WB JEE 2026

Let $a_1, a_2, a_3 \ldots$ are in G.P. such that $n>m, a_n>a_m$ and $a_1+a_n=66, a_2 \cdot a_{n-1}=128$. If $\sum_{r=1}^n a_r=126$, then $n$ is

If $t_n$ denotes the $n^{\text {th }}$ term of an A.P. and $t_p=\frac{1}{q}, t_q=\frac{1}{p}$, then which one of the following options is a root of the equation $(p+2 q-3 r) x^2+(q+2 x-3 p) x+(r+2 p-3 q)=0 ?$

On the set $\mathbb{R}$ of real numbers the relation $\rho$, defined by $\mathrm{x} \rho \mathrm{y}(\mathrm{x}, \mathrm{y} \in \mathbb{R})$ iff

Let $A_1, A_2, \ldots, A_6$ are six sets, each with four elements and $B_1, B_2, \ldots ., B_n$ are $n$ sets, each with two elements. Let $S=A_1 \cup A_2 \cup \ldots \cup A_6=B_1 \cup B_2 \cup \ldots \cup B_n$.
Given that each element of $S...

The ends $A$, $B$ of a straight line segment of constant length $c$ slide upon the fixed rectangular axes $O X, O Y$ respectively. If the rectangle $O A P B$ completed, then the locus of the foot of perpendicular drawn from $P$ to $A B$ is

Intercepts of the plane $\vec{r} \cdot \vec{n}=d(\neq 0)$ on the coordinate axes respectively are

Consider a square $A B C D$ of diagonal length 2a. The square is folded along the diagonal $A C$ so that the plane of $\triangle A B C$ is perpendicular to the plane of $\triangle A D C$. In this case the shortest distance between $A B$ and...

The point of intersection of $\vec{r} \times \vec{a}=\vec{b} \times \vec{a}$ and $\vec{r} \times \vec{b}=\vec{a} \times \vec{b}$, where $\vec{a}=\hat{i}+\hat{j}$ and $\vec{b}=2 \hat{i}-\hat{k}$ is

$\theta$ elimination from the equation $x^2+y^2=\frac{x \cos 3 \theta+y \sin 3 \theta}{\cos ^3 \theta}=\frac{y \cos 3 \theta-x \sin 3 \theta}{\sin ^3 \theta}$ will be

The general solution of the equation $\sin ^{100} \mathrm{x}-\cos ^{100} \mathrm{x}=1$ is

A vector given by $\vec{P}=f(t) \hat{i}+g(t)+\hat{k}$ moves in such a way that it is always parallel to the vector $\vec{Q}=-f^{\prime \prime}(t) \hat{i}+f^{\prime}(t) \hat{j}+\hat{k}$.

Let $\vec{a}=(x, y, z)$ be the vector with $|\vec{a}|=2 \sqrt{3}$, which makes equal angles with the vector $\vec{b}=(y,-2 z, 3 x)$ and $\vec{c}=(2 z, 3 x,-y)$ and is perpendicular to the vector $\vec{d}=(1,-1,2)$. If the angle between $\ve...

Let $\vec{r}=\sin x(\vec{a} \times \vec{b})+\cos y(\vec{b} \times \vec{c})+2(\vec{c} \times \vec{a})$ ，where $\vec{a}, \vec{b}$ and $\vec{c}$ are three non－coplanar vectors． It is given that $\vec{r}$ is perpendicular to $(\vec{a}+\vec{b}+\...

If $\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=\hat{i}-\hat{j}+\hat{k}, \vec{c}=\hat{i}+2 \hat{j}-\hat{k}$, then the value of $\left|\begin{array}{lll}\vec{a} \cdot \vec{a} & \vec{a} \cdot \vec{b} & \vec{a} \cdot \vec{c} \\ \vec{b} \cdot \vec...

The position vectors of two adjacent sides $\overrightarrow{O A}$ and $\overrightarrow{O B}$ of a rectangle $O A C B$ are $\vec{a}$ and $\vec{b}$ respectively, where $O$ is the origin. If $16|\vec{a} \times \vec{b}|=3(|\vec{a}|+|\vec{b}|)^2...

A resistor of resistance＇$R$＇draws power＇$P$＇when connected to an AC source．If an inductance is now placed in series with $R$ ，such that the impedance of the circuit becomes＇$Z$＇，the power drawn will be

The de-Broglie wavelength of an electron in 4th orbit is (where $r=$ radius of the 1st orbit)

A radioactive element ${ }_{92}^{242} \mathrm{X}$ emits two $\alpha$－particles，one electron and two positrons．The transformed nucleus is represented by ${ }_P^{234} Y$ ．The value of $P$ is

A particle of mass $m$ is suspended from a point O by a string of length $R$ ．It is given a velocity $u=3 \sqrt{g R}$ at the bottom．The difference in tension at point $B$ and at the point $C$ is $m$

A body initially at rest and sliding along a frictionless track from a height＇$h$＇（as shown in figure） just completes a vertical circle of diameter $\mathrm{AB}=d$ ．The height＇$h$＇is equal to

Consider a fuse wire of length $l$ and radius $r$ ．The time of heating $(t)$ for passing the maximum current will depend on

Radiation of wavelength $\lambda$ is incident on a photocell．The fastest emitted electron has speed $v$ . If the wavelength is changed to $\frac{3 \lambda}{4}$ ，then the speed of the fastest emitted electron will be

A square of side $L$ lies in the $x-y$ plane，where the magnetic field is given by $B=B_0(2 \hat{i}+3 \hat{j}+4 \hat{k})$ where $B_0$ is constant．The magnetic flux passing through the square is $L$

A uniform but time varying magnetic field is present in a circular region of radius ' $R$ '. The magnetic field is perpendicular and into the plane of loop and the magnitude of field is increasing at a constant rate $\alpha$. There is a str...

