WB JEE 2024

In R, a relation p is defined as follows:
\(\forall a, b \in \mathbb{R}, a p\) holds iff \(a^2-4 a b+3 b^2=0\). Then

Let A be the set of even natural numbers that are < 8 & B be the set of prime integers that are \(<7\) The number of relations from A to B are

For the real numbers \(x\) & \(y\), we write \(x\) p y iff \(x-y+\sqrt{2}\) is an irrational number. Then relation p is

If \((1,5)\) be the midpoint of the segment of a line between the line \(5 x-y-4=0\) and \(3 x+4 y-4=0\), then the equation of the line will be

In \(\triangle \mathrm{ABC}\), co-ordinates of \(\mathrm{A}\) are \((1,2)\) and the equation of the medians through \(\mathrm{B}\) and C are \(x+\mathrm{y}=5\) and \(x=4\) respectively. Then the midpoint of \(\mathrm{BC}\) is

Let \(\Gamma\) be the curve \(\mathrm{y}=\mathrm{be}^{-x / a}\) & \(\mathrm{L}\) be the straight line \(\frac{x}{\mathrm{a}}+\frac{\mathrm{y}}{\mathrm{b}}=1\) where \(\mathrm{a}, \mathrm{b} \in \mathbb{R}\).
Then

A square with each side equal to '\(a\)' above the \(x\)-axis and has one vertex at the origin. One of the sides passing through the origin makes an angle \(\alpha\) \(\left(0<\alpha< \frac{\pi}{4}\right)\) with the positive direction of th...

If \(\mathrm{ABC}\) is an isosceles triangle and the coordinates of the base points are \(B(1,3)\) and \(C(-2,7)\). The coordinates of \(A\) can be

The plane \(2 x-y+3 z+5=0\) is rotated through \(90^{\circ}\) about its line of intersection with the plane \(x+y+z=1\). The equation of the plane in new position is

If the relation between the direction ratios of two lines in \(\mathbb{R}^3\) are given by
\(l+\mathrm{m}+\mathrm{n}=0,2 l \mathrm{~m}+2 \mathrm{mn}-l \mathrm{n}=0\)
then the angle between the lines is
(\(l, \mathrm{~m}, \mathrm{n}\) have t...

Angle between two diagonals of a cube will be

The expression \(\cos ^2 \phi+\cos ^2(\theta+\phi)-2 \cos \theta \cos \phi \cos (\theta+\phi)\) is

If \(0< \theta<\frac{\pi}{2}\) and \(\tan 3 \theta \neq 0\), then \(\tan \theta+\tan 2 \theta+\tan 3 \theta=0\) if \(\tan \theta \cdot \tan 2 \theta=\mathrm{k}\) where \(\mathrm{k}=\)

If \(A\) and \(B\) are acute angles such that \(\sin A=\sin ^2 B\) and \(2 \cos ^2 A=3 \cos ^2 B\), then \((A, B)=\)

A unit vector in XY-plane making an angle \(45^{\circ}\) with \(\hat{i}+\hat{j}\) and an angle \(60^{\circ}\) with \(3 \hat{i}-4 \hat{j}\) is

In a series LCR circuit, the rms voltage across the resistor and the capacitor are \(30 \mathrm{~V}\) and \(90 \mathrm{~V}\) respectively. If the applied voltage is \(50 \sqrt{2} \sin \omega t\), then the peak voltage across the inductor i...

The variation of impedance \(\mathrm{Z}\) of a series \(\mathrm{L C R}\) circuit with frequency of the source is shown in the figure. Which of the following statement(s) is/are true ?

Which of the following statement(s) is/are truc in respect of nuclear binding energy ?
(i) The mass energy of a nucleus is larger than the total mass energy of its individual protons and neutrons.
(ii) If a nucleus could be separated into i...

The equivalent capacitance of a combination of connected capacitors shown in the figure between the points \(\mathrm{P}\) and \(\mathrm{N}\) is

A uniform rod \(\mathrm{AB}\) of length \(1 \mathrm{~m}\) and mass \(4 \mathrm{~kg}\) is sliding along two mutually perpendicular frictionless walls OX and OY. The velocity of the two ends of the \(\operatorname{rod} \mathrm{A}\) and $$\ma...

The position of the centre of mass of the uniform plate as shown in the figure is

Consider a circuit where a cell of emf \(E_0\) and internal resistance \(\mathrm{r}\) is connected across the terminal \(\mathrm{A}\) and \(\mathrm{B}\) as shown in figure. The value of \(\mathrm{R}\) for which the power generated in the c...

Monochromatic light of wavelength \(\lambda=4770 \mathop A\limits^o\) is incident separately on the surfaces of four different metals A, B, C and D. The work functions of A, B, C and D are $$4.2 \mathrm{~eV}, 3.7 \mathrm{~eV}, 3.2 \mathrm{...

A beam of light of wavelength \(\lambda\) falls on a metal having work function \(\phi\) placed in a magnetic field B. The most energetic electrons, perpendicular to the field are bent in circular arcs of radius R. If the experiment is perf...

Two straight conducting plates form an angle \(\theta\) where their ends are joined. A conducting bar in contact with the plates and forming an isosceles triangle with them starts at the vertex at time \(t=0\) and moves with constant veloci...

The electric field of a plane electromagnetic wave in a medium is given by
\(\overrightarrow{\mathrm{E}}(x, y, z, t)=\mathrm{E}_0 \hat{\mathrm{n}} \mathrm{e}^{i k_o[(x+y+z)-c t]}\)
where \(\mathrm{c}\) is the speed of light in free space....

In the given network of AND and OR gates, output Q can be written as (assuming n even)

A \(2 \mathrm{~V}\) cell is connected across the points \(\mathrm{A}\) and \(\mathrm{B}\) as shown in the figure. Assume that the resistance of each diode is zero in forward bias and infinity in reverse bias. The current supplied by the cel...

Consider the integral form of the Gauss' law in electrostatics
\(\oint {\overrightarrow E .d\overrightarrow S } = {Q \over {{\varepsilon _0}}}\)
Which of the following statements are correct?

Three point charges \(\mathrm{q},-2 \mathrm{q}\) and \(\mathrm{q}\) are placed along \(x\) axis at \(x=-{a}, 0\) and $a$ respectively. As \(\mathrm{a} \rightarrow 0\) and \(\mathrm{q} \rightarrow \infty\) while $$\mathrm{q} \mathrm{a}^2=\ma...

A charge Q is placed at the centre of a cube of sides a. The total flux of electric field through the six surfaces of the cube is

When a convex lens is placed above an empty tank, the image of a mark at the bottom of the tank, which is 45 cm from the lens is formed 36 cm above the lens. When a liquid is poured in the tank to a depth of 40 cm, the distance of the image...

If \(\hat{n}_1, \hat{n}_2\) and \(\hat{\mathrm{t}}\) represent, unit vectors along the incident ray, reflected ray and normal to the surface respectively, then

Two convex lens \((\mathrm{L}_1\) and \(\mathrm{L}_2)\) of equal focal length \(\mathrm{f}\) are placed at a distance \(\frac{\mathrm{f}}{2}\) apart. An object is placed at a distance \(4 \mathrm{f}\) in the left of \(\mathrm{L_1}\) as show...

A satellite of mass \(\mathrm{m}\) rotates round the earth in a circular orbit of radius R. If the angular momentum of the satellite is J, then its kinetic energy \((\mathrm{K})\) and the total energy (E) of the satellite are

The internal energy of a thermodynamic system is given by \(U=a s^{4 / 3} V^\alpha\) where \(\mathrm{s}\) is entropy, \(\mathrm{V}\) is volume and '\(\mathrm{a}\)' and '\(\alpha\)' are constants. The value of \(\alpha\) is

The speed distribution for a sample of \(\mathrm{N}\) gas particles is shown below. \(\mathrm{P}(\mathrm{v})=0\) for \(\mathrm{v}>2 \mathrm{v}_0\). How many particles have speeds between \(1.2 \mathrm{v}_0\) and \(1.8 \mathrm{v}_0\) ?

What force \(\mathrm{F}\) is required to start moving this \(10 \mathrm{~kg}\) block shown in the figure if it acts at an angle of \(60^{\circ}\) as shown? \((\mu_s=0.6)\)

A charged particle moving with a velocity \(\vec{v}=v_1 \hat{i}+v_2 \hat{j}\) in a magnetic field \(\vec{B}\) experiences a force \(\vec{F}=F_1 \hat{i}+F_2 \hat{j}\). Here \(v_1, v_2, F_1, F_2\) all are constants. Then $$\overrightarrow{\ma...

The acceleration-time graph of a particle moving in a straight line is shown in the figure. If the initial velocity of the particle is zero then the velocity-time graph of the particle will be

A metal plate of area \(10^{-2} \mathrm{~m}^2\) rests on a layer of castor oil, \(2 \times 10^{-3} \mathrm{~m}\) thick, whose coefficient of viscosity is \(1.55 \mathrm{~Ns} \mathrm{~m}^{-2}\). The approximate horizontal force required to m...

The elastic potential energy of a strained body is

A body floats with \(\frac{1}{n}\) of its volume keeping outside of water. If the body has been taken to height \(\mathrm{h}\) inside water and released, it will come to the surface after time t. Then

Water is filled in a cylindrical vessel of height \(\mathrm{H}\). A hole is made at height \(\mathrm{z}\) from the bottom, as shown in the figure. The value of z for which the range (R) of the emerging water through the hole will be maximum...

A small sphere of mass m and radius r slides down the smooth surface of a large hemispherical bowl of radius R. If the sphere starts sliding from rest, the total kinetic energy of the sphere at the lowest point \(\mathrm{A}\) of the bowl wi...

A small ball of mass m is suspended from the ceiling of a floor by a string of length \(\mathrm{L}\). The ball moves along a horizontal circle with constant angular velocity \(\omega\), as shown in the figure. The torque about the centre (...

The position vector of a particle of mass \(\mathrm{m}\) moving with a constant velocity \(\vec{v}\) is given by \(\vec{r}=x(t) \hat{i}+b \hat{j}\), where \(\mathrm{b}\) is a constant. At an instant, \(\vec{r}\) makes an angle \(\theta\) wi...

A particle of mass '\(m\)' moves in one dimension under the action of a conservative force whose potential energy has the form of \(U(x)=-\frac{\alpha x}{x^2+\beta^2}\) where \(\alpha\) and \(\beta\) are dimensional parameters. The angular ...

Which of the following quantity has the dimension of length ?
(h is Planck's constant, m is the mass of electron and c is the velocity of light)

The Power \((\mathrm{P})\) radiated from an accelerated charged particle is given by \(\mathrm{P} \propto \frac{(q \mathrm{a})^{\mathrm{m}}}{\mathrm{c}^{\mathrm{n}}}\) where \(\mathrm{q}\) is the charge, \(\mathrm{a}\) is the acceleration o...