JEE Main 2026 January 22 Shift 2 Question Paper with Solutions is available. NTA conducted the second shift of JEE Main 2026 on January 22, 2026, from 3:00 PM to 6:00 PM. The paper included questions from Physics, Chemistry, and Mathematics, strictly based on the revised JEE Main syllabus. The overall difficulty level of the Shift 2 paper was reported to be moderate, with Mathematics being comparatively lengthy, while Physics and Chemistry ranged from easy to moderate. Candidates can download the JEE Main 2026 Jan 22 Shift 2 question paper along with detailed solutions to analyze their performance and understand the exam pattern better.
JEE Main 2026 Jan 22 Shift 2 Question Paper with Solutions
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Among the statements
(S1): If \(A(5,-1)\) and \(B(-2,3)\) are two vertices of a triangle whose orthocentre is \((0,0)\), then its third vertex is \((-4,-7)\).
(S2): If positive numbers \(2a, b, c\) are three consecutive terms of an A.P., then the lines \(ax+by+c=0\) are concurrent at \((2,-2)\).
Let \(n\) be the number obtained on rolling a fair die. If the probability that the system \[ \begin{cases} x - ny + z = 6
x + (n-2)y + (n+1)z = 8
(n-1)y + z = 1 \end{cases} \]
has a unique solution is \( \dfrac{k}{6} \), then the sum of \(k\) and all possible values of \(n\) is
Let the domain of the function \[ f(x)=\log_3\!\left[\log_5(7-\log_2(x^2-10x+85))\right] +\sin^{-1}\!\left(\frac{3x-7}{17-x}\right) \]
be \((\alpha,\beta)\). Then \(\alpha+\beta\) is equal to
Let \( [\,\cdot\,] \) denote the greatest integer function, and let \[ f(x) = \min\{\sqrt{2}\,x, x^2\}. \]
Let \[ S = \{x \in (-2,2) : the function g(x) = x[x^2] is discontinuous at x\}. \]
Then \[ \sum_{x \in S} f(x) equals \]
If the mean deviation about the median of the numbers \[ k,\,2k,\,3k,\,\ldots,\,1000k \]
is \(500\), then \(k^2\) is equal to
Let \( P(10, 2\sqrt{15}) \) be a point on the hyperbola \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, \]
whose foci are \( S \) and \( S' \). If the length of its latus rectum is \(8\), then the square of the area of \( \triangle PSS' \) is equal to
The area of the region \[ A = \{(x,y) : 4x^2 + y^2 \le 8 \;and\; y^2 \le 4x\} \]
is
Let the locus of the mid-point of the chord through the origin \(O\) of the parabola \(y^2 = 4x\) be the curve \(S\). Let \(P\) be any point on \(S\). Then the locus of the point, which internally divides \(OP\) in the ratio \(3:1\), is
If \[ X=\begin{bmatrix}x
y
z\end{bmatrix} \]
is a solution of the system of equations \(AX=B\), where \[ adj A= \begin{bmatrix} 4 & 2 & 2
-5 & 0 & 5
1 & -2 & 3 \end{bmatrix} \quad and \quad B=\begin{bmatrix}4
0
2\end{bmatrix}, \]
then \(|x+y+z|\) is equal to
Let \(\alpha, \beta\) be the roots of the quadratic equation \[ 12x^2 - 20x + 3\lambda = 0,\ \lambda \in \mathbb{Z}. \]
If \[ \frac{1}{2} \le |\beta-\alpha| \le \frac{3}{2}, \]
then the sum of all possible values of \(\lambda\) is
Let \(C_r\) denote the coefficient of \(x^r\) in the binomial expansion of \((1+x)^n\), \(n\in\mathbb{N}\), \(0\le r\le n\). If \[ P_n = C_0 - C_1 + \frac{2^2}{3}C_2 - \frac{2^3}{4}C_3 + \cdots + \frac{(-2)^n}{n+1}C_n, \]
then the value of \[ \sum_{n=1}^{25} \frac{1}{2n} P_n \]
equals
The number of elements in the relation \[ R=\{(x,y): 4x^2+y^2<52,\; x,y\in\mathbb{Z}\} \]
is
Let \[ f(x) = [x]^2 - [x+3] - 3,\quad x \in \mathbb{R}, \]
where \([\,\cdot\,]\) denotes the greatest integer function. Then
Let \[ S = \{ z \in \mathbb{C} : 4z^2 + \bar{z} = 0 \}. \]
Then \[ \sum_{z \in S} |z|^2 \]
is equal to
Let \( \vec{a} = 2\hat{i} - \hat{j} + \hat{k} \) and \( \vec{b} = \lambda \hat{j} + 2\hat{k} \), where \( \lambda \in \mathbb{Z} \), be two vectors.
