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The total cost C(x) in rupees associated with the production of x units of an item is given by $C\left(x\right)=0.007x^3-0.003x^2+15x+4000$.

Find the marginal cost when x=17 units are produced.

A plane electromagnetic wave propagating in the $x$-direction has a wavelength of $5.0\text{mm}$. The electric field is in the $y$-direction and its maximum magnitude is $30{\text{Vm}}^{-1}$. Write suitable equations for the electric and magnetic fields as a function of $x$ and $t$.

1. 1.6

Two identical masses are connected to identical springs of spring constant $K$ as shown in the figure. Find the time period of small oscillations.

In an oscillating $LC$ circuit in which $C=4\mu \text{F}$, the maximum potential difference across the capacitor during the oscillation is $1.5\text{V}$ and the maximum current through the inductor is $50\text{mA}$. How much time is required the charge on the capacitor to rise from zero to its maximum value ?

A rod has a total charge $Q$, uniformly distributed along its length $L$. If the rod rotates with angular velocity $\omega$ about its end, compute its magnetic moment.

A block of mass $m$ is dropped onto a spring of spring constant $k$ from a height $h$. The second end of the spring is attached to a second block of mass $M$ as shown in figure. Find the minimum value of $h$ so that the block $M$ bounces off the ground, if the block of mass $m$ sticks to the spring immediately after it comes into contact with it.

two charges are placed on the x-axis + 3 microcoulomb at X equal to zero and a charge of + 5 microcoulomb at X equal to 0.4 metre where must be a third charge placed if the force it experiences to be zero? Please show me how to calculate the answer with full steps

Test on Monday. Please explain faster

A conducting sphere of radius R, and carrying a charge q is joined to a conducting sphere of radius 2R, and carrying a charge – 2q. The charge flown between them will be?

1. 100

If a particle covers half the circle of radius R with constant speed then

A charged capacitor is connected with a resistor. After how many time constants, does the energy of the capacitor become $\frac{1}{10}th$ of its initial value? $(\ln 10=2.303)$

Answer the
following questions:

(a) Quarks
inside protons and neutrons are thought to carry fractional charges
[(+2/3)e ;
(−1/3)e].
Why do they not show up in Millikan’s oil-drop experiment?

(b) What is so
special about the combination e/m?
Why do we not simply talk of e and
m separately?

(c) Why should
gases be insulators at ordinary pressures and start conducting at
very low pressures?

(d) Every metal
has a definite work function. Why do all photoelectrons not come out
with the same energy if incident radiation is monochromatic? Why is
there an energy distribution of photoelectrons?

(e) The energy
and momentum of an electron are related to the frequency and
wavelength of the associated matter wave by the relations:

E
= hν,
p =

But
while the value of λ
is physically significant, the value of ν
(and therefore, the value of the phase speed νλ)
has no physical significance. Why?

A pendulum bob is swinging in a vertical plane such that its angular amplitude is less than ${90}^{\circ }$. If at its extreme position, the string is cut, which of the following trajectory is possible for the bob afterwards?

How
will you ‘weigh the sun’, that is estimate its mass? The
mean orbital radius of the earth around the sun is 1.5 × 10⁸
km.

A small mass slides down an inclined plane of inclination $\theta$ with the horizontal. The coefficient of friciton is $\mu ={\mu }_{0}x$, where $x$ is the distance through which the mass slides down, and ${\mu }_0$ a constant. Then, the speed is maximum after the mass covers a distance of

A long string with charge per unit length λ on it passes through a cube of side a. The minimum and maximum flux through the cube will be

The ratio of the terminal velocities of two drops of same density of radii $R$ and $R/2$ in air is

A chain of mass $10kg$ and length $8m$ is resting on a rough horizontal surface $(\mu =0.2)$. A force $F=15N$ is applied as shown. Find the length (in $m$) of the chain for which no friction force acts.

A current $\left(I\right)$ flows through a uniform wire of diameter $\left(d\right),$ when the mean drift velocity is $V$. If same current flows through a wire of diameter $d/2$ made of the same material, then the mean drift velocity of the electron is

A pair of parallel metal plates are kept with a seperation '$d$'. One plate is at a potential $+V$ and the other is at ground potential. A narrow beam of electrons enters the space between the plates with a velocity $v_0$ and in a direction parallel to the plates. What will be the angle of the beam with the plates after it travels an axial distance $L$?

The mass per unit length of a non-uniform rod $OP$ of length $L$ varies as $m=k\frac{x}{L},$ where $k$ is a constant and $x$ is the distance of point on the rod from end $O$. The distance of the centre of mass of the rod from end $O$ is

A parallel plate capacitor is charged and then isolated. The effect of increasing the plate separation on charge, potential and capacitance respectively are

If $A$ is the area under the graph of the intensity magnetization in ferromagnetic material versus magnetizing field intensity, $V$ is the volume of the magnetic sample and $n$ is the frequency of alternating magnetic field , then the rate of hysteresis loss in the substance is

A body under the action of a force $\overrightarrow{F}=(3\hat{i}-4\hat{j}+\sqrt{11}\hat{k})\text{N}$ acquires an acceleration of $2{\text{m/s}}^2$, the mass of this body must be

A current carrying loop of magnetic moment $16\text{J/T}$ is aligned at ${45}^{\circ }$ with a uniform external magnetic field of $4\text{T}$. What would be the change in potential energy in turning the loop to align its magnetic moment ${135}^{\circ }$ to the magnetic field?

In each situation of column-I, some charge distributions are given with all details explained. The electrostatic potential energy and its nature is given situation in column -II. Match the proper entries from column-II to column-I using the codes given below the columns,

Column-I	Column-II
(P) A thin shell of radius $a$ and having a charge $-Q$ uniformly distributed over its surface as shown	(1) $\frac{1}{8\pi {ϵ}_0}\frac{Q^2}{a}$ in magnitude
(Q) A thin shell of radius $\frac{5a}{2}$ and having a charge $-Q$ uniformly distributed over its surface and a point charge $-Q$ is brought from infinity and placed at its centre as shown.	(2) $\frac{3}{20\pi {ϵ}_0}\frac{Q^2}{a}$ in magnitude
(R) A solid sphere of radius $a$ and having a charge $-Q$ uniformly distributed throughout its volume as shown.	(3) $\frac{27Q}{80\pi {ϵ}_0}\frac{Q^2}{a}$ in magnitude
(S) A solid sphere of radius a and having a charge $-Q$ uniformly distributed throughout its volume. The solid sphere is surrounded by a concentric thin uniformly charged spherical shell of radius 2a and carrying charge $-Q$ as shown	(4) $\frac{5}{27\pi {ϵ}_0}\frac{Q^2}{a}$ in magnitude

The resistance of resistor at temperature $t^{o}C$ can be expressed as $R_t=R_0(1+2\gamma t+\varphi t^2)$, where $R_0$ is the resistance at $0^{o}C$ and $\gamma$,$\varphi$ are constants. The temperature coefficient of the resistance at $t^{o}C$ is