WB JEE 2023

Let A, B, C are subsets of set X. Then consider the validity of the following set theoretic statement:

Let X be a nonvoid set. If \(\rho_1\) and \(\rho_2\) be the transitive relations on X, then
(\(\circ\) denotes the composition of relations)

Let \(\rho\) be a relation defined on set of natural numbers N, as \(\rho = \{ (x,y) \in N \times N:2x + y = 4\}\). Then domain A and range B are

If R and R\(^1\) are equivalence relations on a set A, then so are the relations

A, B are fixed points with coordinates (0, a) and (0, b) (a > 0, b > 0). P is variable point (x, 0) referred to rectangular axis. If the angle \(\angle\)APB is maximum, then

The equation \({r^2}{\cos ^2}\left( {\theta - {\pi \over 3}} \right) = 2\) represents

If \(4{a^2} + 9{b^2} - {c^2} + 12ab = 0\), then the family of straight lines \(ax + by + c = 0\) is concurrent at

The straight lines \(x + 2y - 9 = 0,3x + 5y - 5 = 0\) and \(ax + by - 1 = 0\) are concurrent if the straight line \(35x - 22y + 1 = 0\) passes through the point

The locus of points (x, y) in the plane satisfying \({\sin ^2}x + {\sin ^2}y = 1\) consists of

A rectangle ABCD has its side parallel to the line y = 2x and vertices A, B, D are on lines y = 1, x = 1 and x = \(-\)1 respectively. The coordinate of C can be

If the distance between the plane \(\alpha x - 2y + z = k\) and the plane containing the lines \({{x - 1} \over 2} = {{y - 2} \over 3} = {{z - 3} \over 4}\) and \({{x - 2} \over 3} = {{y - 3} \over 4} = {{z - 4} \over 5}\) is \(\sqrt 6\), ...

The angle between a normal to the plane \(2x - y + 2z - 1 = 0\) and the X-axis is

If \({1 \over 6}\sin \theta ,\cos \theta ,\tan \theta\) are in G.P, then the solution set of \(\theta\) is
(Here \(n \in N\))

The value of 'a' for which the scalar triple product formed by the vectors \(\overrightarrow \alpha = \widehat i + a\widehat j + \widehat k,\overrightarrow \beta = \widehat j + a\widehat k\) and $$\overrightarrow \gamma = a\widehat i ...

If the volume of the parallelopiped with \(\overrightarrow a \times \overrightarrow b ,\overrightarrow b \times \overrightarrow c\) and \(\overrightarrow c \times \overrightarrow a\) as conterminous edges is 9 cu. units, then the volum...

If the potential energy of a hydrogen atom in the first excited state is assumed to be zero, then the total energy of n = \(\infty\) state is,

A uniform magnetic field B exists in a region. An electron of charge q and mass m moving with velocity v enters the region in a direction perpendicular to the magnetic field. Considering Bohr angular momentum quantization, which of the foll...

12 \(\mu\)C and 6 \(\mu\)C charges are given to the two conducting plates having same cross-sectional area and placed face to face close to each other as shown in the figure. The resulting charge distribution in \(\mu\)C on surfaces A, B, ...

The figure shows two identical parallel plate capacitors A and B of capacitances C connected to a battery. The key K is initially closed. The switch is now opened and the free spaces between the plates of the capacitors are filled with a d...

A particle of mass m is projected at a velocity u, making an angle \(\theta\) with the horizontal (x-axis). If the angle of projection \(\theta\) is varied keeping all other parameters same, then magnitude of angular momentum (L) at its max...

There are n elastic balls placed on a smooth horizontal plane. The masses of the balls are \(\mathrm{m}, \frac{\mathrm{m}}{2}, \frac{\mathrm{m}}{2^{2}}, \ldots \frac{\mathrm{m}}{2^{\mathrm{n}-1}}\) respectively. If the first ball hits the s...

In an experiment on a circuit as shown in the figure, the voltmeter shows 8 V reading. The resistance of the voltmeter is,

An amount of charge Q passes through a coil of resistance R. If the current in the coil decreases to zero at a uniform rate during time T, then the amount of heat generated in the coil will be,

The electric field of a plane electromagnetic wave of wave number k and angular frequency \(\omega\) is given \(\vec{E}=E_{0}(\hat{i}+\hat{j}) \sin (k z-\omega t)\). Which of the following gives the direction of the associated magnetic fiel...

A charged particle in a uniform magnetic field \(\vec{B}=B_{0} \hat{k}\) starts moving from the origin with velocity \(v=3 \hat{\mathrm{i}}+4 \hat{\mathrm{k}} ~\mathrm{m} / \mathrm{s}\). The trajectory of the particle and the time \(t\) at ...

A circular coil is placed near a current carrying conductor, both lying on the plane of the paper. The current is flowing through the conductor in such a way that the induced current in the loop is clockwise as shown in the figure. The cur...

A charged particle of charge q and mass m is placed at a distance 2R from the centre of a vertical cylindrical region of radius R where magnetic field varies as \(\vec{B}=\left(4 t^{2}-2 t+6\right) \hat{k}\), where t is time. Then which of ...

In the given circuit, find the voltage drop \(\mathrm{V_L}\) in the load resistance \(\mathrm{R_L}\).

Consider the logic circuit with inputs A, B, C and output Y. How many combinations of A, B and C gives the output Y = 0 ?

Consider a positively charged infinite cylinder with uniform volume charge density \(\rho > 0\). An electric dipole consisting of + Q and \(-\) Q charges attached to opposite ends of a massless rod is oriented as shown in the figure. At th...

A thin glass rod is bent in a semicircle of radius R. A charge is non-uniformly distributed along the rod with a linear charge density \(\lambda=\lambda_0\sin\theta\) (\(\lambda_0\) is a positive constant). The electric field at the centre...

The figure represents two equipotential lines in x-y plane for an electric field. The x-component E\(_x\) of the electric field in space between these equipotential lines is,

An electric dipole of dipole moment \(\vec{p}\) is placed at the origin of the co-ordinate system along the \(\mathrm{z}\)-axis. The amount of work required to move a charge '\(\mathrm{q}\)' from the point \((\mathrm{a}, 0, 0)\) to the poin...

Three identical convex lenses each of focal length \(\mathrm{f}\) are placed in a straight line separated by a distance \(\mathrm{f}\) from each other. An object is located at f/2 in front of the leftmost lens. Then,

Acceleration due to gravity at a height H from the surface of a planet is the same as that at a depth of H below the surface. If R be the radius of the planet, then H vs. R graph for different planets will be,

An earth's satellite near the surface of the earth takes about 90 min per revolution. A satellite orbiting the moon also takes about \(90 \mathrm{~min}\) per revolution. Then which of the following is true?
[where \(\rho_{\mathrm{m}}\) is d...

Six molecules of an ideal gas have velocities 1, 3, 5, 5, 6 and 5 m/s respectively. At any given temperature, if \(\mathrm{\overline V}\) and \(\mathrm{V_{rms}}\) represent average and rms speed of the molecules, then

A given quantity of gas is taken from A to C in two ways; a) directly from A \(\to\) C along a straight line and b) in two steps, from A \(\to\) B and then from B \(\to\) C. Work done and heat absorbed along the direct path A \(\to\) C is ...

Two substances A and B of same mass are heated at constant rate. The variation of temperature \(\theta\) of the substances with time t is shown in the figure. Choose the correct statement.

A cyclic process is shown in p-v diagram and T-S diagram. Which of the following statements is/are true?

A body of mass 2 kg moves in a horizontal circular path of radius 5 m. At an instant, its speed is 2\(\sqrt5\) m/s and is increasing at the rate of 3 m/s\(^2\). The magnitude of force acting on the body at that instant is,

A wire carrying a steady current I is kept in the x-y plane along the curve \(y=A \sin \left(\frac{2 \pi}{\lambda} x\right)\). A magnetic field B exists in the z-direction. The magnitude of the magnetic force in the portion of the wire bet...

A bar magnet falls from rest under gravity through the centre of a horizontal ring of conducting wire as shown in figure. Which of the following graph best represents the speed (v) vs. time (t) graph of the bar magnet?

As shown in the figure, a liquid is at same levels in two arms of a U-tube of uniform cross-section when at rest. If the U-tube moves with an acceleration 'f' towards right, the difference between liquid heights between two arms of the U-t...

As shown in the figure, a pump is designed as horizontal cylinder with a piston having area A and an outlet orifice having an area 'a'. The piston moves with a constant velocity under the action of force F. If the density of the liquid is ...

A mouse of mass m jumps on the outside edge of a rotating ceiling fan of moment of inertia I and radius R. The fractional loss of angular velocity of the fan as a result is,

In a simple harmonic motion, let f be the acceleration and t be the time period. If x denotes the displacement, then |fT| vs. x graph will look like,

In an experiment, the length of an object is measured to be 6.50 cm. This measured value can be written as 0.0650 m. The number of significant figures on 0.0650 m is

A modified gravitational potential is given by \(\mathrm{V}=-\frac{\mathrm{GM}}{\mathrm{r}}+\frac{\mathrm{A}}{\mathrm{r}^{2}}\). If the constant A is expressed in terms of gravitational constant (G), mass (M) and velocity of light (c), then...

A ray of monochromatic light is incident on the plane surface of separation between two media \(\mathrm{X}\) and \(\mathrm{Y}\) with angle of incidence '\(\mathrm{i}\)' in medium $\mathrm{X}$ and angle of refraction 'r' in medium Y. The giv...