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Three rods of same dimensions have thermal conductivities $3\text{K}$, $\text{2K}$ and $\text{K}$. They are arranged as shown in the figure, with their ends at ${100}^{\circ }\text{C},{50}^{\circ }\text{C}$ and $0^{\circ }\text{C}.$ The temperature of their junction point is:

Find the change in internal energy of $3\text{mol}$ of helium gas when its temperature is increased by $2\text{K}$.

A particle of mass m is projected from the ground with an initial speed $u_0$ at an angle $\alpha$ with the horizontal. At the highest point of its trajectory, it makes a completely inelastic collision with another identical particle, which was thrown vertically upward from the ground with the same initial speed $u_0$. The angle that the composite system makes with the horizontal immediately after the collision is ?

Which of the following is not a part of EM wave spectrum:

The rms speed of oxygen molecules in a container is $V$. If the temperature is doubled and oxygen molecules dissociate into oxygen atoms, the rms speed will become :

Find the electric field at a given point ‘O’ due to a semi-circular charged ring having linear charge density $\lambda$.

The objective lens of a compound microscope produces magnification of $10$. In order to get an overall magnification of $100$ when image is formed at $25cm$ from the eye, the focal length of the eye piece should be

The potential energy of a particle executing simple harmonic motion at a distance x from the equilibrium position is proportional to

A tuning fork vibrates at $250\text{Hz}$. Find the length of the shortest closed organ pipe that will resonate with the tuning fork.

[Speed of sound in air $=340\text{m/s}$]

Which of the following are necessary conditions for a particle performing periodic motion?

(i) The particle's path is fixed

(ii) The particle's momentum is constant

(iii) Velocity at any position is same after time t = T time duration where T is particle's Time period.

(iv) Time period is same for every cycle

The random error in the arithmetic mean of $100$ observations of a physical quantity is $x$, then random error in the arithmetic mean of $400$ observations of the same physical quantity will be

A uniform rod of mass $15\text{kg}$ and length $3\text{m}$ hinged at point $\text{O}$, is held stationary initially with the help of a light string (connected at point $\text{A}$) as shown in figure. At $t=0$, if the string breaks, find the hinge force acting on the rod at that instant. (Take $g=10{\text{m/s}}^2)$

A current I flows in circular arc of wire which subtends an angle ${\theta }^{\circ }$ at the centre. If the radius of the circle is r then the magnetic field B at center is:

In the figure given below, the current supplied by the battery is



(Assuming that the resistance of each diode is zero in forward bias and infinity in reverse bias)

Cosider a gravity - free hall in which a tray of mass M , carrying a cubical block of ice of mass m and edge L , is at rest in the middle. If the ice melts, by what distance does the centre of mass of "the tray plus the ice" system descent ?

In the figure the reading of the spring balance will be $:[g=10\frac{m}{s^2}]$

A symmetric double convex lens is cut in two equal parts by a plane containing the principal axis. If the power of the original lens was 4D, the power of a divided lens will be

The direction of Mowgali which is at the position $(4,6)$ with respect to Bageera which is at the position $(2,3)$ is

Pick the wrong answer in the context with rainbow:

Two rods A and B of different materials are welded together as shown in Figure. If their thermal conductivities are $k_1$and $k_2$, the thermal conductivity of the composite rod will be

The adiabatic Bulk modulus of a perfect gas at pressure $P$ is given by

An aeroplane in which the distance between the tips of wings is 50 m is flying horizontally with a speed of 360 km/hr over a place where the vertical component of earth's magnetic field is $2.0$ x ${10}^{-4}weber/m^2$ . The potential difference between the tips of the wings would be

An object is tossed vertically into the air with an initial velocity of $8\text{m/s}$. Using the sign convertion upwards as positive, how does the vertical component of the acceleration $a_y$ of the object (after leaving the hand) vary during the flight of the object?

Find the acceleration of the two blocks of mass $4\text{kg}$ & $5\text{kg},$ if a force of $40\text{N}$ is applied on $4\text{kg}$ block. Friction coefficient between the respective surfaces are shown in figure. (take $g=10{\text{m/s}}^2$).

In a medium in which a transverse progressive wave is travelling, the phase difference between two points with a separation of 1.25 cm is $\left(\frac{\pi }{4}\right)$. If the frequency of wave is 1000 Hz its velocity will be

The three rods of same material and crosssectional area form the sides of a triangle ABC. The points A, B and C are maintained at temperatures T, T$\sqrt{2}and\frac{3T}{(\sqrt{2}+1)}$ respectively. Assuming that only heat conduction takes place and the system is in steady state, find the angle at B. The temperature difference per unit length along CB and CA is equal.

