JEE Main 2026 — 28 January, Morning Shift

If $g(x)=3 x^{2}+2 x-3, f(0)=-3$ and $4 g(f(x))=3 \mathrm{x}^{2}-32 \mathrm{x}+72$, then $f(\mathrm{~g}(2))$ is equal to:

The value of $\sum_{k=1}^{\infty}(-1)^{k+1}\left(\frac{k(k+1)}{k!}\right)$ is :

Let $y=x$ be the equation of a chord of the circle $\mathrm{C}_{1}$ (in the closed half-plane $\mathrm{x} \geq 0$ ) of diameter 10 passing through the origin. Let $\mathrm{C}_{2}$ be another circle described on the given chord as its diameter. If the equation of the chord of the circle $\mathrm{C}_{2}$, which passes through the point $(2,3)$ and is farthest from the center of $\mathrm{C}_{2}$, is $\mathrm{x}+\mathrm{ay}+\mathrm{b}=0$, then $\mathrm{a}-\mathrm{b}$ is equal to :

If $\frac{\tan (\mathrm{A}-\mathrm{B})}{\tan \mathrm{A}}+\frac{\sin ^{2} \mathrm{C}}{\sin ^{2} \mathrm{~A}}=1, \mathrm{~A}, \mathrm{~B}, \mathrm{C} \in\left(0, \frac{\pi}{2}\right)$, then

Let z be a complex number such that $|\mathrm{z}-6|=5$ and $|z+2-6 i|=5$. Then the value of $z^{3}+3 z^{2}-15 z+141$ is equal to

Let ABC be an equilateral triangle with orthocenter at the origin and the side BC on the line $x+2 \sqrt{2} y=4$. If the co-ordinates of the vertex A are $(\alpha, \beta)$, then the greatest integer less than or equal to $|\alpha+\sqrt{2} \beta|$ is

Let $S=\{1,2,3,4,5,6,7,8,9\}$. Let $x$ be the number of 9-digit numbers formed using the digits of the set S such that only one digit is repeated and it is repeated exactly twice. Let $y$ be the number of 9-digit numbers formed using the digits of the set S such that only two digits are repeated and each of these is repeated exactly twice. Then,

Let $S=\left\{x^{3}+a x^{2}+b x+c: a, b, c \in N\right.$ and $a, b, c \leq$ 20 } be a set of polynomials. Then the number of polynomials in S , which are divisible by $\mathrm{x}^{2}+2$, is

A bag contains 10 balls out of which k are red and $(10-\mathrm{k})$ are black, where $0 \leq \mathrm{k} \leq 10$. If three balls are drawn at random without replacement and all of them are found to be black, then the probability that the bag contains 1 red and 9 black balls is :

If $\alpha, \beta$ ，where $\alpha<\beta$ ，are the roots of the equation $\lambda \mathrm{x}^{2}-(\lambda+3) \mathrm{x}+3=0$ such that $\frac{1}{\alpha}-\frac{1}{\beta}=\frac{1}{3}$, then the sum of all possible values of $\lambda$ is ：

Let $f$ be a polynomial function such that $f\left(\mathrm{x}^{2}+1\right)=\mathrm{x}^{4}+5 \mathrm{x}^{2}+2$ ，for all $\mathrm{x} \in \mathbb{R}$ ． Then $\int_{0}^{3} f(\mathrm{x}) \mathrm{dx}$ is equal to

The area of the region $R = \{(x,y) : xy \leq 8, 1 \leq y \leq x^2, x \geq 0\}$ is

The value of $\lim _{x \rightarrow 0} \frac{\log _{e}\left(\sec (e x) \cdot \sec \left(e^{2} x\right) \cdot \ldots \cdot \sec \left(e^{10} x\right)\right)}{e^{2}-e^{2 \cos x}}$ is equal to

The mean and variance of $10$ observations are $9$ and $34.2$ , respectively. If $8$ of these observations are $2,3,5,10,11,13,15,21$, then the mean deviation about the median of all the $10$ observations is

