One end of a horizontal thick copper wire of length $2\mathrm{L}$and radius $2\mathrm{R}$ is welded to an end of another horizontal thin copper wire of length$\mathrm{L}$ and radius$\mathrm{R}$. When the arrangement stretched by applying forces at two ends, the ratio of the elongation in the thin wire to that in the thick wire is 

A massless rod \(B D\) is suspended by two identical massless strings \(A B\) and \(C D\) of equal lengths. A block of mass ' \(m\) ' is suspended point \(P\) such that \(B P\) is equal to ' \(x\) ', if the fundamental frequency of the left wire is twice the fundamental frequency of right wire, then the value of \(x\) is

Two beads, each with charge \(q\) and mass \(m\), are on a horizontal, frictionless, non-conducting, circular hoop of radius \(R\). One of the beads is glued to the hoop at some point, while the other one performs small oscillations about its equilibrium position along the hoop. The square of the angular frequency of the small oscillations is given by
[ \(\varepsilon_0\) is the permittivity of free space.]

The center of a disk of radius \(r\) and mass \(m\) is attached to a spring of spring constant \(k\), inside a ring of radius \(R>r\) as shown in the figure. The other end of the spring is attached on the periphery of the ring. Both the ring and the disk are in the same vertical plane. The disk can only roll along the inside periphery of the ring, without slipping. The spring can only be stretched or compressed along the periphery of the ring, following the Hooke's law. In equilibrium, the disk is at the bottom of the ring. Assuming small displacement of the disc, the time period of oscillation of center of mass of the disk is written as \(T=\frac{2 \pi}{\omega}\). The correct expression for \(\omega\) is ( \(g\) is the acceleration due to gravity):

Two infinitely long straight wires lie in the xy - plane along the lines $x=\pm R$. The wire located at $x = +R$ carries a constant current $I_1$ and the wire located at $x =-R$ carries a constant current $I_2$. A circular loop of radius R is suspended with its centre at $\left(0, 0\sqrt{3}R\right)$ and in a plane parallel to the xy - plane. This loop carries a constant current $I$ in the clockwise direction as seen from above the loop. The current in the wire is taken to be positive if it is in the $+\widehat{j}$ direction. Which of the following statements regarding the magnetic field $\overrightarrow{B}$ is (are) true?

A current carrying infinitely long wire is kept along the diameter of a circular wire loop, without touching it, the correct statement(s) is(are)

A flat surface of a thin uniform disk $A$of radius$R$ is glued to a horizontal table. Another thin uniform disk$B$ of mass $M$and with the same radius $R$rolls without slipping on the circumference of $A$, as shown in the figure. A flat surface of $B$ also lies on the plane of the table. The center of mass of$B$ has fixed angular speed $\omega$about the vertical axis passing through the center of $A$. The angular momentum of $B$ is $nM\omega R^2$with respect to the center of $A$. Which of the following is the value of $n$?

For a reaction, $A\rightleftharpoons P$ , the plots of $\left[A\right]\mathrm{a}\mathrm{n}\mathrm{d}\left[P\right]$ with time at temperatures $T_1\mathrm{a}\mathrm{n}\mathrm{d}{T}_2$ are given below.

If $T_2>T_1$ , the correct statement(s) is (are)
(Assume $∆H^{⊝}\mathrm{a}\mathrm{n}\mathrm{d}∆S^{⊝}$ are independent of temperature and ratio of $\ln K\mathrm{a}\mathrm{t}{T}_1\mathrm{t}\mathrm{o}\ln K\mathrm{a}\mathrm{t}{T}_2$ is greater than $T_2/T_1$ . Here H, S , G and K are enthalpy, entropy, Gibbs energy and equilibrium constant, respectively.)

The figure below is the plot of potential energy versus internuclear distance $\left(\mathrm{d}\right)$ of ${\mathrm{H}}_2$ molecule in the electronic ground state. What is the value of the net potential energy ${\mathrm{E}}_0$ (as indicated in the figure) in $\mathrm{kJ}{\mathrm{mol}}^{-1},$ for $\mathrm{d}={\mathrm{d}}_0$ at which the electron-electron repulsion and the nucleus-nucleus repulsion energies are absent? As reference, the potential energy of $\mathrm{H}$ atom is taken as zero when its electron and the nucleus are infinitely far apart.
Use Avogadro constant as $6.023\times {10}^{23}{\mathrm{mol}}^{-1}.$

Mark total potential energy for 2 H-atom (per mole) as answer.

A normal with slope \(\frac{1}{\sqrt{6}}\) is drawn from the point \((0,-\alpha)\) to the parabola \(x^2=-4 a y\), where \(a>0\). Let \(L\) be the line passing through \((0,-\alpha)\) and parallel to the directrix of the parabola. Suppose that \(L\) intersects the parabola at two points \(A\) and \(B\). Let \(r\) denote the length of the latus rectum and \(s\) denote the square of the length of the line segment \(A B\). If \(r: s=1: 16\), then the value of 24ais ________.

Let \(\mathbb{N}\) denote the set of all natural numbers, and \(\mathbb{Z}\) denote the set of all integers. Consider the functions \(f: \mathbb{N} \rightarrow \mathbb{Z}\) and \(g: \mathbb{Z} \rightarrow \mathbb{N}\) defined by
\(f(n)= \begin{cases}(n+1) / 2 & \text { if } n \text { is odd } \\ (4-n) / 2 & \text { if } n \text { is even }\end{cases}\) And
\(g(n)=\left\{\begin{array}{cc}3+2 n & \text { if } n \geq 0 \\ -2 n & \text { if } n <0\end{array}\right.\)
Define \((g \circ f)(n)=g(f(n))\) for all \(n \in \mathbb{N}\), and \((f \circ g)(n)=f(g(n))\) for all \(n \in \mathbb{Z}\).
Then which of the following statements is (are) TRUE?

Consider two solid spheres P and Q each of density 8 gm $c{m}^{-3}$ and diameters 1 cm and 0.5 cm, respectively. Sphere P is dropped into a liquid of density 0.8 gm $c{m}^{-3}$ and viscosity $\eta =3$ poiseulles. Sphere Q is dropped into a liquid of density 1.6 gm $c{m}^{-\mathrm{3 }}$and viscosity $\eta =2$ poiseulles. the ratio of the terminal velocities of P and Q is.

If the freezing point of a 0.01 molal aqueous solution of a cobalt (III) chloride-ammonia complex (which behaves as a strong electrolyte) is $-{0.0558}^{\mathrm{o}}\mathrm{C}$ , the number of chloride(s) in the coordination sphere of the complex is
$[{\mathrm{K}}_{\mathrm{f}}\mathrm{ }\mathrm{o}\mathrm{f}\mathrm{ }\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}=1.86\mathrm{ }\mathrm{K}\mathrm{ }\mathrm{k}\mathrm{g}\mathrm{ }\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}]$