WB JEE 2025

If the sum of ' $n$ ' terms of an A.P. is $3 n^2+5 n$ and its $m$ th term is 164 , then the value of $m$ is

The number of reflexive relations on a set $A$ of $n$ elements is equal to

Consider three points $P(\cos \alpha, \sin \beta), Q(\sin \alpha, \cos \beta)$ and $R(0,0)$, where $0<\alpha, \beta<\frac{\pi}{4}$. Then

The line parallel to the $x$-axis passing through the intersection of the lines $a x+2 b y+3 b=0$ and $b x-2 a y-3 a=0$ where $(a, b) \neq(0,0)$ is

The straight line $\frac{x-3}{3}=\frac{y-2}{1}=\frac{z-1}{0}$ is

If the equation $\sin ^4 x-(p+2) \sin ^2 x-(p+3)=0$ has a solution, the $p$ must lie in the interval

If $\cos (\theta+\phi)=\frac{3}{5}$ and $\sin (\theta-\phi)=\frac{5}{13}, 0<\theta, \phi<\frac{\pi}{4}$, then $\cot (2 \theta)$ has the value

If $0 \leq a, b \leq 3$ and the equation $x^2+4+3 \cos (a x+b)=2 x$ has real solutions, then the value of $(a+b)$ is

The solution set of the equation $\left(x \in\left(0, \frac{\pi}{2}\right)\right) \tan (\pi \tan x)=\cot (\pi \cot x)$, is

Let $f_n(x)=\tan \frac{x}{2}(1+\sec x)(1+\sec 2 x) \ldots\left(1+\sec 2^n x\right)$, then

If $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar vectors and $\lambda$ is a real number then the vectors $\vec{a}+2 \vec{b}+3 \vec{c}, \lambda \vec{b}+4 \vec{c}$ and $(2 \lambda-1) \vec{c}$ are non-coplanar for

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be vectors of equal magnitude such that the angle between $\vec{a}$ and $\vec{b}$ is $\alpha, \vec{b}$ and $\vec{c}$ is $\beta$ and $\vec{c}$ and $\vec{a}$ is $\gamma$. Then the minimum value of $\cos \a...

If ' $\theta$ ' is the angle between two vectors $\vec{a}$ and $\vec{b}$ such that $|\vec{a}|=7,|\vec{b}|=1$ and $|\vec{a} \times \vec{b}|^2=k^2-(\vec{a} \cdot \vec{b})^2$, then the values of $k$ and $\theta$ are

If $\vec{\alpha}=3 \vec{i}-\vec{k},|\vec{\beta}|=\sqrt{5}$ and $\vec{\alpha} \cdot \vec{\beta}=3$, then the area of the parallelogram for which $\vec{\alpha}$ and $\vec{\beta}$ are adjacent sides is

Let $\vec{a}, \vec{b}, \vec{c}$ be unit vectors. Suppose $\vec{a} \cdot \vec{b}=\vec{a} \cdot \vec{c}=0$ and the angle between $\vec{b}$ and $\vec{c}$ is $\frac{\pi}{6}$. Then $\vec{a}$ is

For a domestic AC supply of 220 V at 50 cycles per sec, the potential difference between the terminals of a two-pin electric outlet in a room is given by

A radioactive nucleus decays as follows :
\(X \xrightarrow{\alpha} X_1 \xrightarrow{\beta} X_2 \xrightarrow{\alpha} X_3 \xrightarrow{\gamma} X_4\)
If the mass number and atomic number of ' $X_4$ ' are 172 and 69 respectively, then the ato...

The minimum wavelength of Lyman series lines is $P$, then the maximum wavelength of these lines is

Let the binding energy per nucleon of nucleus is denoted by ' $E_{b n}$ ' and radius of the nucleus is denoted by ' $r$ '. If mass number of nuclei $A$ and $B$ are 64 and 125 respectively, then

The number of undecayed nuclei $N$ in a sample of radioactive material as a function of time $(t)$ is shown in the figure. Which of the following graphs correctly show the relationship between $N$ and the activity ' $A$ '?

What are the charges stored in the $1 \mu \mathrm{~F}$ and $2 \mu \mathrm{~F}$ capacitors in the circuit as shown in figure once the current (I) become steady?

The variation of density of a solid cylindrical rod of cross sectional area $\alpha$ and length $L$ is $\rho=\rho_0 \frac{x^2}{L^2}$, where $x$ is the distance from one end of the rod. The position of its centre of mass from one end $(x=0)$...

A ball falls from a height $h$ upon a fixed horizontal floor. The co-efficient of restitution for the collision between the ball and the floor is ' $e$ '. The total distance covered by the ball before coming to rest is [neglect the air resi...

An electron in Hydrogen atom jumps from the second Bohr orbit to the ground state and the difference between the energies of the two states is radiated in the form of a photon. This photon strikes a material. If the work function of the mat...

The de-Broglie wavelength of a moving bus with speed $v$ is $\lambda$. Some passengers left the bus at a stoppage. Now when the bus moves with twice of its initial speed, its kinetic energy is found to be twice of its initial value. What is...