Let $\mathrm{A}, \mathrm{B}$ and C be three $2 \times 2$ matrices with real entries such that $\mathrm{B}=(\mathrm{I}+\mathrm{A})^{-1}$ and $\mathrm{A}+\mathrm{C}=\mathrm{I}$. If $\mathrm{BC}=\left[\begin{array}{cc}1 & -5\\ -1 & 2\end{array}\right]$ and $\mathrm{CB}\left[\begin{array}{l}\mathrm{x}_{1}\\ \mathrm{x}_{2}\end{array}\right]=\left[\begin{array}{c}12 \\-6\end{array}\right], \quad$ then $x_{1}+x_{2}$ is

The common difference of the A.P.: $\mathrm{a}_{1}, \mathrm{a}_{2}, \ldots, \mathrm{a}_{\mathrm{m}}$ is $13$ more than the common difference of the A.P.: $\mathrm{b}_{1}, \mathrm{~b}_{2}, \ldots, \mathrm{~b}_{\mathrm{n}}$. If $\mathrm{b}_{31}=-277, \mathrm{~b}_{43}=-385$ and $\mathrm{a}_{78}=$ 327, then $\mathrm{a}_{1}$ is equal to

If the distances of the point $(1,2, a)$ from the line $\frac{x-1}{1}=\frac{y}{2}=\frac{z-1}{1}$ along the lines $\mathrm{L}_{1}: \frac{\mathrm{x}-1}{3}=\frac{\mathrm{y}-2}{4}=\frac{\mathrm{z}-\mathrm{a}}{\mathrm{b}}$ and $\mathrm{L}_{2}: \frac{\mathrm{x}-1}{1}=\frac{\mathrm{y}-2}{4}=\frac{\mathrm{z}-\mathrm{a}}{\mathrm{c}}$ are equal, then $\mathrm{a}+\mathrm{b}+\mathrm{c}$ is equal to

For three unit vectors $\vec{a}, \vec{b}, \vec{c}$ satisfying $|\overrightarrow{\mathrm{a}}-\overrightarrow{\mathrm{b}}|^{2}+|\overrightarrow{\mathrm{b}}-\overrightarrow{\mathrm{c}}|^{2}+|\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{a}}|^{2}=9$ and $|2 \overrightarrow{\mathrm{a}}+\mathrm{k} \overrightarrow{\mathrm{b}}+\mathrm{k} \overrightarrow{\mathrm{c}}|=3$, the positive value of k is :

Let $y=y(x)$ be the solution of the differential equation $x \frac{d y}{d x}-\sin 2 y=x^{3}\left(2-x^{3}\right) \cos ^{2} y, x \neq 0$. If $y(2)=x$, then $\tan (y(1))$ is equal to

In a G.P., if the product of the first three terms is 27 and the set of all possible values for the sum of its first three terms is $\mathbb{R}-(\mathrm{a}, \mathrm{b})$, then $\mathrm{a}^{2}+\mathrm{b}^{2}$ is equal to $\_\_\_\_$ .

For some $\theta \in\left(0, \frac{\pi}{2}\right)$, let the eccentricity and the length of the latus rectum of the hyperbola $x^{2}- \mathrm{y}^{2} \sec ^{2} \theta=8$ be $\mathrm{e}_{1}$ and $\ell_{1}$, respectively, and let the eccentricity and the length of the latus rectum of the ellipse $x^{2} \sec ^{2} \theta+y^{2}=6$ be $e_{2}$ and $\ell_{2}$, respectively. If $e_{1}^{2}=e_{2}^{2}\left(\sec ^{2} \theta+1\right), \quad$ then $\left(\frac{\ell_{1} \ell_{2}}{\mathrm{e}_{1} \mathrm{e}_{2}}\right) \tan ^{2} \theta$ is equal to $\_\_\_\_$ .

