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If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are unit vectors, then $|3\overrightarrow{a}-2\overrightarrow{b}{|}^2+|3\overrightarrow{b}-2\overrightarrow{c}{|}^2+|3\overrightarrow{c}-2\overrightarrow{a}{|}^2$ does not exceed ?

A block of mass $m$ is on an inclined plane of angle $\theta$. The coefficient of friction between the block and the plane is $\mu$ and $\tan \theta >\mu$. The block is held stationary by applying a force $P$ parallel to the plane. The direction of force pointing up the plane is taken to be positive. As $P$ is varied from $P_1=mg(\sin \theta -\mu \cos \theta )$ to $P_2=mg(\sin \theta +\mu \cos \theta )$, the frictional force $f$ versus $P$ graph will look like:

Inside a long straight uniform wire of circular cross - section there is a long cylindrical cavity of radius $r$ whose axis is parallel to the axis of the wire. Axis of cavity is displaced from the axis of wire by a distance $L$ as shown in figure. A direct current density $J$ flows through the wire. Then

(I) Magnetic field inside the cavity is non-uniform

(II) Magnetic field inside the cavity is uniform

(III) Field inside the cavity is directly proportional to $L$

(IV) Magnetic field inside the cavity is independent of $L$

A pendulum clock loses 12 seconds a day if the temperature is ${40}^{\circ }C$ and goes fast by 4 seconds a day if the temperature is ${20}^{\circ }C$. Find the temperature at which the clock will show correct time, and the coefficient of linear expansion of the metal of the pendulum shaft.

A bar of iron is 10 cm at 20°C. At 19°C it will be (a of iron = 11 ´ 10^-6/°C)

A satellite is projected vertically upwards from an earth station. AT what height above the earth's surface will the force on the satellite due to the earth be reduced to half its value at the earth station ? (Radius of the earth is 6400 km.)

A ball collides elastically with another ball of the same mass. The collision is oblique and initially second ball was at rest. After the collision, the two balls move with same speeds. What will be the angle between the velocities of the balls after the collision?

A particle is subjected to two simple harmonic motions in the same direction having equal amplitudes and equal frequency. If the resultant amplitude is equal to the amplitude of the individual motions, find the phase difference between the individual motions.

A series circuit consists of a resistance, inductor and capacitance. The applied voltage and the current at any instant are given by $E=141.4\cos (5000t–{10}^{\circ })\text{V}$ and $I=5\cos (5000t–{370}^{\circ })\text{A}$. The inductance of inductor is $0.01\text{H}$. Calculate the value of capacitance.

A small piece of Cesium metal having work function $(\varphi =1.9\text{eV})$ is kept at a distance of $20\text{cm}$ from a large metal plate having a charge density of $1.0\times {10}^{-9}{\text{C m}}^{-2}$ on the surface facing the small Cesium piece. A monochromatic light of wavelength $400\text{nm}$ is incident on the Cesium piece. Find the minimum and the maximum kinetic energy of the photoelectrons reaching the large metal plate. Neglect any change in electric field due to the small piece of cesium present.$(hc=1240\text{eV nm})$

Which of the following is a correct graph for molar heat capacity (at constant pressure) versus temperature for an ideal di-atomic gas ?

Figure shows three regions of magnetic field, each of area A, and in each region magnitude of the field decreases at a constant rate $‘{\alpha }^{\prime }$. If $\overrightarrow{E}$ is induced electric field then value of line integral $\oint \overrightarrow{E}.d\overrightarrow{r}$ along the given loop is equal to

Two forces 12N and 16 N are acting on a body. What can be max. And minimum force on the body?

An object moves in a straight line according to the equation, x = 2 $t^3$ - 3 $t^2$ - 6t, where x is in meter and t is in second. At what time is the acceleration of the object zero?

A simple pendulum has a time period $T_1$ when on the earth's surface and $T_2$ when taken to a height $2R$ above the earth's surface, where $R$ is the radius of the earth. The value of $\left(\frac{T_1}{T_2}\right)$ is:

A steel wire is suspended vertically from a rigid support. When loaded with a weight in air, it extends by $l_a$ and when the weight is immersed completely in water, the extension is reduced to $l_w$. Then the relative density of material of the weight is

A particle moving in uniform acceleration covers a distance of ,3 meter in 8th sec and in 16th s during its motion

Velocity of the particle of the end of 12th s

Total distance travelled by the body in 20 s

For two particles A and B, graphs are plotted for $\sqrt{V}$ against the de broglie wavelengths $\lambda$, where $V$ is the potential on the particles. Which of the following relation is correct about the mass of the particles?

A tube has two areas of cross-section as shown in the figure. The diameter of the bigger section is double the diameter of the smaller section. Find the range of water falling on the horizontal surface, if piston is moving with a constant velocity $0.4\text{m/s}$ and $h=2.5\text{m}(g=10{\text{m/s}}^2)$

The potential energy function for the force between two atoms in a diatomic molecule which is approximately given as,

$U\left(x\right)=\frac{a}{x^{12}}-\frac{b}{x^6}$. Where $a\&b$ are constants and $x$ is the distance between the atoms. The disassociation energy of the molecule, $D=\left[U\right(x=\infty )-U_{equilibrium}]$ is

A particle of mass m is kept a fixed, smooth sphere of radius R at a position, where the radius through the particle makes an angle of ${30}^{\circ }$ with the vertical. The: particle is released from this position. (a) What is the force exerted by the sphere on the particle just after the release ? (b) Find the distance travelled by the particle before it leaves contact with the sphere.

One mole of an ideal gas undergoes a process $P=P_0{\left(1+{\left(\frac{2V_0}{V}\right)}^2\right]}^{-1},$ where$P_0,V_0$ are constants. Change in temperature of the gas when volume is changed from $V=V_0$ to $V=2V_0$ is

A stone of mass $m$ is thrown with a velocity $u$ at a block of mass $M$ which is initially at rest. The collision between the bodies is elastic. If the final velocities of the stone and the block are in the ratio $-4:1$, then what is the ratio of the mass of the stone to the mass of the block?

Two identical heater wires are first connected in series and then in parallel with a source of electricity.



The ratio of heat produced in the first case to heat produced in second case is:

A system of three charges are placed as shown in the figure.





If $D>>d$, the potential energy of the system is best given by

In a mass-haul diagram, the distance (D) of the centre of massof excavation or embankment from the line representing the volumeto be hauled over is given by

Light of wavelength $\lambda$ strikes a photoelectric surface and electrons are ejected with kinetic energy $K.$ If $K$ is to be increased to exactly twice its original value, the wavelength must be changed to $\lambda$ such that

Work done per mole in an isothermal change is