$10^{20}$ photons of wavelength 660 nm are emitted per second from a lamp. The wattage of the lamp is (Planck's constant $=6.6 \times 10^{-34} \mathrm{Js}$ )

A diode is connected in parallel with a resistance as shown in Figure. The most probable current (I) - voltage (V) characteristic is

Manufacturers supply a zener diode with zener voltage $\mathrm{V}_{\mathrm{z}}=5.6 \mathrm{~V}$ and maximum power dissipation $P_{\mathrm{z}, \max }=\frac{1}{4} \mathrm{~W}$. This zener diode is used in the following circuit. Calculate the ...

Which logic gate is represented by the following combinations of logic gates?

Consider a particle of mass 1 gm and charge 1.0 Coulomb is at rest. Now the particle is subjected to an electric field $E(t)=E_0 \sin \omega t$ in the $x$-direction, where $E_0=2$ Newton/Coulomb and $\omega=1000 \mathrm{rad} / \mathrm{sec}$...

Two charges $+q$ and $-q$ are placed at points $A$ and $B$ respectively which are at a distance $2 L_{\mathrm{p} p a t}$ $C$ is the mid point of $A$ and $B$. The workdone in moving a charge $+Q$ along the semicircle $\operatorname{CSD}\left...

Figure shows the graph of angle of deviation $\delta$ versus angle of incidence i for a light ray striking a prism. The prism angle is

Let $\bar{V}, V_{m s}, V_p$ denotes the mean speed, root mean square speed and most probable speed of the molecules each of mass $m$ in an ideal monoatomic gas at absolute temperature $T$ Kelvin. Which statement(s) is/are correct?

For an ideal gas, a cyclic process ABCA as shown in P-T diagram, when presented in P-V plot, would be

Temperature of a body $\theta$ is slightly more than the temperature of the surrounding $\theta_0$. Its rate of cooling $(R)$ versus temperature of the body $(\theta)$ is plotted. Its shape would be

The minimum force required to start pushing a body up a rough (having co-efficient of friction $\mu$ ) inclined plane is $\vec{F}_1$ while the minimum force needed to prevent it from sliding is $\overrightarrow{F_2}$. If the inclined plane ...

A force $\vec{F}=a \hat{i}+b \hat{j}+c \hat{k}$ is acting on a body of mass $m$. The body was initially at rest at the origin. The co-ordinates of the body after time ' $t$ ' will be

Two spheres $S_1$ and $S_2$ of masses $m_1$ and $m_2$ respectively collide with each other. Initially $S_1$ is at rest and $S_2$ is moving with velocity $v$ along $x$-axis. After collision $S_2$ has a velocity $\frac{v}{2}$ in a direction p...

A particle of charge ' $q$ ' and mass ' $m$ ' moves in a circular orbit of radius ' $r$ ' with angular speed ' $\omega$ '. The ratio of the magnitude of its magnetic moment to that of its angular momentum depends on

Ruma reached the metro station and found that the escalator was not working. She walked up the stationary escalator with velocity $v_1$ in time $t_1$. On other day if she remains stationary on the escalator moving with velocity $v_2$, then ...

Acceleration-time $(a-t)$ graph of a body is shown in the figurd. Corresponding velocity-time $(v-t)$ graph is

One end of a stecl wire is fixed to the ceiling of an elevator moving up with an acceleration $2 \mathrm{~m} / \mathrm{s}^2$ and a load of 10 kg hangs from the other end. If the cross section of the wire is $2 \mathrm{~cm}^2$, then the long...

The apparent coefficient of expansion of a liquid, when heated in a copper vessel is $C$ and when heated in silver vessel is $S$. If $A$ is linear coefficient of expansion of copper, then linear coefficient of expansion of silver is

A piece of granite floats at the interface of mercury and water contained in a beaker as in figure. If the densities of granite, water and mercury are $\rho, \rho_1$ and $\rho_2$ respectively, the ratio of the volume of granite in water to ...

Three different liquids are filled in a U-tube as shown in figure. Their densities are $\rho_1, \rho_2$ and $\rho_3$ respectively. From the figure we may conclude that

A simple pendulum is taken at a place where its distance from the earth's surface is equal to the radius of the earth. Calculate the time period of small oscillations if the length of the string is 4.0 m . (Take $g=\pi^2 \mathrm{~ms}^{-2}$ ...

The variation of displacement with time of a simple harmonic motion (SHM) for a particle of mass $m$ is represented by $y=2 \sin \left(\frac{\pi t}{2}+\phi\right) \mathrm{cm}$. The maximum acceleration of the particle is

A quantity $X$ is given by $\varepsilon_0 L \frac{\Delta V}{\Delta t}$, where $\varepsilon_0$ is the permittivity of free space, $L$ is the length, $\Delta V$ is a potential difference and $\Delta t$ is a time interval. The dimension of $X$...

The resistance $\mathrm{R}=\frac{\mathrm{V}}{\mathrm{I}}$ where $\mathrm{V}=(25 \pm 0.4)$ Volt and $\mathrm{I}=(200 \pm 3)$ Ampere. The percentage error in ' $R$ ' is

If the dimensions of length are expressed as $G^x C^y h^z$, where $G, C$ and $h$ are the universal gravitational constant, speed of light and Planck's constant respectively, then