The displacement current flows through a capacitor when the voltage across its plates

For Boolean variables $A$ and $B, A \oplus B=A \bar{B}+\bar{A} B$ ．Then，which of the following statements is / are correct？

The inputs to a digital circuit are as shown below. The output Y is

The I-V characteristics graph shown below is exhibited by

The circuit has two oppositely connected ideal diodes in parallel as shown in the figure．What is the current flowing in the circuit？

A simple pendulum of length $l$ has a bob of mass $m$ ，with a charge $q$ ．On it a vertical sheet of charge， with surface charge density＇$\sigma$＇passes through the point of suspension．At equilibrium，if the string makes an angle $\theta$ wit...

A person has a minimum distance of distinct vision of 50 cm ．The power of lenses required to read a book at a distance of 25 cm is

A plano－convex lens fits exactly into a plano－concave lens．Their plane surfaces are parallel to each other．If lenses are made of different materials of refractive indices $\mu_1$ and $\mu_2$ and $R$ is the radius of curvature of the curved ...

An electromagnetic wave，whose wave normal makes an angle of $45^{\circ}$ with the vertical，is travelling in air and strikes a horizontal liquid surface．While travelling through the liquid，it gets deviated by $15^{\circ}$ ．If the speed of el...

A ray of light travelling in air is incident on one face of a parallel glass slab of thickness $t$ and refractive index $\mu$ at an angle of incidence $i$ ．Total time spent by the ray inside the slab is

2 moles of an ideal gas with $\frac{C_p}{C_v}=\frac{5}{3}$ are mixed with 3 moles of another ideal gas with $\frac{C_p}{C_v}=\frac{4}{3}$ ．The value of $\frac{C_p}{C_v}$ for the mixture is

Two identical metal bars are heated in two different temperatures and allowed to cool in the same surroundings. Which one of the following figures correctly shows their cooling curves?

Three blocks of masses $m_1=2 \mathrm{~kg}, m_2=3 \mathrm{~kg}$ and $m_3=5 \mathrm{~kg}$ are placed on a horizontal frictionless surface and a force of 30 N pulls the system as shown below．The value of tension $T$ will be

A circular coil, carrying current, has radius $R$. The distance from the centre of the coil on the axis where the magnetic induction will be $\frac{1}{27}$ th of its value at the centre of the coil is

The magnetic moment of an iron bar is $M$. It is now bent in such a way that it forms an arc section of a circle subtending an angle of $60^{\circ}$ at the centre. The magnetic moment of the arc section is

There is a ring of radius $r$ having linear charge density $\lambda$ and rotating with a uniform angular velocity $\omega$. The magnitude of the magnetic field produced by this ring at its own centre would be ( $\mu_0=$ permeability of air)...

Which of the velocity-time $(v-t)$ graph(s) can possibly represent one-dimensional motion of a particle?

From a tower of height $H$ ，a particle is thrown vertically upwards with a speed $u$ ．The time taken by the particle to hit the ground is $n$ times that taken by it to reach the highest point of its path．The relation between $H, u$ and $n$ ...

A body of density＇$\rho$＇is dropped slowly on the surface of a lake of depth $d$ ．If the density of the lake water be＇$\rho^{\prime \prime}\left(\rho^{\prime}<\rho\right)$ then the time taken by the body to reach the bottom of the lake is ‘

Two spherical soap bubbles of radii $r_1$ and $r_2$ in vaccum coalesce under isothermal condition．The newly formed bubble has a radius（ $r$ ）given by

Which one of the following graphs represents the velocity-time $(v-t)$ graph of a small spherical body falling in a viscous liquid?

A pipe $A$ is connected with other pipes $B$ and $C$ as shown in the figure．The areas of cross－section of $A, B$ and $C$ are respectively $\alpha, \frac{\alpha}{2}$ and $\frac{\alpha}{4}$ ．If the velocities of flow of water through $A$ and ...

The moment of inertia of a thin disc about axes $a, b, c, d$ are $\mathbf{I}_1, \mathbf{I}_2, \mathbf{I}_3$ and $\mathbf{I}_4$ respectively，as shown in figure．If the moment of inertia about an axis passing through the centre and perpendicul...

A uniform $\operatorname{rod} A B$ is suspended from a point $P$ ，at a variable distance $x$ ，from $A$ ，as shown in figure． To make the rod horizontal，a mass＇$m$＇is suspended from its end $A$ ．Which set of variables will give a straight lin...

Density and volume of a body are given as $(20 \pm 4) \mathrm{gm} / \mathrm{cm}^3$ and $(10 \pm 1) \mathrm{cm}^3$ respectively. The absolute error in measurement of mass is

The velocity $v$ of a particle at time $t$ is given by $v=a t+\frac{b}{t+c}$, where $a, b$ and $c$ are constants. The dimension of $a, b$ and $c$ are, respectively