JEE Advanced 2026 Question Paper with Solutions

A solid cylinder of radius R rolls without slipping with a center of mass speed $v_0 = \sqrt{gR/3}$ on a horizontal surface with a vertical edge, as shown in the figure. Here, g is the acceleration due to gravity. At the moment when the cylinder loses contact with the surface due to rotation around the corner, the speed of its center of mass is:

Consider a large disk of radius R and two smaller disks, each of radius $r = R/50$, lying on its circumference, as shown in the figure. The smaller disks are initially in contact with each other, with an angular separation __ between their center. They are made to roll without slipping in opposite directions, with constant angular velocities _ and 2_ while the large disk is held stationary. The time t at which the smaller disks are again in contact is:

List-I shows four planar structures made of uniform solid rods each of mass m and length l. In the List-II the possible moment of inertia of these structures about an axis OO', which lies in the plane of the structures, are given. Choose the option that describes the correct match between the entries in List-I to those in List-II.\n$\quad$List-I: (P), (Q), (R), (S) [four rod structures].\n$\quad$List-II: (1) $(5/4)ml^2$; (2) $(1/6)ml^2$; (3) $(1/12)ml^2$; (4) $(2/3)ml^2$; (5) $(1/3)ml^2$.

List-I shows four configurations made of straight and semi-circular narrow tubes containing air. A sound wave of wavelength _ = 0.29 m enters these structures at the point S and a sound detector is placed at D. Between the points S and D the sound travels only through the tubes. The List-II contains the possible smallest values of l (refer to the figures) for which the detector D records maximum amplitude. Ignore effects of sharp corners. [Given cos(15�) = 0.97]. Choose the option that best describes the match between the entries in List-I to those in List-II. List-I: (P), (Q), (R), (S). List-II: (1) 1.32 m; (2) 1.19 m; (3) 0.51 m; (4) 0.29 m; (5) 0.

A particle is thrown with a speed v from a point O at an angle _ with the horizontal plane such that it passes through the point P at a height of 1 m and horizontal distance of 5 m from O, as shown in the figure. If acceleration due to gravity is g m/s$^2$, then the correct statement(s) is/are:

Match the major products obtained in the reactions given in List-I with the corresponding structures in List-II and choose the correct option.\n$\quad$ List-I: (P) $\ce{C6H5-CH2-CN} \xrightarrow{\text{aqueous NaOH}}$; (Q) $\ce{C6H5-CH2-CN} \xrightarrow{(1)\ (CH_3CO)_2O,\ (2)\ Na_2CO_3}$; (R) $\ce{p-O2N-C6H4-CH2-Br} \xrightarrow{\text{aqueous NaOH}}$; (S) $\ce{p-O2N-C6H4-CH2-Br} \xrightarrow{\text{aqueous } Na_2CO_3}$.\n$\quad$ List-II: (1) $\ce{p-O2N-C6H4-CH2-OH}$; (2) $\ce{p-O2N-C6H4-CH2-NH2}$; (3) $\ce{C6H5-CH2-COOH}$; (4) $\ce{C6H5-CH2-CONH2}$; (5) $\ce{p-O2N-C6H4-CH2-NHCOCH3}$.

The List-II contains products obtained from the reaction of compounds in List-I with O3/Zn�H2O followed by cyclization (via more stable enolate) in the presence of aqueous NaOH. Match each entry in List-I with appropriate entry in List-II and choose the correct option.

Consider the following species: List-I contains different molecular shapes and List-II contains total number of species with the same molecular shapes from the given species. Match each entry in List-I with the appropriate entry in List-II and choose the correct option. Given species: $\ce{SOCl2}$, $\ce{XeOF4}$, $\ce{ClF3}$, $\ce{ClF5}$, $\ce{XeF5+}$, $\ce{SO3^2-}$, $\ce{XeF4}$, $\ce{SF4}$.\n$\quad$List-I: (P) See-saw; (Q) T-Shaped; (R) Trigonal Planar; (S) Square Pyramidal.\n$\quad$List-II: (1) One; (2) Two; (3) Three; (4) Four; (5) Zero.

List-I contains various physical/chemical processes, and List-II contains combinations of changes in enthalpy (_H) and entropy (_S). Match each entry in List-I to the appropriate entry in List-II, and choose the correct option.\n$\quad$List-I: (P) Physisorption; (Q) Diamond _ Graphite; (R) Denaturation of protein; (S) Propene _ Cyclopropane.\n$\quad$List-II: (1) _H > 0 and _S > 0; (2) _H 0 and _S 0.

Considering LiBH4 reduces an ester group to the corresponding alcohol and does not reduce a carboxylic acid group, the correct statement about the major products P, Q, R and S is

Match each entry in List-I to the correct entry in List-II and choose the correct option.\n$\quad$ List-I: (P) The circle with centre (1, 2) and touching the straight line $3x + 4y = 1$, passes through; (Q) The common tangent to the circle $x^2 + y^2 = 2$ and the parabola $y^2 = 8x$ with positive slope, passes through; (R) Let M be the end point of the latus rectum of the ellipse $3x^2 + 4y^2 = 48$ such that M lies in the first quadrant. Then the normal to the ellipse drawn at M passes through; (S) Let H be the hyperbola whose centre is at the origin, one of the foci is at (5,0), and one directrix is $5x + 16 = 0$. Then H passes through.\n$\quad$ List-II: (1) the point (1, 1); (2) the point (7, 9); (3) the point (3, 2); (4) the point (2, 5); (5) the point $(8, 3\sqrt{3})$.

Match each entry in List-I to the correct entry in List-II and choose the correct option.\n$\quad$ List-I: (P) The number of elements in the set $\{x \in [-\pi, \pi] : \sin x + \cos x = 1\}$; (Q) The number of elements in the set $\{x \in [-\pi/2, \pi/2] : \sin^6 x + \cos^6 x = 1\}$; (R) The number of elements in the set $\{x \in [-\pi, \pi] : \cos^2(x/2) - \sin x = 1/2\}$; (S) The number of elements in the set $\{x \in [-2\pi, 2\pi] : 6\sin^2(x/2) - \cos 3x = 3\}$.\n$\quad$ List-II: (1) is 1; (2) is 2; (3) is 3; (4) is 4; (5) is 5.

Match each entry in List-I to the correct entry in List-II and choose the correct option.\n$\quad$ List-I: (P) If _ and _ are the distinct roots of the equation $x^2 + x + 1 = 0$, then the quadratic equation with roots $1/(\alpha+1)^{2026}$ and $1/(\beta+1)^{2026}$ is; (Q) If _ and _ are the distinct roots of the equation $x^2 + x + 1 = 0$, then the quadratic equation with roots $1/(\alpha+1)^{\max}$ and $1/(\beta+1)^{\max}$ is; (R) If _ and _ are the distinct roots of the equation $x^2 - x + 1 = 0$, then the value of $1/(\gamma-1)^{\max}$ is; (S) If _ and _ are the distinct roots of the equation $x^2 + x - 1 = 0$, then the value of $1/(\sigma+1)^d$ is.\n$\quad$ List-II: (1) $x^2 + x + 1 = 0$; (2) $x^2 - x + 1 = 0$; (3) $x^2 + x - 1 = 0$; (4) $-1$; (5) $-4$.

Let P be the point on the parabola $y = x^2$ such that the slope of the tangent to the parabola at the point P is 4. Let Q be the point in the first quadrant lying on the circle $x^2 + y^2 = 2$ such that the slope of the tangent to the circle at the point Q is $-1$. Let R be the point in the first quadrant lying on the ellipse $x^2 + 4y^2 = 8$ such that the slope of the tangent to the ellipse at the point R is $-1/2$. Then the radius of the circle passing through the points P, Q and R is

Let $\alpha = \left(1 - 2\cos\left(\frac{\pi}{11}\right)\right)\left(1 - 2\cos\left(\frac{3\pi}{11}\right)\right)\left(1 - 2\cos\left(\frac{9\pi}{11}\right)\right)\left(1 - 2\cos\left(\frac{27\pi}{11}\right)\right)\left(1 - 2\cos\left(\frac{81\pi}{11}\right)\right)$. Then the value of $5 - \alpha^2$ is \underline{\hspace{2cm}}.

Consider a circuit consisting of a capacitor of capacitance C and a coil with N turns per unit length, cross sectional area S and length d, where $d^2 \gg S$. There is another coil of length $d/2$, cross sectional area $S/2$ and $2N$ turns per unit length completely inside the larger coil, as shown in the figure. The ends of this smaller coil are connected with each other by an isolated conducting wire. The self-inductance of the larger coil is L. Neglecting edge effects and all the Ohmic resistances, the resonant frequency of the circuit is:

List-I contains four conducting loops lying in the XY plane, as shown in the figures. The loops are rotating about the axis passing through the point O with time period T in clockwise direction. The region x > 0 contains a uniform magnetic field B in the +z direction. List-II contains the qualitative variation of the induced current i(t) for each of these loops. Choose the option which describes the correct match between the entries in List-I to those in List-II.

In List-I, four optical effects are mentioned. The physical phenomena of light which are essential to describe these optical effects are given in List-II. Choose the option which describes the correct match between the entries in List-I to those in List-II.\n$\quad$List-I: (P) Colorful sky in north polar region (Aurora Borealis); (Q) Partially polarized sun light; (R) Rainbow; (S) Dark and bright fringes.\n$\quad$List-II: (1) Dispersion and reflection; (2) Total internal reflection; (3) Diffraction; (4) Scattering of light by molecules in the atmosphere; (5) Emission of radiation from oxygen and nitrogen atoms excited by charged particles.