If $\quad k=\tan \left(\frac{\pi}{4}+\frac{1}{2} \cos ^{-1}\left(\frac{2}{3}\right)\right)+\tan \left(\frac{1}{2} \sin ^{-1}\left(\frac{2}{3}\right)\right)$ then the number of solutions of the equation $\sin ^{-1}(\mathrm{k} \mathrm{x}-1)=\sin ^{-1} \mathrm{x}-\cos ^{-1} \mathrm{x}$ is $\_\_\_\_$ .

The value of $\sum_{\mathrm{r}=1}^{20}\left(\left|\sqrt{\pi\left(\int_{0}^{\mathrm{r}} \mathrm{x}|\sin \pi \mathrm{x}| \mathrm{dx}\right)}\right|\right)$ is $\_\_\_\_$ .

Let PQR be a triangle such that $\overrightarrow{\mathrm{PQ}}=-2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}}$ and $\overrightarrow{\mathrm{PR}}=a \hat{i}+b \hat{j}-4 \hat{k}, a, b \in \mathbb{Z}$. Let $S$ be the point on QR , which is equidistant from the lines PQ and PR. If $|\overrightarrow{\mathrm{PR}}|=9$ and $\overrightarrow{\mathrm{PS}}=\hat{\mathrm{i}}-7 \hat{\mathrm{j}}+2 \hat{\mathrm{k}}$, then the value of $3 a-4 b$ is $\_\_\_\_$ .

10 kg of ice at $-10^{\circ} \mathrm{C}$ is added to 100 kg of water to lower its temperature from $25^{\circ} \mathrm{C}$. Consider no heat exchange to surroundings. The decrement to the temperature of water is $\_\_\_\_$ ${ }^{\circ} \mathrm{C}$. (specific heat of ice $=2100 \mathrm{~J} / \mathrm{Kg} .{ }^{\circ} \mathrm{C}$, specific heat of water $=4200 \mathrm{~J} / \mathrm{Kg} .{ }^{\circ} \mathrm{C}$, latent heat of fusion of ice $=3.36 \times 10^{5} \mathrm{~J} / \mathrm{Kg}$ )

The electric current in the circuit is given as $i=i_{\mathrm{o}}(t / T)$. The r.m.s current for the period $t=0$ to $t=T$ is $\_\_\_\_$

In the potentiometer, when the cell in the secondary circuit is shunted with $4 \Omega$ resistance, the balance is obtained at the length 120 cm of wire. Now when the same cell in shunted with $12 \Omega$ resistance, the balance is shifted to a length of 180 cm . The internal resistance of cell is $\_\_\_\_$ $\Omega$.

Water drops fall from a tap on the floor, 5 m below, at regular intervals of time, the first drop strikes the floor when the sixth drop begins to fall. The height at which the fourth drop will be from ground, at the instant when the first drop strikes the ground is $\_\_\_\_$ m. $\left(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}\right)$

The electric field of an electromagnetic wave travelling through a medium is given by $\vec{E}(x, t)=25 \sin \left(2.0 \times 10^{15} t-10^{7} x\right) \hat{n}$ then the refractive index of the medium is $\_\_\_\_$。 (All given measurement are in SI units)

Assuming in forward bias condition there is a voltage drop of 0.7 V across a silicon diode, the current through diode $\mathrm{D}_{1}$ in the circuit is $\_\_\_\_$ mA . (Assume all diodes in the given circuit are identical)

The magnitudes of power of a biconvex lens (refractive index 1.5) and that of a plano-concave lens (refractive index $=1.7$ ) are same. If the curvature of plano-concave lens exactly matches with the curvature of back surface of the biconvex lens, then ratio of radius of curvature of front and back surface of the biconvex lens is $\_\_\_\_$。

In the following $p-V$ diagram the equation of state along the curved path is given by $(V-2)^{2}=4$ ap where $a$ is a constant. The total work done in the closed path is

For the two cells having same EMF E and internal resistance $r$, the current passing through the external resistor $6 \Omega$ is same when both the cells are connected either in parallel or in series. The value of internal resistance $r$ is $\_\_\_\_$ $\Omega$.