Let \( \vec{c} = \vec{a} \times \vec{b} \) and \( \vec{d} \) be a vector of magnitude \(2\) in the \(yz\)-plane.
If \( |\vec{c}| = \sqrt{53} \), then the maximum possible value of \( (\vec{c}\cdot\vec{d})^2 \) is equal to
If \(y=y(x)\) satisfies the differential equation \[ 16(\sqrt{x}+9\sqrt{x})(4+\sqrt{9+\sqrt{x}})\cos y\,dy=(1+2\sin y)\,dx,\quad x>0 \]
and \[ y(256)=\frac{\pi}{2},\quad y(49)=\alpha, \]
then \(2\sin\alpha\) is equal to
If \[ \lim_{x\to 0} \frac{e^{(a-1)x}+2\cos bx+(c-2)e^{-x}} {x\cos x-\log_e(1+x)} =2, \]
then \(a^2+b^2+c^2\) is equal to
Let \(f\) and \(g\) be functions satisfying \[ f(x+y) = f(x)f(y), \quad f(1) = 7 \] \[ g(x+y) = g(xy), \quad g(1) = 1, \]
for all \(x,y \in \mathbb{N}\). If \[ \sum_{x=1}^{n} \left(\frac{f(x)}{g(x)}\right) = 19607, \]
then \(n\) is equal to
Let \( L \) be the line \[ \frac{x+1}{2} = \frac{y+1}{3} = \frac{z+3}{6} \]
and let \( S \) be the set of all points \( (a,b,c) \) on \( L \), whose distance from the line \[ \frac{x+1}{2} = \frac{y+1}{3} = \frac{z-9}{0} \]
along the line \( L \) is \( 7 \).
Then \[ \sum_{(a,b,c)\in S} (a+b+c) \]
is equal to
Let \(S\) and \(S'\) be the foci of the ellipse \[ \frac{x^2}{25} + \frac{y^2}{9} = 1 \]
and \(P(\alpha,\beta)\) be a point on the ellipse in the first quadrant.
If \[ (SP)^2 + (S'P)^2 - SP \cdot S'P = 37, \]
then \(\alpha^2 + \beta^2\) is equal to
Suppose \(a, b, c\) are in A.P. and \(a^2, 2b^2, c^2\) are in G.P. If \(a < b < c\) and \(a+b+c = 1\), then \(9(a^2+b^2+c^2)\) is equal to ______.
Let \(S\) be the set of the first 11 natural numbers. Then the number of elements in \[ A = \{ B \subseteq S : n(B) \ge 2 and the product of all elements of B is even \} \]
is ______.
Let \( \cos(\alpha + \beta) = -\dfrac{1}{10} \) and \( \sin(\alpha - \beta) = \dfrac{3}{8} \), where \( 0 < \alpha < \dfrac{\pi}{3} \) and \( 0 < \beta < \dfrac{\pi}{4} \).
If \[ \tan 2\alpha = \frac{3(1 - r\sqrt{5})}{\sqrt{11}(s + \sqrt{5})}, \quad r, s \in \mathbb{N}, \]
then the value of \( r + s \) is _______.
Let \([\,]\) be the greatest integer function. If \[ \alpha=\int_{0}^{64}\left(x^{1/3}-[x^{1/3}]\right)\,dx, \]
then \[ \frac{1}{\pi}\int_{0}^{\alpha\pi}\frac{\sin^2\theta}{\sin^6\theta+\cos^6\theta}\,d\theta \]
is equal to ______.
