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A particle of mass ${10}^{-2}\text{kg}$ is moving along the positive $x$-axis under the influence of a force $F\left(x\right)=-\frac{\alpha }{8x^2}$, where $\alpha ={10}^{-2}{\text{N.m}}^2$. At time $t=0$, it is at $x=1.0\text{m}$ and velocity is $v=0$. Find the velocity when it reaches $x=0.5\text{m}$. Consider the magnitude along with the sign to represent the velocity.

If $\overrightarrow{A}-\overrightarrow{B}$ is a unit vector along x - axis and $\overrightarrow{A}=\hat{i}-\hat{j}+\hat{k}$, then what is $\overrightarrow{B}$?

Consider a 10-cm long piece of a wire which carries a current of 10 A.Find the magnitude of the magnetic field due to the piece at a point which makes an equilateral triangle with the ends of the piece.

For a pure substance, a plot of Gibbs' free energy (G) against temperature at constant pressure and Gibbs' free energy (G) against pressure and constant temperature are respectively:

Both the springs shown in figure are unstretched. If the block is displaced by a distance x and released, what will be the initial acceleration ?

The given graph shows a relation between the number of games won (y) by each player having a particular age (x).





The equation of the line is

The given diagram shows an electron gun of a hot cathode ray tube.

X and Y in the diagram respectively represent

State Bohr’s postulate of hydrogen atom which successfully explains the emission lines in the spectrum of hydrogen atom. Use Rydberg formula to determine the wavelength of $H_{\alpha }$line.[Given: Rydberg constant R=$1.03\times {10}^7{m}^{-1}]$

An uncharged parallel plate capacitor of capacitance $C$ completely filled with a dielectric slab of dielectric constant $K$ is connected across a battery of emf $E$. Then heat generated in the wires till the capacitor is fully charged is

A particle starts from rest with a time varying acceleration $a=(2t-4)$. Here $t$ is in $s$ and $a$ in $m/s^2$. At what time, particle will have maximum velocity in negative direction?

A box, when hung from a spring balance shows a reading of 50 kg. If the same box is hung from the same spring balance inside an evacuated chamber, the reading on the scale will be

In a single slit diffraction experiment, first minimum for red light $\left(660\text{nm}\right)$ coincides with first maximum of some other light of wavelength $\lambda$. The value of $\lambda$ in $\text{nm}$ is

A $2\text{g}$ bullet moving with a speed of $210\text{m/s}$ brought to a sudden stoppage by an obstacle. The total heat produced goes to the bullet. If the specific heat of the bullet is $0.03{\text{cal/g}}^{\circ }\text{C}$ the rise in its temperature will be

A uniform magnetic field $B$ exists in a direction perpendicular to the plane of a square loop made of a metal wire. The wire has a diameter of $4\text{mm}$ and a total length of $30\text{cm}$. The magnetic field changes with time at a steady rate $\frac{dB}{dt}=0.032{\text{Ts}}^{-1}.$ The induced current in the loop is close to (Resistivity of the metal wire is $\rho =1.23\times {10}^{-8}\Omega \text{m}$)

A particle executes simple harmonic motion about the point X =0 . At time t =0 it has displacement cm and zero velocity. If the frequency of motion is $0.25se{c}^{-2}$, find

(a) the time period,

(b) angular frequency,

(c) the amplitude,

(d) maximum speed,

(e) the displacement at t =3 sec and

(f) the velocity at t =3 sec.

If $N_0$ is the original mass of the substance of half-life period $t_{\frac{1}{2}}$ = 5 years, then the amount of substance left after 15 years is

For a reaction the rate constants k₁ and k₂ calculated at two different temperatures T₁=500 K and T₂= 700 K are 0.025 s^-1 and 0.07s^-1 respectively. The values of A is:

Ship $\text{A}$ is sailing towards north-east with velocity vector $\overrightarrow{v}=30\hat{i}+50\hat{j}\text{km/hr}$ where $\hat{i}$ points east and $\hat{j}$ north. Ship $B$ is at a distance of $80\text{km}$ east and $150\text{km}$ north of Ship $A$ and is sailing towards west at $10\text{km/hr}$. $A$ will be at minimum distance from $B$ in,

What are the three steps of scientific method?

A collar of mass 10 kg is released from rest from position A, when spring is in natural length and slides along a rough vertical fixed rod. The block comes to rest at point B, 4 m below the point A. The spring constant of spring is 100 N/m and natural length of spring is 3 m. Find work done by friction (‘O’ is a fixed point)

Find the value of 1+cos2A.

For the shown part of a circuit, the current $\left(i\right)$ as a function of time $\left(t\right)$ is, $i=2t^2+5$, where $t$ is in $\text{second}$. The displacement current at $t=1\text{s}$ is -

If a unit vector is represented by $0.5\hat{i}+0.8\hat{j}+c\hat{k}$, then the value of ‘c’ is

A body cools from ${50}^{\circ }C$ to ${40}^{\circ }C$ in $5$ minute. Surrounding temperature is ${20}^{\circ }C$. Then what will be its temperature $5$ minutes after reaching ${40}^{\circ }C$?

Separation between the plates of a parallel plate capacitor is and the area of each plate is A. When a slab of material of dielectric constant and thickness t(t<d) is introduced between the plates, its capacitance becomes

For the given semi-infinite rod of uniformly distributed line charge, find the net electric field at point $\text{P}$.