The electric field associated with an electromagnetic wave travelling in vacuum is given by $E_0\sin(3y + 4z + \omega t)$, where _ is the angular frequency. All quantities are in SI units. The correct statement(s) about this wave is/are: [Given: speed of light in vacuum $c = 3 \times 10^8$ m/s]

Consider a hydrogen atom with $v_k$, $r_k$, and $K_k$ denoting the velocity, orbital radius and kinetic energy of the electron in the kth orbit, respectively. The electron undergoes a transition from the nth orbit, emitting radiation corresponding to the Lyman series. Considering h to be the Planck's constant and $\varepsilon_0$ the permittivity of the free space, the correct statement(s) is/are:

For real numbers _, _, _, _ and _, consider the matrix $M = \begin{bmatrix} \alpha & 1/\sqrt{2} & -1/\sqrt{2} \\ 1/\sqrt{3} & \beta & 1/\sqrt{3} \\ \gamma & \delta & \mu \end{bmatrix}$. Suppose that $MM^T = I$, where $M^T$ is the transpose of the matrix M, and I is the $3 \times 3$ identity matrix. Let $\vec{u} = \alpha\hat{i} + 1/\sqrt{3}\hat{j} + \gamma\hat{k}$, $\vec{v} = 1/\sqrt{2}\hat{i} + \beta\hat{j} + \delta\hat{k}$ and $\vec{w} = -1/\sqrt{2}\hat{i} + 1/\sqrt{3}\hat{j} + \mu\hat{k}$. Match each entry in List-I to the correct entry in List-II and choose the correct option.\n$\quad$List-I: (P) The value of $\gamma^2 + \delta^2$ is; (Q) If $x\vec{u} + y\vec{v} + z\vec{w} = \hat{j}$ for some real numbers x, y and z, then the value of x is; (R) The value of $[\vec{u} \cdot (\vec{v} \times \vec{w})]$ is; (S) The value of $[\vec{u} \times (\vec{v} \times \vec{w})]$ is.\n$\quad$List-II: (1) 0; (2) 1; (3) $1/\sqrt{2}$; (4) $1/\sqrt{3}$; (5) $2/3$.

Considering only the principal values of the inverse trigonometric functions, the value of $\cot^{-1}(\cot(-11)) + 10\sin(2\cos^{-1}(1/\sqrt{2})) + 10\sin(2\tan^{-1}(2))$ is

Which one of the following matrices can be obtained by performing elementary row transformations on the $3 \times 3$ identity matrix?

Consider the function $f : (0, \infty) \to (-\infty, \infty)$ given by $f(x) = \sqrt{x}\log_a(x) - x + 1$. Then which one of the following statements is TRUE?

Consider the matrix $M = \begin{bmatrix} 2 & -1 \\ 1 & 0 \end{bmatrix}$. Let p, q, r, s, a, b, c and d be integers such that $M^{26} = \begin{bmatrix} p & q \\ r & s \end{bmatrix}$ and $\sum_{k=1}^{26} M^k = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$. Then which of the following statements is (are) TRUE?

For a reversible reaction $R \rightleftharpoons P$, at constant temperature, both the forward and the backward reactions are first order elementary reactions with rate constants $k_f$ and $k_b$, respectively. At time zero, the concentration of R is $[R]_0$ and the concentration of P is zero. At any given time, $[R]$ and $[P]$ are the concentrations of R and P, respectively. If $k_b = 4k_f$, the correct graphical representation of the reaction is

Reaction of PtF$_6$ with oxygen (O$_2$) gas results in the formation of an ionic compound, $X^+[Y]^-$. Correct statement(s) is(are)

Correct statement(s) about the compounds X, Y and Z is(are) [Given: MnO$_2$ + HCl _ MnCl$_2$ + X (greenish yellow gas); X + H$_2$O _ Y + Z; NH$_3$ (excess) + X _ Z]

As shown in the figure, five Carnot engines, each with efficiency _ and same number of cycles per unit time, are operating between six heat reservoirs. The amount of heat released per cycle by one engine is completely absorbed by the next engine. Consider $Q_0$ to be the amount of heat absorbed per cycle by the first engine and W as the amount of total work done by all the engines per cycle, then the net efficiency of the system is found to be $\eta_{net} = \frac{W}{Q_0} = \frac{211}{243}$. The value of _ is:

A tank contains two immiscible liquids of densities $6\rho$ and $2\rho$. The higher density liquid is filled up to a height $L/2$ from the bottom. A thin rod of density _ and length L is fully immersed and hinged at the bottom so that it can oscillate freely, as shown in the figure. If the rod is slightly disturbed from its equilibrium, the time period of small oscillations is $\frac{2\pi}{n}\sqrt{\frac{L}{g}}$ where g is the acceleration due to gravity. The value of n is: