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A concave lens forms the image of an object such that the distance between the object and image is 10 cm and the magnification produced is 1/4.The focal length of the lens will be

The instantaneous values of alternating current and voltage in a circuit are given as

$i=\frac{1}{\sqrt{2}}\text{sin}\left(100\pi t\right)\text{ampere}$

$e=\frac{1}{\sqrt{2}}\text{sin}(100\pi t+\frac{\pi }{3})\text{volt}$

The average power in watts consumed in the circuit is

For an RLC circuit driven with voltage of amplitude $v_m$ and frequency ${\omega }_0=\frac{1}{\sqrt{LC}}$, the current exhibits resonance. The quality factor, Q, is given by

When a solid sphere of moment of inertia ‘I’ moves down an inclined plane, the percentage of rotational energy to total energy is approximately

A solid iron ball and a solid aluminium ball of the same diameter are released together at the surface of a deep lake. Which ball will reach the bottom first?

A 50Hz AC current of crest value 1A flows through the primary coil of transformer. If mutual induction between primary and secondary is 1.5H, then the crest voltage induced in the secondary coil is:

If R, C and L denote resistance, capacitance and inductance. Which of the following does NOT have the dimensions of frequency?

The electric potential varies in space according to the relation $V=3x+4y.$ A particle of mass $0.1kg$ starts from rest from point $(2,3.2)$ under the influence of this field. The charge on the particle is $+1\mu C.$ Assume V and $(x,y)$ are in S.I. units
The component of electric field in the y - direction $\left(E_y\right)$ is

In the circuit shown in the given figure, the resistances $R_1$ and $R_2$ are respectively.

For an enclosure maintained at 2000K, the maximum radiation occurs at wavelength ${\lambda }_m$. If the temperature is raised to 3000 K, the peak will shift to

A current carrying wire heats a metal rod. The wire produces a constant power $P$ to the rod. The metal rod is enclosed in an insulated conatainer. It is observed that the temperature $\left(T\right)$ in the metal rod changes with time $t$ as $T\left(t\right)=T_0(1+\beta t^{1/4})$ where $\beta$ is a constant with appropriate dimension of temperature. The heat capacity of the metal is:

In the circuit given, power developed across 1Ω,2Ω and 3Ω resistances are in the ratio of

A coil of wire having finite inductance and resistance has a conducting ring placed coaxially within it. The coil is connected to a battery at time t = 0, so that a time-dependent current $I_1\left(t\right)$ starts flowing through the coil. If $I_2\left(t\right)$ is the current induced in the ring and B(t) is the magnetic field at the axis of the coil due to $I_1\left(t\right)$, then the product of $I_2\left(t\right)$ and B(t)

A layer of ice at $0^{\circ }\text{C}$ of thickness $x_1$ is floating on a surface of water. $L,\rho$ and $K$ respectively are the latent heat of fusion of ice, density of ice and thermal conductivity of ice. If the atmospheric temperature is $-T^{\circ }\text{C}$, the time taken for the thickeness of the layer of ice to increase from $x_1$ to $x_2$ is given by

The tungsten filament of an electric lamp has a length of $0.5\text{m}$ and a diameter $6\times {10}^{-5}\text{m}$. The power rating of the lamp is $60\text{W}$. Assuming the radiation from the filament is equivalent to 80% of a perfect black body radiator at the same temperature, estimate the steady state temperature of the filament (Stefan's constant $=5.7\times {10}^{-8}{\text{W m}}^{-2}{\text{K}}^{-4}$)

Two pendulums have time periods $T$ and $5T/4$. They start SHM at the same time from the mean position. After how many oscillations of the smaller pendulum they will be again in the same phase

A person walking at $3\text{kmph}$ finds rain falling vertically. When he walks in the opposite direction with the same speed, the rain appears to fall at ${45}^{\circ }$ to the vertical. What is the speed of rain ?

An experiment takes $10\text{min}$ for an electric kettle to heat a certain quantity of water from $0^{\circ }\text{C}$ to ${100}^{\circ }\text{C}$. If it takes $54\text{min}$ to convert this water at ${100}^{\circ }\text{C}$ into steam, then latent heat of steam is:

As shown below, a uniform disc is mounted to a fixed axle and is free to rotate without friction. A thin uniform rod is rigidly attached to the disc so that it will rotate with the disc. A block is attached to the rod.

Disc : mass = 3m, radius = R, moment of inertia about center $I_D=\frac{3}{2}m{R}^2$

Rod : mass = m, length = 2R, moment of inertia about one end $I_R=\frac{4}{3}m{R}^2$

Block : mass 2m

The system is held in equilibrium with the rod at an angle ${\theta }_0$ to the vertical, as shown above, by a horizontal string of negligible mass one end attached to the disc and the other to a wall.