Two wires $A$ and $B$ made of different materials of length 6.0 cm and 5.4 cm , respectively and area of cross sections $3.0 \times 10^{-5} \mathrm{~m}^{2}$ and $4.5 \times 10^{-5} \mathrm{~m}^{2}$, respectively are stretched by the same magnitude under a given load. The ratio of the Young's modulus of $A$ to that of $B$ is $\mathrm{x}: 3$. The value of x is $\_\_\_\_$。

A block of mass 5 kg is moving on an inclined plane which makes an angle of $30^{\circ}$ with the horizontal. Friction coefficient between the block and inclined plane surface is $\frac{\sqrt{3}}{2}$. The force to be applied on the block so that the block will move down without acceleration is $\_\_\_\_$ N.

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

Statement-I : A plane wave after passing through prism remains as plane wave but passing through small pin hole may become spherical wave.

Statement-II : The curvature of a spherical wave emerging from a slit will increase for increasing slit width.

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

When both jaws of vernier callipers touch each other, zero mark of the vernier scale is right to zero mark of main scale, $4^{\text {th }}$ mark on vernier scale coincides with certain mark on the main scale. While measuring the length of a cylinder, observer observes 15 divisions on main scale and $5^{\text {th }}$ division of vernier scale coincides with a min scale division. Measured length of cylinder is $\_\_\_\_$ mm . (Least count of Vernier calliper $=0.1 \mathrm{~mm}$ )

The magnetic field at the centre of a current carrying circular loop of radius $R$ is $16 \mu \mathrm{~T}$. The magnetic field at a distance $x=\sqrt{3} R$ on its axis from the centre is $\_\_\_\_$ $\mu \mathrm{T}$.

Two point charges of 1 nC and 2 nC are placed at the two corners of equilateral triangle of side 3 cm . The work done in bringing a charge of 3 nC from infinity to the third corner of the triangle is $\_\_\_\_$ $\mu \mathrm{J}$. $\frac{1}{4 \pi \epsilon_{0}}=9 \times 10^{9} \mathrm{~N} . \mathrm{m}^{2} / \mathrm{C}^{2}$

A particle of mass m falls from rest through a resistive medium having resistive force, $\mathrm{F}=-\mathrm{kv}$, where v is the velocity of the particle and k is a constant. Which of the following graphs represents velocity ( v ) versus time ( t ) ?

The ratio of de Broglie wavelength of a deutron with kinetic energy $E$ to that of an alpha particle with kinetic energy $2 E$, is $n: 1$. The value of $n$ is $\_\_\_\_$ . (Assume mass of proton $=$ mass of neutron) :

A convex lens of refractive index 1.5 and focal length $f=18 \mathrm{~cm}$ is immersed in water. The difference in focal lengths of the given lens when it is in water and in air is $\alpha \times \mathrm{f}$. The value of $\alpha$ is $\_\_\_\_$ . (refractive index of water $=4 / 3$ )

The equivalent resistance between the points $A$ and $B$ in the following circuit is $\frac{x}{5} \Omega$. The value of $x$ is $\_\_\_\_$ .

$20.0 \mathrm{dm}^{3}$ of an ideal gas ' X ' at 600 K and 0.5 MPa undergoes isothermal reversible expansion until pressure of the gas is 0.2 MPa . Which of the following option is correct? (Given: $\log 2=0.3010$ and $\log 5=0.6989$ )

CORRECT order of stability for the following is $\mathrm{CH}_{2}=\mathrm{CH}^{-}, \mathrm{CH}_{3}-\mathrm{CH}_{2}^{-}, \mathrm{CH} \equiv \mathrm{C}^{-}$

At $\mathrm{T}(\mathrm{K}), 2$ moles of liquid A and 3 moles of liquid $B$ are mixed. The vapour pressure of ideal solution formed is 320 mm Hg . At this stage, one mole of A and one mole of $B$ are added to the solution. The vapour pressure is now measured as 328.6 mm Hg . The vapour pressure (in mm Hg ) of A and B are respectively:

Method used for separation of mixture of products ( B and C ) obtained in the following reaction is:

Consider a weak base ' B ' of $\mathrm{pK}_{\mathrm{b}}=5.699$. ' x ' mL of 0.02 M HCl and ' y ' mL of 0.02 M weak base ' B ' are mixed to make 100 mL of a buffer of pH 9 at $25^{\circ} \mathrm{C}$. The values of ' x ' and ' y ' respectively are: [Given: $\log 2=0.3010, \log 3=0.4771, \log 5=0.699$ ]

An organic compound undergoes first order decomposition. The time taken for decomposition to $\left(\frac{1}{8}\right)^{\text {th }}$ and $\left(\frac{1}{10}\right)^{\text {th }}$ of its initial concentration are $t_{1 / 8}$ and $t_{1 / 10}$ respectively.

What is the value of $\frac{\mathrm{t}_{1 / 8}}{\mathrm{t}_{1 / 10}} \times 10$ ? $(\log 2=0.3)$

Given below are the four isomeric compounds (P, Q, R, S)

Identify correct statements from below.

A. Q, $R$ and $S$ will give precipitate with 2, 4 - DNP.

B. P and Q will give positive Bayer's test. C. Q and R will give sooty flame.

D. R and S will give yellow precipitate with $\mathrm{I}_{2} / \mathrm{NaOH}$.

E. Q alone will deposit silver with Tollen's reagent

Choose the correct option.

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 $\mathrm{BF}_{4}^{-}, \mathrm{SiF}_{4}, \mathrm{XeF}_{4}$ and $\mathrm{SF}_{4}$, that have unequal $\mathrm{E}-\mathrm{F}$ bond lengths is two. Here, E is the central atom.

Statement II: Among $\mathrm{O}_{2}^{-}, \mathrm{O}_{2}^{2-}, \mathrm{F}_{2}$ and $\mathrm{O}_{2}^{+}, \mathrm{O}_{2}^{-}$has the highest bond order.

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

Regarding the hydrides of group 15 elements $\mathrm{EH}_{3}$ ( $\mathrm{E}=\mathrm{N}, \mathrm{P}, \mathrm{As}, \mathrm{Sb}$ ), select the correct statement from the following:

A. The stability of hydrides decreases down the group.

B. The basicity of hydrides decreases down the group.

C. The reducing character increases down the group.

D. The boiling point increases down the group.

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: Griss-Ilosvay test is used for the detection of nitrite ion, which involves the use of sulphanilic acid and $\alpha$-naphthylamine reagent.

Statement II: In the above test, sulphanilic acid is diazotized by the acidified nitrite ion, which on further coupling with $\alpha$-naphthylamine forms an azo-dye.

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

$\mathrm{Ph}-\mathrm{CH}=\mathrm{CH}_{2} \xrightarrow[\mathrm{HBr}]{(\mathrm{PhCOO})_{2}}$ Product

Consider the above reaction

A. The reaction proceeds through a more stable radical intermediate.

B. The role of peroxide is to generate $\dot{H}$ (Hydrogen radical).

C. During this reaction, benzene is formed as a biproduct.

D. 1-Bromo-2-phenylethane is fanned as the minor product.

E. The same reaction in absence of peroxide proceeds via carbocation intermediate.

Identify the correct statements. Choose the correct answer from the options given below:

The wave numbers of three spectral lines of H-atom are considered. Identify the set of spectral lines belonging to Balmer series. ( $\mathrm{R}=$ Rydberg constant)

Figure 1. Electron probability density for 2s orbital

Figure 2. Wave function for 2s orbital

Which of the following point in Figure 2 most accurately represents the nodal surface as shown in Figure 1 ?

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, from the following, in which both the ions are coloured in aqueous solution is 3 . $\left[\mathrm{Sc}^{3+}, \mathrm{Ti}^{3+}\right],\left[\mathrm{Mn}^{2+}, \mathrm{Cr}^{2+}\right],\left[\mathrm{Cu}^{2+}, \mathrm{Zn}^{2+}\right]$ and $\left[\mathrm{Ni}^{2+}, \mathrm{Ti}^{4+}\right]$

Statement II : $\mathrm{Th}^{4+}$ is the strongest reducing agent among $\mathrm{Th}^{4+}, \mathrm{Ce}^{4+}, \mathrm{Gd}^{3+}$ and $\mathrm{Eu}^{2+}$.

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

In period 4 of the periodic table, the elements with highest and lowest atomic radii are respectively.

The correct statement among the following is :

500 mL of 1.2 M KI solution is mixed with 500 mL of $0.2 \mathrm{M} \mathrm{KMnO}_{4}$ solution in basic medium. The liberated iodine was titrated with standard $0.1 \mathrm{M} \mathrm{Na}_{2} \mathrm{~S}_{2} \mathrm{O}_{3}$ solution in the presence of starch indicator till the blue color disappeared. The volume (in L ) of $\mathrm{Na}_{2} \mathrm{~S}_{2} \mathrm{O}_{3}$ consumed is $\_\_\_\_$ . (Nearest integer)

Consider the following redox reaction taking place in acidic medium $\mathrm{BH}_{4}^{-}(\mathrm{aq})+\mathrm{ClO}_{3}^{-}(\mathrm{aq}) \rightarrow \mathrm{H}_{2} \mathrm{BO}_{3}^{-}(\mathrm{aq})+\mathrm{Cl}^{-}(\mathrm{aq})$ If the Nernst equation for the above balanced reaction is $\mathrm{E}_{\text {cell }}=\mathrm{E}_{\text {cell }}^{\mathrm{o}}-\frac{\mathrm{RT}}{\mathrm{nF}} \ell \mathrm{n} \mathrm{Q}$, Then the value of n is $\_\_\_\_$ . (Nearest integer)

X is the number of geometrical isomers exhibited by $\left[\mathrm{Pt}\left(\mathrm{NH}_{3}\right)\left(\mathrm{H}_{2} \mathrm{O}\right) \mathrm{BrCl}\right]$.

Y is the number of optically inactive isomer(s) exhibited by $\left[\mathrm{CrCl}_{2}(\mathrm{ox})_{2}\right]^{3-}$

Z is the number of geometrical isomers exhibited by $\left[\mathrm{Co}\left(\mathrm{NH}_{3}\right)_{3}\left(\mathrm{NO}_{2}\right)_{3}\right]$

The value of $\mathrm{X}+\mathrm{Y}+\mathrm{Z}$ is $\_\_\_\_$。

$0.53 \mathrm{~g}$ of an organic compound ( x ) when heated with excess of nitric acid (concentrated) and then with silver nitrate gave 0.75 g of silver bromide precipitate. 1.0 g of $(\mathrm{x})$ gave 1.32 g of $\mathrm{CO}_{2}$ gas on combustion. The percentage of hydrogen in the compound ( x ) is $\_\_\_\_$ $\%.$ [Nearest Integer]

[Given : Molar mass in $\mathrm{g} \mathrm{mol}^{-1} \mathrm{H}: 1, \mathrm{C}: 12, \mathrm{Br}:$ 80, Ag : 108, O : 16; Compound (x) : $\mathrm{C}_{\mathrm{x}} \mathrm{H}_{\mathrm{y}} \mathrm{Br}_{\mathrm{z}}$ ]

Consider the dissociation equilibrium of the following weak acid $\mathrm{HA} \rightleftharpoons \mathrm{H}^{+}(\mathrm{aq})+\mathrm{A}^{-}(\mathrm{aq})$ If the pKa of the acid is 4 , then the pH of 10 mM HA solution is $\_\_\_\_$ . (Nearest integer) [Given : The degree of dissociation can be neglected with respect to unity]