Let ${\text{S}}_{\text{n}}=\underset{\text{k}=1}{\overset{4\text{n}}{\Sigma }}{\left(-1\right)}^{\frac{\text{k}\left(\text{k}+1\right)}{2}}{\text{k}}^2$. Then S_n can take value(s)

Let \(z=x+i y\) be a complex number, where \(x\) and \(y\) are integers. Then, the area of the rectangle whose vertices are the roots of the equation \(z \bar{z}^3+\bar{z} z^3=350\) is

The function \(f:[0,3] \rightarrow[1,29]\), defined by \(f(x)=2 x^{3}-15 x^{2}+36 x+1\), is

Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be two unit vectors such that $\overrightarrow{a}.\overrightarrow{b}=0.$ For some $x,y\in \mathrm{R,}$ let $\overrightarrow{c}=x\overrightarrow{a}+y\overrightarrow{b}+\left(\overrightarrow{a}\times \overrightarrow{b}\right).$ If $\left|\overrightarrow{c}\right|=2$ and the vector $\overrightarrow{c}$ is inclined at the same angle $\alpha$ to both $\overrightarrow{a}$ and $\overrightarrow{b},$ then the value of $8{\cos }^2\alpha$ is

If \(\vec{a}, \vec{b}\) and \(\vec{c}\) are unit vectors satisfying \(|\vec{a}-\vec{b}|^{2}+|\vec{b}-\vec{c}|^{2}+|\vec{c}-\vec{a}|^{2}=9\), then \(|2 \vec{a}+5 \vec{b}+5 \vec{c}|\) is

Let $\mathrm{S}$ be the sample space of all $3\times 3$ matrices with entries from the set $\left\{0,\mathrm{ }1\right\}.$ Let the events ${\mathrm{E}}_1$ and ${\mathrm{E}}_2$ be given by
${\mathrm{E}}_1=\left\{\mathrm{A}\in \mathrm{S}:\mathrm{d}\mathrm{e}\mathrm{t}\mathrm{A}=0\right\}$ and
${\mathrm{E}}_2=\{\mathrm{A}\in \mathrm{S}:$ sum of entries of $\mathrm{A}$ is $7\}.$
If a matrix is chosen at random from $\mathrm{S},$ then the conditional probability $\mathrm{P}\left({\mathrm{E}}_1|{\mathrm{E}}_2\right)$ equals____

A plane passes through \((1,-2,1)\) and is perpendicular to two planes \(2 x-2 y+z=0\) and \(x-y+2 z=4\), then the distance of the plane from the point \((1,2,2)\) is

An object and a concave mirror of focal length $f=10 \mathrm{cm}$ both move along the principal axis of the mirror with constant speeds. The object moves with speed $V_0=15 \mathrm{cm} {\mathrm{s}}^{-1}$ towards the mirror with respect to a laboratory frame. The distance between the object and the mirror at a given moment is denoted by $u$. When $u=30 \mathrm{cm}$, the speed of the mirror $V_m$ is such that the image is instantaneously at rest with respect to the laboratory frame, and the object forms a real image. The magnitude of $V_m$ is _______ $\mathrm{cm} {\mathrm{s}}^{-1}$.

The given graphs/data I, II, III and IV represent general trends observed for different physisorption and chemisorption processes under mild conditions of temperature and presure. Which of the following

An ideal gas is in thermodynamic equilibrium. The number of degrees of freedom of a molecule of the gas is $n$. The internal energy of one mole of the gas is $U_n$ and the speed of sound in the gas is $v_n$. At a fixed temperature and pressure, which of the following is the correct option?

One mole of a monatomic ideal gas is taken through a cycle \(A B C D A\) as shown in the \(p\)-V diagram. Column II gives the characteristics involved in the cycle. Match them with each of the processes given in Column I.

Two lines ${\text{L}}_1\text{ : }\text{x}=5\text{,}\frac{\text{y}}{3-\alpha }=\frac{\text{z}}{-2}$ and ${\text{L}}_2\text{ : }\text{x}=\alpha \text{,}\frac{\text{y}}{-1}=\frac{\text{z}}{2-\alpha }$ are coplanar. Then $\alpha$ can take value(s)

Paragraph :
A special metal \(S\) conducts electricity without any resistance. A closed wire loop, made of \(S\), does not allow any change in flux through itself by inducing a suitable current to generate a compensating flux. The induced current in the loop cannot decay due to its zero resistance. This current gives rise to a magnetic moment which in turn repels the source of magnetic field or flux. Consider such a loop, of radius \(a\), with its center at the origin. A magnetic dipole of moment \(m\) is brought along the axis of this loop from infinity to a point at distance \(r(\gg a)\) from the center of the loop with its north pole always facing the loop, as shown in the figure below.

The magnitude of magnetic field of a dipole \(m\), at a point on its axis at distance \(r\), is \(\frac{\mu_{0}}{2 \pi} \frac{m}{r^{3}}\), where \(\mu_{0}\) is the permeability of free space. The magnitude of the force between two magnetic dipoles with moments, \(m_{1}\) and \(m_{2}\), separated by a distance \(r\) on the common axis, with their north poles facing each other, is \(\frac{k m_{1} m_{2}}{r^{4}}\), where \(k\) is a constant of appropriate dimensions. The direction of this force is along the line joining the two dipoles.

Question :
The work done in bringing the dipole from infinity to a distance$r$from the centre of the loop by the given process is proportional to: