JEE Main 2026 — 21 January, Morning Shift

If the domain of the function $f(x)=\cos ^{-1}\left(\frac{2 x-5}{11-3 x}\right)+\sin ^{-1}\left(2 x^{2}-3 x+1\right)$ is the interval $[\alpha, \beta]$, then $\alpha+2 \beta$ is equal to :

The area of the region, inside the ellipse $x^{2}+4 y^{2}=4$ and outside the region bounded by the curves $\mathrm{y}=|\mathrm{x}|-1$ and $\mathrm{y}=1-|\mathrm{x}|$, is :

The number of relations, defined on the set $\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}\}$, which are both reflexive and symmetric, is equal to:

Let a point A lie between the parallel lines $\mathrm{L}_{1}$ and $\mathrm{L}_{2}$ such that its distances from $\mathrm{L}_{1}$ and $\mathrm{L}_{2}$ are 6 and 3 units, respectively. Then the area (in sq. units) of the equilateral triangle ABC , where the points B and C lie on the lines $\mathrm{L}_{1}$ and $\mathrm{L}_{2}$ respectively, is :

Let $\vec{a}=-\hat{i}+2 \hat{j}+2 \hat{k}, \vec{b}=8 \hat{i}+7 \hat{j}-3 \hat{k}$ and $\vec{c}$ be a vector such that $\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{b}}$. If $\overrightarrow{\mathrm{c}} \cdot(\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}})=4$, then $|\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{c}}|^{2}$ is equal to :

Let $\mathrm{a}_{1}, \mathrm{a}_{2}, \mathrm{a}_{3}, \ldots$. be a G.P. of increasing positive terms such that $a_{2} \cdot a_{3} \cdot a_{4}=64$ and $a_{1}+a_{3}+a_{5}=\frac{813}{7}$.Then $\mathrm{a}_{3}+\mathrm{a}_{5}+\mathrm{a}_{7}$ is equal to :

Let $\overrightarrow{\mathrm{c}}$ and$\overrightarrow{\mathrm{d}}$ be vectors such that $|\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}|=\sqrt{29}$ and $\overrightarrow{\mathrm{c}} \times(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}})=(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}) \times \overrightarrow{\mathrm{d}}$. If $\lambda_{1}, \lambda_{2}\left(\lambda_{1}>\lambda_{2}\right)$ are the possible values of $(\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}) \cdot(-7 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}})$, then the equation$\mathrm{K}^{2} \mathrm{x}^{2}+\left(\mathrm{K}^{2}-5 \mathrm{~K}+\lambda_{1}\right) \mathrm{xy}+\left(3 \mathrm{~K}+\frac{\lambda_{2}}{2}\right) \mathrm{y}^{2}-8 \mathrm{x}+12 \mathrm{y}+\lambda_{2}=0$ represents a circle, for k equal to :

Let $y=y(x)$ be the solution curve of the differential equation $\left(1+x^{2}\right) d y+\left(y-\tan ^{-1} x\right) d x=0$, $y(0)=1$. Then the value of $y(1)$ is :

The number of strictly increasing functions f from the set $\{1,2,3,4,5,6\}$ to the set $(1,2,3, \ldots, 9)$ such that $\mathrm{f}(\mathrm{i}) \neq \mathrm{i}$ for $1 \leq \mathrm{i} \leq 6$, is equal to :

Let $f: \mathrm{R} \rightarrow(0, \infty)$ be a twice differentiable function such that $\mathrm{f}=18, \mathrm{f}^{\prime}=0$ and $\mathrm{f}^{\prime \prime}=4$.Then $\lim _{x \rightarrow 1}\left(\log _{e}\left(\frac{f(2+x)}{f}\right)^{\frac{18}{(x-1)^{2}}}\right)$ is equal to :

Let the foci of hyperbola coincide with the foci of the ellipse $\frac{x^{2}}{36}+\frac{y^{2}}{16}=1$. If the eccentricity of the hyperbola is 5 , then the length of its latus rectum is:

The value of $\int_{-\pi / 6}^{\pi / 6}\left(\frac{\pi+4 x^{11}}{1-\sin (|x|+\pi / 6)}\right) d x$ is equal to

Let the mean and variance of 7 observations $2,4,10, \mathrm{x}, 12,14, \mathrm{y}, \mathrm{x}>\mathrm{y}$, be 8 and 16 respectively. Two numbers are chosen from $\{1,2,3, \mathrm{x}-4, \mathrm{y}, 5\}$ one after another without replacement, then the probability, that the smaller number among the two chosen numbers is less than 4 , is:

Let $(\alpha, \beta, \gamma)$ be the co-ordinates of the foot of the perpendicular drawn from the point $(5,4,2)$ on the line $\vec{r}=(-\hat{i}+3 \hat{j}+\hat{k})+\lambda(2 \hat{i}+3 \hat{j}-\hat{k})$.Then the length of the projection of the vector $\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}$ on the vector $6 \hat{i}+2 \hat{j}+3 \hat{k}$ is :

Let PQ and MN be two straight lines touching the circle $x^{2}+y^{2}-4 x-6 y-3=0$ at the points A and B respectively. Let O be the centre of the circle and $\angle \mathrm{AOB}=\pi / 3$. Then the locus of the point of intersection of the lines PQ and MN is:

If the coefficient of $x$ in the expansion of $\left(a x^{2}+b x+c\right)(1-2 x)^{26}$ is -56 and the coefficients of $\mathrm{x}^{2}$ and $\mathrm{x}^{3}$ are both zero, then $\mathrm{a}+\mathrm{b}+\mathrm{c}$ is equal to

If $R$ is the smallest equivalence relation on the set $\{1,2,3,4\}$ such that $\{(1,2),(1,3)\} \subset R$, then the number of elements in $R$ is $\qquad$

The value of $\operatorname{cosec} 10^{\circ}-\sqrt{3} \sec 10^{\circ}$ is equal to:

The sum of all the roots of the equation $(\mathrm{x}-1)^{2}-5|\mathrm{x}-1|+6=0$, is:

Let O be the vertex of the parabola $x^{2}=4 y$ and Q be any point on it. Let the locus of the point P , which divides the line segment OQ internally in the ratio 2: 3 be the conic C . Then the equation of the chord of C , which is bisected at the point $(1,2)$, is:

Let $\mathrm{f}: \mathrm{R} \rightarrow \mathrm{R}$ be a twice differentiable function such that the quadratic equation $\mathrm{f}(\mathrm{x}) \mathrm{m}^{2}-2 \mathrm{f}^{\prime}(\mathrm{x}) \mathrm{m}+\mathrm{f}^{\prime}(\mathrm{x})=0$ in m , has two equal roots for every $\mathrm{x} \in \mathrm{R}$. If $\mathrm{f}(0)=1$, $\mathrm{f}^{\prime}(0)=2$ and $(\alpha, \beta)$ is the largest interval in which the function $\mathrm{f}\left(\log _{\mathrm{e}} \mathrm{x}-\mathrm{x}\right)$ is increasing, then $\alpha+\beta$ is equal to

Let $\mathrm{a}_{1}=1$ and for $\mathrm{n} \geq 1, \mathrm{a}_{\mathrm{n}+1}$ $=\frac{1}{2} a_{n}+\frac{n^{2}-2 n-1}{n^{2}(n+1)^{2}}$. Then $\left|\sum_{n=1}^{\infty}\left(a_{n}-\frac{2}{n^{2}}\right)\right|$ is equal to $\_\_\_\_$ .

Let $S=\{(m, n): m, n \in\{1,2,3, \ldots \ldots, 50\}\}$. If the number of elements ( $\mathrm{m}, \mathrm{n}$ ) in S such that $6^{\mathrm{m}}+9^{\mathrm{n}}$ is a multiple of 5 is p and the number of elements $(\mathrm{m}, \mathrm{n})$ in S such that $\mathrm{m}+\mathrm{n}$ is a square of a prime number is q , then $\mathrm{p}+\mathrm{q}$ is equal to... $\_\_\_\_$

For some $\alpha, \beta \in R$, let $A=\left[\begin{array}{ll}\alpha & 2 \\1 & 2\end{array}\right]$ and $B=\left[\begin{array}{ll}1 & 1 \\1 & \beta\end{array}\right]$ be such that $A^{2}-4 A+2 I=B^{2}-3 B+I=O$. Then $\left(\operatorname{det}\left(\operatorname{adj}\left(\mathrm{A}^{3}-\mathrm{B}^{3}\right)\right)\right)^{2}$ is equal to $\_\_\_\_$

$6 \int_{0}^{\pi}|(\sin 3 x+\sin 2 x+\sin x)| d x$ is equal to........

A gas based geyser heats water flowing at the rate of 5.0 litres per minute from $27^{\circ} \mathrm{C}$ to $87^{\circ} \mathrm{C}$. The rate of consumption of the gas is $\_\_\_\_$ g/s. (Take heat of combustion of gas $=5.0 \times 10^{4} \mathrm{~J} / \mathrm{g}$ ) specific heat capacity of water $=4200 \mathrm{~J} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}$

A conducting circular loop of area $1.0 \mathrm{~m}^{2}$ is placed perpendicular to a magnetic field which varies as $\mathrm{B}=\sin (100 \mathrm{t})$ Tesla. If the resistance of the loop is $100 \Omega$, then the average thermal energy dissipated in the loop in one period is $\_\_\_\_$ J.

Water flows through a horizontal tube as shown in the figure. The difference in height between the water columns in vertical tubes is 5 cm and the area of cross-sections at $A$ and $B$ are $6 \mathrm{~cm}^{2}$ and $3 \mathrm{~cm}^{2}$ respectively. The rate of flow will be $\_\_\_\_$ $\mathrm{cm}^{3} / \mathrm{s}$. (take $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$ )

In an experiment the values of two spring constants were measured as $\mathrm{k}_{1}=(10 \pm 0.2) \mathrm{N} / \mathrm{m}$ and $\mathrm{k}_{2}=(20 \pm 0.3) \mathrm{N} / \mathrm{m}$. If these springs are connected in parallel, then the percentage error in equivalent spring constant is :

A 4 kg mass moves under the influence of a force $\vec{F}=\left(4 t^{3} \hat{i}-3 t \hat{j}\right) N$ where $t$ is the time in second. If mass starts from origin at $\mathrm{t}=0$, the velocity and position after $\mathrm{t}=2 \mathrm{~s}$ will be :

Consider a modified Bernoulli equation.

If t has the dimension of time then the dimensions of A and B are $\_\_\_\_$ , $\_\_\_\_$ respectively.

A current carrying is placed vertically and a particle of mass m with charge Q is released from rest. The particle moves along the axis of solenoid. If g is acceleration due to gravity then the acceleration (a) of the charged particle will satisfy :

A parallel plate capacitor has capacitance C , when there is vacuum within the parallel plates. A sheet having thickness $\left(\frac{1}{3}\right)^{\text {rd }}$ of the separation between the plates and relative permittivity K is introduced between the plates. The new capacitance of the system is :

The electric field a plane electromagnetic wave is given by : $\mathrm{E}_{\mathrm{y}}=69 \sin \left[0.6 \times 10^{3} \mathrm{x}-1.8 \times 10^{11} \mathrm{t}\right] \mathrm{V} / \mathrm{m}$. The expression for magnetic field associated with this electromagnetic wave is $\_\_\_\_$ T.

In a double slit experiment the distance between the slits is 0.1 cm and the screen is placed at 50 cm from the slits plane. When one slit is covered with a transparent sheet having thickness $t$ and refractive index $\mathrm{n}(=1.5)$, the central fringe shifts by 0.2 cm . The value of t is $\_\_\_\_$ cm .

A light wave described by $\mathrm{E}=60 \sin \left(3 \times 10^{15}\right) \mathrm{t}+ \sin \left(12 \times 10^{15}\right)$ t] (in SI units) falls on a metal surface of work function 2.8 eV . The maximum kinetic energy of ejected photoelectron is (approximately) $\_\_\_\_$ eV. $\quad\left(\mathrm{h}=6.6 \times 10^{-34} \mathrm{~J}-\mathrm{s} . \quad\right.$ and $\mathrm{e}=1.6 \times 10^{-19} \mathrm{C}$ )

If an alpha particle with energy 7.7 MeV is bombarded on a thin gold foil, the closest distance from nucleus it can reach is $\_\_\_\_$ m . (Atomic number of gold $=79$ and $\frac{1}{4 \pi \epsilon_{0}}=9 \times 10^{9}$ in SI units)

An aluminium and steel rods having same lengths and cross-sections are joined to make total length of 120 cm at $30^{\circ} \mathrm{C}$. The coefficient of linear expansion of aluminium and steel are $24 \times 10^{-6} /{ }^{\circ} \mathrm{C}$ and $1.2 \times 10^{-5} /{ }^{\circ} \mathrm{C}$, respectively. The length of this composite rod when its temperature is raised to $100^{\circ} \mathrm{C}$, is $\_\_\_\_$ cm.

A 1 m long metal rod AB completes the circuit as shown in figure. The area of circuit is perpendicular to the magnetic field of 0.10 T . If the resistance of the total circuit is $2 \Omega$ then the force needed to move the rod towards right with constant speed (v) of $1.5 \mathrm{~m} / \mathrm{s}$ is $\_\_\_\_$ N.

Two strings (A, B) having linear densities $\mu_{\mathrm{A}}=2 \times 10^{-4} \mathrm{~kg} / \mathrm{m}$ and $\mu_{\mathrm{B}}=4 \times 10^{-4} \mathrm{~kg} / \mathrm{m}$ and lengths $\mathrm{L}_{\mathrm{A}}=2.5 \mathrm{~m}$ and $\mathrm{L}_{\mathrm{B}}=1.5 \mathrm{~m}$ respectively are joined. Free ends of A and B are tied to two rigid supports C and D , respectively creating a tension of 500 N in the wire. Two identical pulses, sent from C and D ends, take time $\mathrm{t}_{1}$ and $\mathrm{t}_{2}$, respectively, to reach the joint. The ratio $\mathrm{t}_{1} / \mathrm{t}_{2}$ is :

Initially a satellite of 100 kg is in a circular orbit of radius $1.5 \mathrm{R}_{\mathrm{E}}$. This satellite can be moved to a circular orbit of radius $3 \mathrm{R}_{\mathrm{E}}$ by supplying $\alpha \times 10^{6} \mathrm{~J}$ of energy. The value of $\alpha$ is $\_\_\_\_$ . (Take Radius of Earth $\mathrm{R}_{\mathrm{E}}=6 \times 10^{6} \mathrm{~m}$ and $\mathrm{g}=10 \mathrm{m} / \mathrm{s}^{2}$ )

A point charge of $10^{-8} \mathrm{C}$ is placed at origin. The work done in moving a point charge $2 \mu \mathrm{C}$ from point $\mathrm{A}(4,4,2) \mathrm{m}$ to $\mathrm{B}(2,2,1) \mathrm{m}$ is $\_\_\_\_$ J. ( $\frac{1}{4 \pi \epsilon_{0}}=9 \times 10^{9}$ in SI units)

A collimated beam of light of diameter 2 mm is propagating along x -axis. The beam is required to be expanded in a collimated beam of diameter 14 mm using a system of two convex lenses. If first lens has focal length 40 mm , then the focal length of second lens is $\_\_\_\_$ mm .

The heat generated in 1 minute between points A and B in the given circuit, when a battery of 9 V with internal resistance of $1 \Omega$ is connected across these points is $\_\_\_\_$ J.

Two identical thin rods of mass M kg and length L m are connected as shown in figure. Moment of inertia of the combined rod system about an axis passing through point P and perpendicular to the plane of the rods is $\frac{X}{2} M^{2} \mathrm{~kg} \mathrm{~m}^{2}$. The value of $x$ is $\_\_\_\_$ .

10 mole of oxygen is heated at constant volume from $30^{\circ} \mathrm{C}$ to $40^{\circ} \mathrm{C}$. The change in the internal energy of the gas is $\_\_\_\_$ cal. (The molecular specific heat of oxygen at constant pressure, $\mathrm{C}_{\mathrm{p}}=7 \mathrm{cal} . / \mathrm{mol}^{\circ} \mathrm{C}$ and $\mathrm{R}=2 \mathrm{cal} . / \mathrm{mol}^{\circ} \mathrm{C}$.)

In a microscope the objective is having focal length $f_{0}=2 \mathrm{~cm}$ and eye-piece is having focal length $f_{\mathrm{e}}=4 \mathrm{~cm}$. The tube length is 32 cm . The magnification produced by this microscope for normal adjustment is $\_\_\_\_$ .

Consider the following reactions.

$\mathrm{PbCl}_{2}+\mathrm{K}_{2} \mathrm{CrO}_{4} \rightarrow \mathrm{~A}+2 \mathrm{KCl}$ (Hot solution)

$\mathrm{A}+\mathrm{NaOH} \rightleftharpoons \mathrm{B}+\mathrm{Na}_{2} \mathrm{CrO}_{4}$

$\mathrm{PbSO}_{4}+4 \mathrm{CH}_{3} \mathrm{COONH}_{4} \rightarrow\left(\mathrm{NH}_{4}\right)_{2} \mathrm{SO}_{4}+\mathrm{X}$

In the above reactions, $\mathrm{A}, \mathrm{B}$ and X are respectively.

Which of the following represents the correct trend for the mentioned property?

A. $\mathrm{F > P > S > B}$ — First Ionization Energy

B. $\mathrm{Cl > F > S > P}$ — Electron Affinity

C. $\mathrm{K > Al > Mg > B}$ — Metallic character

D. $\mathrm{K_2O > Na_2O > MgO > Al_2O_3}$ — Basic character

Choose the correct answer from the option given below:

A hydrocarbon ' P ' $\left(\mathrm{C}_{4} \mathrm{H}_{8}\right)$ on reaction with HCl gives an optically active compound ' Q ' $\left(\mathrm{C}_{4} \mathrm{H}_{9} \mathrm{Cl}\right)$ which on reaction with one mole of ammonia gives compound ' R ' $\left(\mathrm{C}_{4} \mathrm{H}_{11} \mathrm{~N}\right)$. ' R ' on diazotization followed by hydrolysis gives ' S '. Identify $\mathrm{P}, \mathrm{Q}, \mathrm{R}$ and S .

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

Statement I: The number of pairs among [ $\mathrm{SiO}_{2}, \mathrm{CO}_{2}$ ], $\left[\mathrm{SnO}, \mathrm{SnO}_{2}\right],\left[\mathrm{PbO}, \mathrm{PbO}_{2}\right]$ and $\left[\mathrm{GeO}, \mathrm{GeO}_{2}\right]$, which contain oxides that are both amphoteric is 2 .

Statement II : $\mathrm{BF}_{3}$ is an electron deficient molecule can act as a lewis acid, forms adduct with $\mathrm{NH}_{3}$ and has a trigonal planar geometry.

In the light of the above statement, choose the correct answer from the option given below.

$80\,\mathrm{mL}$ of a hydrocarbon on mixing with $264\,\mathrm{mL}$ of oxygen in a closed U-tube undergoes complete combustion. The residual gases after cooling to $273\,\mathrm{K}$ occupy $224\,\mathrm{mL}$. When the system is treated with $\mathrm{KOH}$ solution, the volume decreases to $64\,\mathrm{mL}$. The formula of the hydrocarbon is:

$14.0\,\mathrm{g}$ of calcium metal is allowed to react with excess $\mathrm{HCl}$ at $1.0\,\mathrm{atm}$ pressure and $273\,\mathrm{K}$. Which of the following statements is incorrect?

[Given: Molar mass in $\mathrm{g\,mol^{-1}}$ — $\mathrm{Ca}=40$, $\mathrm{Cl}=35.5$, $\mathrm{H}=1$]

In Carius method, 0.75 g of an organic compound gave 1.2 g of barium sulphate, find percentage of sulphur (molar mass $32 \mathrm{~g} \mathrm{~mol}^{-1}$ ). Molar mass of barium sulphate is $233 \mathrm{~g} \mathrm{~mol}^{-1}$.

Elements P and Q form two types of non-volatile, non-ionizable compounds PQ and $\mathrm{PQ}_{2}$. When 1 g of PQ is dissolved in 50 g of solvent ' A '. $\Delta \mathrm{T}_{\mathrm{b}}$ was 1.176 K while when 1 g of $\mathrm{PQ}_{2}$ is dissolved in 50 g of solvent ' A ', $\Delta \mathrm{T}_{\mathrm{b}}$ was 0.689 K . ( $\mathrm{K}_{\mathrm{b}}$ of ' A ' $=5 \mathrm{~K} \mathrm{kg} \mathrm{mol}^{-1}$ ). The molar masses of elements P and Q (in $\mathrm{g} \mathrm{mol}^{-1}$ ) respectively, are :

From the following, the least stable structure is :

$\mathrm{MnO}_{4}^{2-}$, in acidic medium, disproportionates to :

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

Statement I: The number of species among $SF_4,\ NH_4^+,\ [NiCl_4],\ XeF_4,\ [PtCl_4]^{2-},\ SeF_4$ and $[Ni(CN)_4]^{2-}$ that have tetrahedral geometry is 3.

Statement II: In the set $NO_2,\ BeH_2,\ BF_3,\ AlCl_3$, all the molecules have incomplete octet around the central atom.

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: Among $\left[\mathrm{Cu}\left(\mathrm{NH}_{3}\right)^{2+},\left[\mathrm{Ni}(\mathrm{en})_{3}\right)\right]^{2+}$, $\left[\mathrm{Ni}\left(\mathrm{NH}_{3}\right)_{6}\right]^{2+}$ and $\left[\mathrm{Mn}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+},\left[\mathrm{Mn}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{2+}$, has the maximum number of unpaired electrons.