Let a vector \[ \vec a=\sqrt{2}\,\hat i-\hat j+\lambda \hat k,\ \lambda>0, \]
make an obtuse angle with the vector \[ \vec b=-\lambda^2\hat i+4\sqrt{2}\hat j+4\sqrt{2}\hat k \]
and an angle \( \theta \), \( \frac{\pi}{6}<\theta<\frac{\pi}{2} \), with the positive \(z\)-axis. If the set of all possible values of \( \lambda \) is \((\alpha,\beta)-\{\gamma\}\), then \( \alpha+\beta+\gamma \) is equal to ______.
Which of the following are true for a single slit diffraction?
A. Width of central maxima increases with increase in wavelength keeping slit width constant.
B. Width of central maxima increases with decrease in wavelength keeping slit width constant.
C. Width of central maxima increases with decrease in slit width at constant wavelength.
D. Width of central maxima increases with increase in slit width at constant wavelength.
E. Brightness of central maxima increases for decrease in wavelength at constant slit width.
In an open organ pipe \( \nu_3 \) and \( \nu_6 \) are 3rd and 6th harmonic frequencies, respectively.
If \( \nu_6 - \nu_3 = 2200\ Hz \), then the length of the pipe is \hspace{1cm mm.
(Take velocity of sound in air as \(330\ m s^{-1\).)
The correct truth table for the given input data of the following logic gate is:
Given below are two statements:
Statement I: An object moves from position \( \vec{r}_1 \) to position \( \vec{r}_2 \) under a conservative force field \( \vec{F} \).
The work done by the force is \[ W = -\int_{\vec{r}_1}^{\vec{r}_2} \vec{F} \cdot d\vec{r}. \]
Statement II: Any object moving from one location to another location can follow infinite number of paths. Therefore, the amount of work done by the object changes with the path it follows for a conservative force.
In the light of the above statements, choose the correct answer from the options given below:
Light is incident on a metallic plate having work function \(110 \times 10^{-20}\,J\). If the produced photoelectrons have zero kinetic energy, then the angular frequency of the incident light is \hspace{2cm rad/s.
(\(h = 6.63 \times 10^{-34\,J·s\))
Using a simple pendulum experiment \(g\) is determined by measuring its time period \(T\). Which of the following plots represent the correct relation between the pendulum length \(L\) and time period \(T\)?
The smallest wavelength of Lyman series is \(91\ nm\). The difference between the largest wavelengths of Paschen and Balmer series is nearly __________ nm.
An electric power line having total resistance of \(2\,\Omega\), delivers \(1\,kW\) of power at \(250\ V\). The percentage efficiency of the transmission line is __________.
Figure shows the circuit that contains three resistances (\(9\,\Omega\) each) and two inductors (\(4\,mH\) each). The reading of ammeter at the moment switch \(K\) is turned ON, is __________ A.
The wavelength of light while it is passing through water is \(540\,nm\). The refractive index of water is \( \frac{4}{3} \). The wavelength of the same light when it is passing through a transparent medium having refractive index of \( \frac{3}{2} \) is __________ nm.
Given below are two statements:
Statement I: For a mechanical system of many particles, total kinetic energy is the sum of kinetic energies of all the particles.
Statement II: The total kinetic energy can be the sum of kinetic energy of the center of mass with respect to the origin and the kinetic energy of all the particles with respect to the center of mass as reference.
In the light of the above statements, choose the correct answer from the options given below:
Five positive charges each having charge \(q\) are placed at the vertices of a regular pentagon as shown in the figure. The electric potential \(V\) and the electric field \(\vec{E}\) at the center \(O\) of the pentagon due to these five positive charges are
Consider two boxes containing ideal gases \(A\) and \(B\) such that their temperatures, pressures and number densities are same. The molecular size of \(A\) is half of that of \(B\) and mass of molecule \(A\) is four times that of \(B\). If the collision frequency in gas \(B\) is \(32\times10^8\ s^{-1}\), then collision frequency in gas \(A\) is \hspace{1.5cm\,\(s^{-1\).
Given below are two statements:
Statement I: A satellite is moving around earth in an orbit very close to the earth surface. The time period of revolution of satellite depends upon the density of earth.
Statement II: The time period of revolution of the satellite is \[ T = 2\pi \sqrt{\frac{R_e}{g}} \]
(for satellite very close to the earth surface), where \(R_e\) is the radius of earth and \(g\) is acceleration due to gravity.
