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Figure shows a parallel plate capacitor having square plates of edge length $a$ and separation $d$. The gap between the plates is filled with a dielectric of dielectric constant $K$ which varies parallel to an edge as, $K=K_o+\alpha x$

Where, $K$ and $\alpha$ are constants and $x$ is the distance from the left end. Calculate the capacitance of system.

Six vectors $\overrightarrow{a}$ to $\overrightarrow{f}$ have directions as indicated in the figure. Which of the following statements may be true?

The value of the definite integral $\underset{-\pi /4}{\overset{\pi /4}{\int }}\frac{dx}{(1+e^{x\cos x})({\sin }^{4}x+{\cos }^{4}x)}$ is equal to

Two particles $\text{A}$ and $\text{B}$ of masses $1\text{kg}$ and $2\text{kg}$ respectively are projected with speed $u_A=200{\text{ms}}^{-1}$ and $u_B=50{\text{ms}}^{-1}$ in the directions as shown in figure. Initially they were $90\text{m}$ apart. They collide in the midair and stick with each other. Find the maximum height attained by the centre of mass of the system.

[Assume acceleration due to gravity to be constant as
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$g=10{\text{ms}}^{-2}$]

The distance and displacement covered respectively by a particle after half a revolution of circular path of radius $r$ are

A current $I$ is flowing through the sides of the rectangle as shown below. The magnetic field at the crossing point of the diagonal is

A ball is thrown upward with an initial velocity of $100{\text{ms}}^{-1}$. Find the height of the ball after $15\text{s}$. Take $g=10{\text{ms}}^{-2}$

For changing the capacitance of a given parallel plate capacitor, a dielectric material of dielectric constant $K$ is used, which has the same area as the plates of the capacitor. The thickness of the dielectric slab is $\frac{3}{4}d,$ where ${}^{\prime }{d}^{\prime }$ is the separation between the plates of the parallel plate capacitor. The new capacitance $\left(C^{\prime }\right)$ in terms of the original capacitance $\left(C_0\right)$ is given by the following relation:

Suppose an experiment where a monochromatic light is passed through $3$ slits as shown in the figure. $($Consider $3$ parallel rays as shown from the $3$ slits$)$ What is the path difference between rays number $1$ and $3$?

1. 0.07

1. 6.83

If the two dielectrics of dielectric constant $K_1$ & $K_2$ with thickness $t_1$ & $t_2$ respectively, were to be replaced by a single dielectric completely filling the space between the two plates, then its equivalent dielectric constant will be

A charge $Q$ is kept at the corner of a cube. If the net flux through the cube is $\varphi$, the electric flux passing through one of those faces not touching the charge will be

A difference of 2.3 eV
separates two energy levels in an atom. What is the frequency of
radiation emitted when the atom makes a transition from the upper
level to the lower level?

An 8-bit successive approximation type ADC has a resolution of 30 mV. The digital output for an analog input of 2.86 V is

How can u say that speed=time by length

Calculate the mass of a thin rod of length L whose mass distribution is uniform throughout, with linear mass density ${\lambda }_0$.

Find the value of $r$ in terms of $R$, so that power in external circuit is maximum.

The velocity of a particle varies with time at $V\left(t\right)=e^{-2\pi t}$. Find the diaplacement of the particle till $t=(\frac{1}{2\pi })$ sec.

Which of the following graphs correctly represents the variation of molar heat capacity at constant volume with temperature for an ideal monoatomic gas?

The value of $\underset{1/e}{\overset{\tan x}{\int }}\frac{tdt}{1+t^2}+\underset{1/e}{\overset{\cot x}{\int }}\frac{dt}{t(1+t^2)}$, where $x\in (\frac{\pi }{6},\frac{\pi }{3}),$ is

The equations of motion of two waves propagating in the same direction in a medium is given by
$y_1=Asin(\omega t-kx)$
and $y_2=Asin(\omega t-kx-\theta )$.
Then amplitude of the particle in this medium is

Two conducting spheres of radii $r_1$ and r₂ are at the same potential. Their charges are in the ratio of

A stone is thrown vertically upwards with an initial velocity of $25\text{m/s}$. The maximum height reached by the stone is

$[$Take $g=10{\text{m/s}}^2]$

A block of weight $W$ is kept on an inclined surface whose angle of inclination is $\theta$ with horizontal. The component of weight perpendicular to the incline is

Suppose a wavefront originates from a point source as shown in the figure. Which of the following arrows correctly indicates the direction of propagation of light at point A?

Two containers of sand are arranged like the block as shown in figure. The containers alone have negligible mass; the sand in these containers has a total mass $M_{tot}$; the sand in the hanging container $H$ has mass $m$.

To measure the magnitude of the acceleration of the system, a large number of experiments have been carried out where $m$ varies from experiment to experiment but $M_{tot}$ does not; that is, sand is shifted between the containers before each trial.



Which of the curves in the graph correctly gives the acceleration magnitude as a function of the ratio $m/M_{tot}$ (vertical axis is for acceleration)?

The magnetic moment of a magnet of dimensions $(10\text{cm}\times 4\text{cm}\times 1\text{cm})$ is $4{\text{A-m}}^2$. Its intensity of magnetisation is :

The energy of a hydrogen atom in the first excited state, if the potential energy is assumed to be zero in the ground state is

Question 63 (iv)



How many faces does the following solid have?



Octahedron