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The temperatures of the two outer surfaces of a composite slab, consisting of two materials having coefficients of thermal conductivity K and 2K and thickness x and 4x, respectively, are $T_2$ and $T_1$ ($T_2>T_1$). The rate of heat transfer through the slab, in a steady state is $\left(\frac{A(T_2-T_1)K}{x}\right)f$, with f equal to

A person throws balls into air after every second. The next ball is thrown when the velocity of the first ball is zero. How high do the ball rise above his hand?

In the figure the string has a mass $4.5\text{g}$. How much time will it take for a mass for a traverse disturbance produced at the floor to reach the pulley? (Take $g=10{\text{ms}}^{-2}$)

Two pendulums of same amplitude but time period $3\text{s}$ and $7\text{s}$ start oscillating simultaneously from two opposite extreme positions. After how much time they will be in phase.

Density of a solid sphere is given by the expression $\rho ={\rho }_0(1+\frac{x}{R})$ where $x$ is the distance from the centre of the sphere and $0\le x\le R$, $R$ is the radius of the solid sphere. Moment of inertia of the solid sphere about an axis passing through its centre will be: (Given ${\rho }_0$ is a constant)

A uniform constant magnetic field $B$ is directed at an angle ${45}^o$ to the $x$-axis in the $xy$ plane. $\text{PQRS}$ is a rigid square frame carrying a constant current $I_o$, with its centre at origin $\text{O}$. At time $t=0$, the frame is at rest in the position shown in figure with its sides parallel to $X$ & $Y$ axes. The sides of the square has length $L$. The torque $\overrightarrow{\tau }$ about $\text{O}$ acting on the frame will be:

(Given, the coil has single turn)

An electromagnetic wave of frequency $1\times {10}^{14}$ hertz is propagating along z - axis. The amptitude of electric field is 4 V/m. If ${ϵ}_0=8.8\times {10}^{-12}{C}^2/N-m^2$, then average energy density of electric field will be:

A monochromatic light of wavelength $\lambda$ is incident on the slits. The interference pattern is observed on a screen and central maxima is formed at point $O$. Two thin glass plates of thickness $t_1$ and $t_2$ and refractive indices ${\mu }_1$ and ${\mu }_2$ respectively, are placed in front of each of the two slits. It is observed that at point $O$, the central maxima is formed again. Then $\frac{t_1}{t_2}=?$

The distance of the closest approach of an alpha particle fired at a nucleus with kinetic energy K is $r_0$. The distance of the closest approach when the alpha particle is fired at the same nucleus with kinetic energy 2K will be

Two particles are of the same mass $m$. One particle is dropped from a height of $10\text{m}$ and the other particle is projected up with a velocity of $10\text{m/s}$. If both stick to each other at some height, find the maximum height reached by their $COM$. [Take $g=10{\text{m/s}}^2$]

An object 20 cm from a convex mirror give rise to a virtual image 15 cm behind the mirror. The focal length of the mirror is

A laser beam has intensity $2.5\times {10}^{14}$${\text{Wm}}^{-2}$. The amplitudes of electric and magnetic fields in the beam are :

The velocity – time graph of a particle moving along a straight line is as shown in fig. The rate of acceleration and deceleration is constant and it is equal to $5{\text{ms}}^{-2}$. If the average velocity during the motion is $20{\text{ms}}^{-1}$, then



The value of $t$ is

A rubber ball is released from a height of $5\text{m}$ above the floor. It bounces back repeatedly, always rising to $\frac{81}{100}$ of the height through which it falls. Find the average speed of the ball.

(Take $g=10{\text{m s}}^{-2})$

A block rests on a rough plane whose inclination angle $\theta$ with horizontal is increased (gradually), then contact force(C) between block & plane varies with $\theta$ (consider interval before block slips)

A cuboidal body of dimensions 2 m $\times$ 3 m $\times$ 4 m is acted upon by a perpendicular force of 50 N on the top surface (2 m $\times$ 3 m) as shown in the figure. Calculate the pressure due to the force on that surface.