The gas having average speed four times as that of $S{O}_2$ at same temperature is

(Molar mass of $S{O}_2=64g$)

A thin ring of radius $r$ carries a uniformly distributed charge. The ring rotates at a constant angular speed of $n$ revolutions per second about an axis passing through centre and perpendicular to its plane. If $B$ is the magnitude of magnetic field at the centre of ring, then the charge carried by the ring is

The rigid bodies $A$ and $B$ of masses $2\text{kg}$ and $4\text{kg}$ respectively are rigidly connected to a spring of the constant $400\text{N/m}$. The body $B$ rests on a horizontal table, from the rest position, the body $A$ is compressed by $2\text{cm}$ and then released. Find the frequency of oscillation.

A conic surface is placed in a uniform electric field $E$ as shown such that field is perpendicular to the surface on the side AB. The base of the cone is of radius $R$ and height of the cone is $h$. The angle of cone is $\theta$ as shown. Find the magnitude of that flux which enters the cone's curved surface on the left side. Don’t count the outgoing flux.

Two tall buildings are $40\text{m}$ apart. With what speed a ball must be thrown horizontally from a window $145\text{m}$ above the ground in one building, so that it will enter a window $22.5\text{m}$ above the ground in the other ?

(Take $g=10m/s^2$)

Find the differentiation of a function represented by $y=e^{x^3}$ with respect to $x$.

A particle moves in a circular path of radius $20\text{cm}$ as shown. Find angular velocity of particle w.r.t an axis passing through centre of the circle and perpendicular to plane of the circle.

The e.m.f. of a cell is ε and its internal resistance is r. Its terminals are connected to a resistance R. The potential difference between the terminals is 1.6 V for $R=4\Omega$ and 1.8 V for $R=9\Omega$. Then,

Let $\left[{\in }_0\right]$ denote the dimensional formula of the permittivity of the vacuum and $\left[{\mu }_0\right]$ that of the permeability of the vacuum. If $M=$ mass, $L=$ length, $T=$ time and $I=$ electric current, then:

Given $F=\frac{1}{4\pi {\in }_0}.\frac{q_1{q}_2}{r^2}$, where $q_1$ and $q_2$ are charges and r is the distance. Also,

$c=\frac{1}{\sqrt{{\in }_0{\mu }_0}}$, where $c$ is speed.

A liquid of mass $m$ is set into oscillations in a U-tube of cross section $A$. Its time period recorded is $T$, where $T=2\pi \sqrt{l/2g}$ and $l$ is the length of the liquid column. If the liquid of same mass is set into oscillations in U-tube of cross-section $A/16$, then determine time period of oscillation.

Three positive charges of equal value q are placed at the vertices of an equilateral triangle. The resulting lines of force should be sketched as in

The initial velocity of a particle is u (at t=0) and the acceleration f is given by $f=at$ where $a$ is a constant. Which of the following relation is valid?

A large slab of mass 5 kg lies on a smooth horizontal surface, with a block of mass 4 kg lying on top of it. The coefficient of friction between the block and the slab is 0.25. If the block is pulled horizontally by a force of F = 6 N, the work done by the force of friction on the slab, between the instants t=2 s and t=3 s, is (g = 10 m$s^{-2}$)

Find the density of the oil inside the U-tube as shown in figure. (Take density of water ${\rho }_w=1000{\text{kg/m}}^3$)

Find velocity v(t) if the rod’s initial velocity was $v_0$ and mass is m.

A particle of mass 0.1 kg is subjected to a force which varies with distance as shown in the figure. If it starts its journey from rest at x = 0, its velocity at x = 12 m is

Which of the following gates will have an output of 1

A force of ${10}^3$ newton stretches the length of a hanging wire by 1 millimetre. The force required to stretch a wire of same material and length but having four times the diameter by 1 millimetre is

A transparent solid cylindrical rod has a refractive index of $\frac{2}{\sqrt{3}}$. It is surrounded by air. A light ray is incident at the midpoint of one end of the rod as shown in the figure.



The incident angle $\theta$ for which the light ray grazes along the wall of the rod is

A body is executing simple harmonic motion. At a displacement x, its potential energy is $E_1$ and at a displacement $y$, its potential energy is $E_2$. The potential energy $E$ at a displacement $(x+y)$ is