In the light of the above statements, choose the correct answer from the options given below.
In parallax method for the determination of focal length of a concave mirror, the object should always be placed:
When a part of a straight capillary tube is placed vertically in a liquid, the liquid rises upto certain height \( h \).
If the inner radius of the capillary tube, density of the liquid and surface tension of the liquid decrease by \(1%\) each, then the height of the liquid in the tube will change by ____ %.
A uniform bar of length \(12\,cm\) and mass \(20m\) lies on a smooth horizontal table. Two point masses \(m\) and \(2m\) are moving in opposite directions with the same speed \(v\) and in the same plane as the bar, as shown in the figure. These masses strike the bar simultaneously and get stuck to it. After collision the entire system is rotating with angular frequency \(\omega\). The ratio of \(v\) and \(\omega\) is
A laser beam has intensity of \(4.0\times10^{14}\ W/m^2\). The amplitude of magnetic field associated with the beam is \hspace{1.5cm\ T.
(Take \(\varepsilon_0=8.85\times10^{-12\ C^2/N m^2\) and \(c=3\times10^8\ m/s\))
Three small identical bubbles of water having same charge on each coalesce to form a bigger bubble.
Then the ratio of the potentials on one initial bubble and that on the resultant bigger bubble is:
If \(\varepsilon_0\), \(E\) and \(t\) represent the free space permittivity, electric field and time respectively, then the unit of \[ \frac{\varepsilon_0 E}{t} \]
will be
A capacitor \(P\) with capacitance \(10\times10^{-6}\,F\) is fully charged with a potential difference of \(6.0\,V\) and disconnected from the battery. The charged capacitor \(P\) is connected across another capacitor \(Q\) with capacitance \(20\times10^{-6}\,F\). The charge on capacitor \(Q\) when equilibrium is established will be \( \alpha \times 10^{-5}\,C \) (assume capacitor \(Q\) does not have any charge initially). The value of \( \alpha \) is _____.
A conducting circular loop is rotated about its diameter at a constant angular speed of \(100 \, rad s^{-1}\) in a magnetic field of \(0.5 \, T\), perpendicular to the axis of rotation. When the loop is rotated by \(30^\circ\) from the horizontal position, the induced EMF is \(15.4 \, mV\). The radius of the loop is _______ mm.
(Take \( \pi = \dfrac{22}{7} \))
A cylindrical conductor of length \(2 \, m\) and area of cross-section \(0.2 \, mm^2\) carries an electric current of \(1.6 \, A\) when its ends are connected to a \(2 \, V\) battery. Mobility of electrons in the conductor is \( \alpha \times 10^{-3} \, m^2/V s. \) The value of \( \alpha \) is _______.
(Electron concentration \(= 5 \times 10^{28} \, m^{-3}\), electron charge \(= 1.6 \times 10^{-19} \, C\))
Two masses \(m\) and \(2m\) are connected by a light string going over a pulley (disc) of mass \(30m\) with radius \(r=0.1\,m\). The pulley is mounted in a vertical plane and is free to rotate about its axis. The \(2m\) mass is released from rest and its speed when it has descended through a height of \(3.6\,m\) is ______ m/s. (Assume string does not slip and \(g=10\,m s^{-2}\)).
An insulated cylinder of volume \(60\,cm^3\) is filled with a gas at \(27^\circC\) and \(2\) atmospheric pressure. The gas is then compressed making the final volume \(20\,cm^3\) while allowing the temperature to rise to \(77^\circC\). The final pressure is ______ atmospheric pressure.
[Ni(PPh\(_3\))\(_2\)Cl\(_2\)] is a paramagnetic complex. Identify the \underline{INCORRECT statements about this complex.
A. The complex exhibits geometrical isomerism.
B. The complex is white in colour.
C. The calculated spin-only magnetic moment of the complex is \(2.84\) BM.
D. The calculated CFSE (Crystal Field Stabilization Energy) of Ni in this complex is \(-0.8\Delta_o\).
E. The geometrical arrangement of ligands in this complex is similar to that in Ni(CO)\(_4\).
Choose the correct answer from the options given below:
Consider the following reaction:
The product \(Y\) formed is:
The IUPAC name of the following compound is:
Given below are two statements:
Statement I: The first ionization enthalpy of Cr is lower than that of Mn.
