its maximum value is $\alpha \pi \times \frac{1}{\mathrm{e}^{2}} \mathrm{~J} / \mathrm{m}^{3}$. The value of $\alpha$ is $\_\_\_\_$ . $\left(\mu_{0}=4 \pi \times 10^{-7} \mathrm{Tm} / \mathrm{A}\right)$.

Consider the transition metal ions $\mathrm{Mn}^{3+}, \mathrm{Cr}^{3+}, \mathrm{Fe}^{3+}$ and $\mathrm{Co}^{3+}$ and all form low spin octahedral complexes. The correct decreasing order of unpaired electrons in their respective d-orbitals of the complexes is

The formal changes on the atoms marked as (1) to (4) in the Lewis representation of $\mathrm{HnO}_{3}$ molecule respectively are

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

Statement I: Phenol on treatment with $\mathrm{CHCl}_{3} /$ aq. KOH under refluxing condition, followed by acidification produces $p$-hydroxy benzladehyde as the major product and $o$-hydroxy benzaldehyde as the minor product.

Statement II : The mixture of $p$-hydroxybenzaldehyde and $o$-hydroxybenzaldehyde can be easily separated through steam distillation.

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

The energy required by electrons, present in the first Bohr orbit of hydrogen atom to be excited to second Bohr orbit is $\_\_\_\_$ $\mathrm{J} \mathrm{mol}^{-1}$.

(Given : $\mathrm{R}_{\mathrm{H}}=2.18 \times 10^{-11}$ ergs)

A 'p'-block element (E) and hydrogen form a binary cation $(\mathrm{EH_x})^+$, while $\mathrm{EH_3}$ on treatment with $\mathrm{K_2HgI_4}$ in alkaline medium gives a precipitate of basic mercury(II)amido-iodine. Given below are first ionisation enthalpy values $(kJ\,mol^{-1})$ for first element each from group 13, 14, 15 and 16. Identify the correct first ionisation enthalpy value for element E.

In the reaction, $\mathrm{2Al(s) + 6HCl(aq) \rightarrow 2Al^{3+}(aq) + 6Cl^{-}(aq) + 3H_2(g)}$. Which of the following statement is correct?

Consider a solution of $\mathrm{CO}_{2}(\mathrm{~g})$ dissolved in water in a closed container.

Which one of the following plots correctly represents variation of log (partial pressure of CO2 in vapour phase above water) [y-axis] with log (mole fraction of $\mathrm{CO}_{2}$ in water) [ x -axis] at $25^{\circ} \mathrm{C}$ ?

A first row transition metla (M) does not liberate $\mathrm{H}_{2}$ gas from dilute HCl .1 mol of aqueous solution of $\mathrm{MSO}_{4}$ is treated with excess of aqueous KCN and then $\mathrm{H}_{2} \mathrm{~S}(\mathrm{~g})$ is passed through the solution. The amount of MS (metal sulphide) formed from the above reaction is $\_\_\_\_$ mol.

The correct order of reactivity of $\mathrm{CH}_{3} \mathrm{Br}$ in methanol with the following nucleophiles is $\mathrm{F}^{-}, \mathrm{I}^{-}, \mathrm{C}_{2} \mathrm{H}_{5} \mathrm{O}^{-}$and $\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}^{-}$

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

Statement I : Benzene is nitrated to give nitrobenzene, which on further treatment with $\mathrm{CH}_{3} \mathrm{COCl} / \mathrm{AlCl}_{3}$ will give

Statement II : $\mathrm{NO}_{2}$ group is a $m$-directing, and deactivating group.

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

A → products (First order reaction).

Three sets of experiment were performed for a reaction under similar experimental conditions. Run $1 \Rightarrow 100 \mathrm{~mL}$ of 10 M solution of reactant A Run $2 \Rightarrow 200 \mathrm{~mL}$ of 10 M solution of reactant A Run $3 \Rightarrow 100 \mathrm{~mL}$ of 10 M solution of reactant $\mathrm{A}+$ 100 mL of $\mathrm{H}_{2} \mathrm{O}$ added. The correct variation of rate of reaction is

Match the LIST-I with LIST-II

Choose the correct answer from the options given below

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

Statement I: The halogen that makes longest bond with hydrogen in HX, has the smallest covalent radius in its group.

Statement II: A group 15 element's hydride $\mathrm{EH}_{3}$ has the lowest boiling point among corresponding hydrides of other group 15 elements. The maximum covalency of that element E is 4 .

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

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

Statement I: Sucrose is dextrorotatory. However sucrose upon hydrolysis gives a solution having mixture of products. This solution shows laevorotation.

Statement II: Hydrolysis of sucrose gives glucose and fructose. Since the laevorotation of glucose is more than the dextrorotation of fructose the resulting solution becomes laevorotatory.

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

Two p-block elements X and Y form fluroides of the type $\mathrm{EF}_{3}$. The fluoride compound $\mathrm{XF}_{3}$ is a Lewis acid and $\mathrm{YF}_{3}$ is a Lewis base. The hybridization of the central atoms of $\mathrm{XF}_{3}$ and $\mathrm{YF}_{3}$ respectively are

As compared with chlorocyclohexane, which of the following statements correctly apply to chlorobenzene?

A. The magnitude of negative charge is more on chlorine atoms

B. The $\mathrm{C}-\mathrm{Cl}$ bond has partial double bond character

C. $\mathrm{C}-\mathrm{Cl}$ bond is less polar

D. $\mathrm{C}-\mathrm{Cl}$ bond is longer due to repulsion between delocalised electrons of the aromatic ring and lone pairs of electrons of chlorine.