The string is now cut, and the disc-rod-block system is free to rotate. As disc rotates, the rod passes the horizontal position shown below. The linear speed of the end of the rod for the instant the rod is in the horizontal position is : (The arrow represent the direction of vector quantities)

The mass numbers of helium and sulphur are 4 and 32 respectively. The ratio of the nuclear radii of sulphur and helium will be

The mass and volume of a body are $4.237\text{g}$ and $2.5{\text{cm}}^3$ respectively. The density of the material of the body in correct significant figures is

Using differentials, find the approximate value of the following up to 3 places of decimal.
$(26{)}^{\frac{1}{3}}$

On a hot day in Jaipur, an oil trucker loaded 40 kL (kilolitres) of diesel fuel. On his way to Shimla, he encounters a temperature drop of ${20}^{\circ }C$, where he stopped and delivered the entire load. How many litres did he deliver? The γ for diesel is $9.50\times {10}^{-4}{/}^{\circ }C$ and α for his steel truck is $11\times {10}^{-6}{/}^{\circ }C$. If you find that the volume has decreased, think about who is paying for the "missing” diesel.

A homogeneous block having its cross section as a parallelogram of sides $a$ and $b$ as shown, is lying at rest and is in equilibrium on a smooth horizontal surface. Then, find the value of acute angle $\theta$ for which it will be in equlibrium

What is dielectric medium

A quantity of heat required to change the unit mass of a solid substance, from solid state to liquid state, while the temperature remains constant, is known as

The resultant of two vectors A and B is perpendicular to the vector A and its magnitude is equal to half the magnitude of vector B. The angle between A and B is

A common emitter amplifier is designed with n-p-n transistor (α =0.99). The input impedance is 1 KΩ and the load is 10 KΩ. The voltage gain will be.

How Vernier scale help in measurement of high

A particle of mass m moving in the x-direction with speed 2v is hit by another particle of mass 2m moving in the y-direction with speed v. If the collision is perfectly inelastic, the percentage loss in the energy during the collision is close to

If sides of a rectangle with given perimeter are $a$ & $b$, then find the relation between $a$ & $b$ for which area of the given rectangle is maximum.

Let R = { (P,Q) : OP = OQ , O being the origin} be an equivalence relation on A . The equivalence class [( 1,2)] is

A string of length $2\text{m}$ is fixed at both ends. If the speed of transverse wave on the string be $500\text{m/s}$, then what will be the frequency of vibration of string in its fifth harmonic and corresponding wavelength of the wave?

Natural water contains a small amont of tritium ${(}_1^{3}H)$. This isotope beta-decays with a half-life of 12.5 years. A mountaineer while climbing towards a difficult peak finds debris of some earlier unsuccessful attempt. Among other things he finds a sealed bottle of whisky. On return he analyses the whisky and finds that it contains only 1.5 per cent of the ${(}_1^{3}H)$. radioactivity as compared to a recently purchased bottle marked '8 years old'. Estimate the time of that unsuccessful attempt.

A mild-steel wire of length $2L$ and cross-sectional area $A$ is stretched, well within the elastic limit, horizontally between two pillars. A mass $m$ is suspended from the midpoint of the wire. Strain in the wire is

Under forced oscillation, the phase of harmonic motion of the particle:

A body of mass 1 kg tied to one end of string is revolved in a horizontal circle of radius 0.1 m with a speed of 3 revolution/sec, assuming the effect of gravity is negligible, then linear velocity, acceleration and tension in the string will be

Two particles $A$ and $B$ are projected simultaneously from the top of two towers of height $10\text{m}$ and $20\text{m}$ respectively. Particle $A$ is projected upwards at an angle of ${45}^{\circ }$ with horizontal at a speed of $10\sqrt{2}\text{m/s}$ and particle $B$ is projected horizontally at a speed of $10\text{m/s}$ towards $A$. What is the distance between towers if particles collide in the air ?Take $g=10{\text{m/s}}^2$.

Find the cut off wavelength for the continuous X-rays coming from an X-ray tube operating at 30 kV.

The number of spectral lines that are possible when electrons in fourth energy level in different hydrogen atoms returns to the ground state is

When a mass $M$ hangs from a spring of length $L$, it stretches the spring by a distance $x$. Now the spring is cut in two parts of length $\frac{L}{3}$ and $\frac{2L}{3}$, and the two springs thus formed are connected to a straight rod of mass $M$ which is horizontal in the configuration as shown in the figure. Find the stretch in each of the spring.

With respect to a rectangular cartesian co-ordinate system three vectors are expressed as $\overrightarrow{a}=4\hat{i}-\hat{j},\overrightarrow{b}=-3\hat{i}+2\hat{j}$ and $\overrightarrow{c}=-\hat{k}$ where $\hat{i},\hat{j}$ and $\hat{k}$ are unit vectors along the x, y, z axes respectively. The unit vector along the direction of sum of these vectors is