Statement II: The second and third ionization enthalpies of Cr are higher than those of Mn.
In the light of the above statements, choose the correct answer from the options given below:
At \(T\) K, \(100\,g\) of \(98%\) \(H_2SO_4\) (w/w) aqueous solution is mixed with \(100\,g\) of \(49%\) \(H_2SO_4\) (w/w) aqueous solution. What is the mole fraction of \(H_2SO_4\) in the resultant solution?
(Given: Atomic mass \(H = 1\,u,\; S = 32\,u,\; O = 16\,u\).
Assume that temperature after mixing remains constant.)
The compound \(A\), C\(_8\)H\(_8\)O\(_2\), reacts with acetophenone to form a single product via cross-aldol condensation. The compound \(A\) on reaction with conc. NaOH forms a substituted benzyl alcohol as one of the two products. The compound \(A\) is:
Correct statements regarding Arrhenius equation among the following are:
[label=\Alph*.]
Factor \(e^{-E_a/RT}\) corresponds to fraction of molecules having kinetic energy less than \(E_a\).
At a given temperature, lower the \(E_a\), faster is the reaction.
Increase in temperature by about \(10^\circC\) doubles the rate of reaction.
Plot of \(\log k\) vs \(\dfrac{1}{T}\) gives a straight line with slope \(= -\dfrac{E_a}{R}\).
Choose the correct answer from the options given below:
Which of the following mixture gives a buffer solution with \( pH = 9.25 \)?
Given: \( \mathrm{p}K_b(\mathrm{NH_4OH}) = 4.75 \)
Among \( \mathrm{H_2S}, \mathrm{H_2O}, \mathrm{NF_3}, \mathrm{NH_3} \) and \( \mathrm{CHCl_3} \), identify the molecule \( (X) \) with lowest dipole moment value.
The number of lone pairs of electrons present on the central atom of the molecule \( (X) \) is
Given below are two statements:
Statement I:
Elements \(X\) and \(Y\) are the most and least electronegative elements, respectively, among \(N\), \(As\), \(Sb\) and \(P\). The nature of the oxides \(X_2O_3\) and \(Y_2O_3\) is acidic and amphoteric, respectively.
Statement II: \(BCl_3\) is covalent in nature and gets hydrolysed in water. It produces \([B(OH)_4]^-\) and \([B(H_2O)_6]^{3+}\) in aqueous medium.
In the light of the above statements, choose the correct answer from the options given below:
Match List-I with List-II.
\begin{tabular{|c|l||c|l|
\hline
List-I & Reaction of Glucose with & List-II & Product formed
\hline
A. & Hydroxylamine & I. & Gluconic acid
B. & Br\(_2\) water & II. & Glucose pentaacetate
C. & Excess acetic anhydride & III. & Saccharic acid
D. & Concentrated HNO\(_3\) & IV. & Glucoxime
\hline
\end{tabular
Choose the correct answer from the options given below:
Consider the following reduction processes:
\[ Al^{3+} + 3e^- \longrightarrow Al(s), \quad E^\circ = -1.66\,V \] \[ Fe^{3+} + e^- \longrightarrow Fe^{2+}, \quad E^\circ = +0.77\,V \] \[ Co^{3+} + e^- \longrightarrow Co^{2+}, \quad E^\circ = +1.81\,V \] \[ Cr^{3+} + 3e^- \longrightarrow Cr(s), \quad E^\circ = -0.74\,V \]
The tendency to act as reducing agent decreases in the order:
3,3-Dimethyl-2-butanol cannot be prepared by:
\(A + 2B \longrightarrow AB_2\)
\(36.0\,g\) of \(A\) (Molar mass \(= 60\,g mol^{-1}\)) and \(56.0\,g\) of \(B\) (Molar mass \(= 80\,g mol^{-1}\)) are allowed to react. Which of the following statements are correct?
[A.] \(A\) is the limiting reagent.
[B.] \(77.0\,g\) of \(AB_2\) is formed.
[C.] Molar mass of \(AB_2\) is \(140\,g mol^{-1}\).
[D.] \(15.0\,g\) of \(A\) is left unreacted after completion of reaction.
Choose the correct answer from the options given below:
When \(1\) g of compound \((X)\) is subjected to Kjeldahl's method for estimation of nitrogen, \(15\) mL of \(1\) M \(\mathrm{H_2SO_4}\) was neutralized by ammonia evolved. The percentage of nitrogen in compound \((X)\) is:
The energy of first (lowest) Balmer line of H atom is \(x\) J.
The energy (in J) of second Balmer line of H atom is:
Identify the correct statements:
A. Hydrated salts can be used as primary standard.
B. Primary standard should not undergo any reaction with air.
C. Reactions of primary standard with another substance should be instantaneous and stoichiometric.
D. Primary standard should not be soluble in water.
E. Primary standard should have low relative molar mass.
Choose the correct answer from the options given below:
Given below are two statements:
Statement I: \( \mathrm{C < O < N < F} \) is the correct order in terms of first ionization enthalpy values.
Statement II: \( \mathrm{S > Se > Te > Po > O} \) is the correct order in terms of the magnitude of electron gain enthalpy values.
In the light of the above statements, choose the correct answer from the options given below:
The dibromo compound \(P\) (molecular formula: \(C_9H_{10}Br_2\)) when heated with excess sodamide followed by treatment with dilute HCl gives \(Q\). On warming \(Q\) with mercuric sulphate and dilute sulphuric acid yields \(R\), which gives a positive iodoform test but a negative Tollens' test. The compound \(P\) is:
If the enthalpy of sublimation of Li is \(155\,kJ mol^{-1}\), enthalpy of dissociation of \( \mathrm{F_2} \) is \(150\,kJ mol^{-1}\), ionization enthalpy of Li is \(520\,kJ mol^{-1}\), electron gain enthalpy of F is \(-313\,kJ mol^{-1}\), and standard enthalpy of formation of LiF is \(-594\,kJ mol^{-1}\), then the magnitude of lattice enthalpy of LiF is ______ kJ mol\(^{-1}\) (Nearest integer).
Consider \( A \xrightarrow{k_1} B \) and \( C \xrightarrow{k_2} D \) are two reactions. If the rate constant (\(k_1\)) of the \( A \rightarrow B \) reaction can be expressed by the following equation
\[ \log_{10} k = 14.34 - \frac{1.5 \times 10^{4}}{T/K} \]
and activation energy of \( C \rightarrow D \) reaction (\(E_{a2}\)) is \( \dfrac{1}{5} \)th of the \( A \rightarrow B \) reaction (\(E_{a1}\)), then the value of (\(E_{a2}\)) is _______ kJ mol\(^{-1}\) (Nearest Integer).
Among the following oxides of 3d elements, the number of mixed oxides are ______.
\[ \mathrm{Ti_2O_3,\ V_2O_4,\ Cr_2O_3,\ Mn_3O_4,\ Fe_3O_4,\ Fe_2O_3,\ Co_3O_4} \]
Consider the following electrochemical cell:
\[ Pt | \ \mathrm{O_2(g,1\,bar)} \ | \ \mathrm{HCl(aq)} \ || \ \mathrm{M^{2+}(aq,1.0\,M)} \ | \ \mathrm{M(s)} \]
The pH above which oxygen gas would start to evolve at the anode is ______ (nearest integer).
Given:
\[ E^\circ_{\mathrm{M^{2+}/M}} = 0.994\,V \] \[ E^\circ_{\mathrm{O_2/H_2O}} = 1.23\,V \] \[ \frac{RT}{F}(2.303)=0.059\,V \]
The mass of benzanilide obtained from the benzoylation reaction of \(5.8 \, g\) of aniline, if yield of product is \(82%\), is _______ g (nearest integer).
(Given molar mass in g mol\(^{-1}\): H : 1, C : 12, N : 14, O : 16)
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Based on initial analysis, JEE Main Jan 22nd Shift 2 was Easy to Semi-Difficult, where Chemistry remains the lengthiest section. Students can access the JEE Main Jan 22 Shift 2 official questions with answer keys here.
Students can check the detailed paper analysis for JEE Main Jan 22nd Shift 2 here.
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