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Moment of Inertia of a thin uniform rod rotating about the perpendicular axis passing through its centre is I. If the same rod is bent into a ring and its moment of inertia about its diameter is I’, then the ratio $\frac{I}{I^{\prime }}$ is

If $\overrightarrow{a}$ and $\overrightarrow{b}$are position vectors of $A$ and $B$ respectively, then the position vector of $C$ in $\overrightarrow{AB}$ produced such that $\overrightarrow{AC}=3\overrightarrow{AB}$ is

A pair of coils of turns $n_1$ & $n_2$ are kept close to each other. If a current passing through the first is reduced at the rate ‘r’, an emf of 2mV is developed across the other coil. If the second coil carries current which is the reduced at the rate 3r, the emf produced across the first coil will be

A convex driving mirror of focal length $20\text{cm}$, is fitted in a motor car. If the second car $2\text{m}$ broad and $1.6\text{m}$ high is $6\text{m}$ away from the first car and overtakes the first car at a relative speed of $15\text{m/s}$, then how fast relatively in $\text{m/s}$ will the image be moving? Answer upto 3 decimal places

A particle executes SHM with a time period of $4\text{s}$. Find the time taken by the particle to go directly from its mean position to half of its amplitude.

The system is released from rest. Derive the relation between acceleration of the masses involved.

Excess charge flowing through the battery after closing the switch is

A particle is projected at an angle with the horizontal such that it follows a trajectory given by the equation $y=6x–2x^2.$ Find the maximum height attained by it.

What
type of substances would make better permanent magnets, ferromagnetic
or ferrimagnetic. Justify your answer.

Three rods of copper, brass and steel are welded together to form a Y - shaped structure. Area of cross - section of each rod $=4{\text{cm}}^2$. End of copper rod is maintained at ${100}^{\circ }C$ where as ends of brass and steel rods are kept at $0^{\circ }C$. Lengths of the copper, brass and steel rods are $46,13\text{and}12\text{cm}$ respectively. The rods are thermally insulated from surroundings except at ends. Thermal conductivities of copper, brass and steel are $0.92,0.26$ and $0.12$ CGS units respectively. Rate of heat flow through copper rod is

Two particles moving at right angles have their de-Broglie wavelengths ${\lambda }_1$ and ${\lambda }_2$. If their collision is perfectly inelastic, what is the final wavelength $\lambda$, in terms of ${\lambda }_1$ and ${\lambda }_2?$

For the network shown in the figure the value of the current $i$ is : -

The variation of pressure with volume of the gas at different temperatures can be graphically represented as shown in figure. On the basis of this graph answer the following questions.

(i) How will the volume of a gas change if its presuure is increased at constant temperature?

(ii) At a constant pressure, how will the volume of a gas change if the temperature is increased from 200 K to 400 K?

The velocity of the rotating body is given by $\overrightarrow{V}=\overrightarrow{\omega }\times \overrightarrow{r},$ where $\overrightarrow{\omega }$ is the angular velocity and $\overrightarrow{r}$ is the radius vector. The angular velocity of a body is $\overrightarrow{\omega }=\hat{i}-2\hat{j}+2\hat{k}$ and the radius vector $\overrightarrow{r}=4\hat{j}-3\hat{k},$ then $|\overrightarrow{V}|$ is

A popular game in Indian village is goli which is played with small glass balls called golis. The goli of one player is situated at a distance of 2.0 m from the goli of the second player. This second played has to project his goli by keeping the thumb of the left hand at the place of his goli, holding the goli between his two middle fingers and making the throw, if the projected goli hits the goli of the first player, the second player wins. If the height from which the goli projected is 19.6 m from the ground and the goli is to be projected horizontally, with what speed should it be projected so that it directly hits the stationary goli without falling on the ground earlier.

Find the maximum wavelength of electromagnetic radiation which can create a hole - electron pair in germanium. The band gap in germanium is $0.65\text{eV}$.

$(hc=1242\text{eV nm})$

A particle moves in a circle of radius 2.0 cm at a speed given by v = 4t, where v is in cm/s and t is in seconds.

Find the tangential acceleration at t =1 s.

Find total acceleration at t =1 s.

For a plano convex lens the radius of curvature of the convex part is $14\text{cm}$. If the refractive index of the material of the lens is $1.7$, calculate the focal length of the lens.

A wire (fixed at both ends) of length $l$ having tension $T$ and radius $r$ vibrates with natural frequency $f$. Another wire of same metal with length $2l$ having tension $2T$ and radius $2r$ will vibrate with natural frequency

A rod of length $l$ and mass $m$ is bent in shape of a ring. The moment of inertia of the ring about any of its diameter is:

A cantilever-type micro-accelerometer is designed by scaling down each dimension of a macro-accelerometer by a factor of 100. If the natural frequency of the macro-accelerometer is $\omega$, then the natural frequency of the micro-accelerometer will be

A parallel beam of visible light consisting of wavelengths ${\lambda }_1$ & ${\lambda }_2$ is incident on a standard YDSE apparatus with $d=1\text{mm},D=1\text{m}.$ $P$ is a point on the screen at a distance $y$ from center of screen $O.y=y_1$ is the nearest point above $O$ where the two maxima coincide and $y=y_2$ is the nearest point above $O$ where the two minima coincide. ${\beta }_1$ & ${\beta }_2$ are fringe width corresponding to wave length ${\lambda }_1$ & ${\lambda }_2.$



$\begin{array}{cc}\text{Column\_I}& \text{Column-II}\\ \text{(A)}{\beta }_1=0.3\text{mm},{\beta }_2=0.5\text{mm}& \text{(P) The}{2}^{nd}\text{nearest point above}O\text{where two maxima coincide is}y=2y_1\\ \text{(B)}{\beta }_1=0.3\text{mm},{\beta }_2=0.4\text{mm}& \text{(Q)}{y}_2\text{has no finite value}\\ \text{(C)}{\beta }_1=0.2\text{mm},{\beta }_2=0.4\text{mm}& \text{(R) The}{2}^{nd}\text{nearest point above}O\text{where the two minima coincide is}y=3y_2\\ \text{(D)}{\beta }_1=0.2\text{mm},{\beta }_2=0.6\text{mm}& \text{(S)}{y}_1=\text{LCM of}{\beta }_1\text{and}{\beta }_2\\ & \text{(T) The}{2}^{nd}\text{nearest point above}O\text{where two minima coincide is}y=2y_2\end{array}$

Which of the following option has the correct combination considering column-I and column-II ?

A thin prism made of glass is dipped in water, then minimum deviation with respect to air by it will be $[given{}^a{\mu }_g=\frac{3}{2}and{}^a{\mu }_w=\frac{4}{3}]$:

The shear stress at a point in a liquid is found to be $0.03{\text{N/m}}^2$. The velocity gradient for the fluid flow is $0.15{\text{s}}^{-1}$. What will be its viscosity (in $\text{Poise}$) ?

A disc rotating about its axis with angular speed ω_ois
placed lightly (without any translational push) on a perfectly
frictionless table. The radius of the disc is R. What are the
linear velocities of the points A, B and C on the disc shown in Fig.
7.41? Will the disc roll in the direction indicated?

Voltage rating of a parallel plate capacitor is $500V$. Its dielectric can withstand a maximum electric field of ${10}^{6}V/m$. The plate area is ${10}^{-4}{m}^2$. What is the dielectric constant if the capacitance is $15pF$?

(given ${ϵ}_0=8.86\times {10}^{-12}{C}^2/N{m}^2)$.

What is the angle between the following vectors?

$\overrightarrow{A}=\hat{i}+\hat{j}+\hat{k}$ and $\overrightarrow{B}=-2\hat{i}-2\hat{j}-2\hat{k}$.

Calculate the tension in the string shown in figure. The pulley and the string are light and all surfaces are frictionless. Take g = 10 $m/s^2$.

A special metal $S$ conducts electricity without any resistance. A closed wire loop, made of $S$, does not allow any change in flux through itself by inducing a suitable current to generate a compensating flux. The induced current in the loop cannot decay due to its zero resistance. This current gives rise to a magnetic moment which in turn repels the source of magnetic field or flux. Consider such a loop, of radius $a$, with its center at the origin. A magnetic dipole of moment $m$ is brought along the axis of this loop from infinity to a point at distance $r(>>a)$ from the center of the loop with its north pole always facing the loop, as shown in the figure below.



The magnitude of magnetic field of a dipole $m$, at a point on its axis at distance $r$, is $\frac{{\mu }_{0}m}{2\pi r^3}$, where ${\mu }_0$ is the permeability of free space. The magnitude of the force between two magnetic dipoles with moments, $m_1$ and $m_2$, separated by a distance $r$ on the common axis, with their north poles facing each other, is $\frac{k{m}_1{m}_2}{r^4}$, where $k$ is a constant of appropriate dimensions. The direction of this force is along the line joining the two dipoles.

The work done in brining the dipole from infinity to a distance $r$ from the center of the loop by the given process is proportional to