The amplitude of a damped oscillator becomes one-fourth in one minute. The amplitude after 16 minutes will be ${\frac{1}{x}}^{th}$ of its original amplitude. Find the value of $x.$

A particle is moving in a circle of radius $20\text{cm}$. Its linear speed is given by $v=2t$, where $t$ is in $\text{second}$ and $v$ is in $\text{m/s}$. Find the radial acceleration at $t=3\text{s}$.

A uniform wire of length 8 metre has a resistance R. Now a portion of the wire of length x metre is elongated carefully so that the final length of the wire becomes 12 metre and total resistance of the wire becomes 4R. Then x is equal to

The slope of tangent(s) to curve $y=\sqrt{18-x^2}$ at the point(s) where its ordinate and abscissa are equal is/are

The force F is given in terms of time t and displacement x by the equation F = A cos Bx + C sin Dt. The dimensional formula of $\frac{D}{B}$ is

Don't hate me for asking this question, as it will need you to work your calculator! So, in the radioactive decay -

${}^{238}Pu{\to }^{234}U+\alpha ,$

What will be the kinetic energy of the outgoing α-particle?

Things you will need:

A totally reflecting, small plane mirror placed horizontally faces a parallel beam of light as shown in figure (42-E1). The mass of the mirror is 20g. Assume that there is no absorption in the lens and that 30${}^{\%}$ of the light emitted by the source goes through the lens. Find the power of the source needed to support the weight of the mirror. Take g = 10 $m^{-s}$

figure (42-E1)

A short bar magnet of moment $4{\text{Am}}^2$ is placed in the magnetic merdian with its south-pole pointing geographic north. The distance between two null points is found to be $20\text{cm}$.



$\left(i\right)$The value of horizontal component of the earth's magnetic field at the place is

A $7.5$ wide two-lane road on a plain terrain is to be laid along a horizontal curve of radius $500m$. For a design speed of $100kmph$, super elevation is provided as per $IRC:73-1980$. Consider acceleration due to gravity as $9.81m/s^2$. The level difference between the inner and outer edges of the road will be

An
air chamber of volume V has
a neck area of cross section a into
which a ball of mass m just
fits and can move up and down without any friction (Fig.14.33). Show
that when the ball is pressed down a little and released, it executes
SHM. Obtain an expression for the time period of oscillations
assuming pressure-volume variations of air to be isothermal [see Fig.
14.33].

Consider a body shown in the figure below, consisting of two identical balls, each of mass 'M' connected by a light rigid rod of length 'L'. If an impulse, J = Mv is imparted to the body at one of its ends, what would be its angular velocity?

A point source of light $B$ is placed at a distance $L$ in front of the center of a mirror of width $d$ hung vertically on a wall. A man walks in front of the mirror along a line parallel to the mirror at a distance $2L$ from it as shown. The greatest distance over which he can see the image of the light source in the mirror is

In the equation $(P+\frac{a}{V^2})(V-b)=\text{constant}$, the unit of $a$ is

The mean unconfined compressive strength of a purely cohensive soil was found to be $50kN/m^2$, The ultimate bearing capacity of a square footing calculated by Terzaghi's concept (bearing capacity factor $N_c$ = 5.7) will be

A metallic block carrying current $I$ is subjected to a uniform magnetic induction $\overrightarrow{B}$ as shown in the figure. Find the force experience $\overrightarrow{F}$ by the moving charges. Also find the face, whose potential is lowered result of this force. Assume the speed of the charge carriers to be $v$.

A train is moving along a straight line with a constant acceleration $a$. A boy standing in the train throws a ball upwards with a speed of $10\text{m/s}$ at an angle ${30}^{\circ }$ to the horizontal. The boy has to move forward by $5\text{m}$ inside the train to catch the ball back at its initial height. The acceleration of the train in ${\text{m/s}}^2$ is: