The work functions of silver and sodium are 4.6 and 2.3 eV, respectively. The ratio of the slope of the stopping potential versus frequency plot for silver to that of sodium is

The instantaneous voltages at three terminals marked $X, Y$ and $Z$ are given by
$V_X=V_0\sin \omega t$
$V_Y=V_0\sin \left(\omega t+\frac{2\pi }{3}\right)$ and
$V_Z=V_0\sin \left(\omega t+\frac{4\pi }{3}\right).$
An ideal voltmeter is configured to read rms value of the potential difference between its terminals. It is connected between points $X$ and $Y$ and then between $Y$ and $Z$ . The reading(s) of the voltmeter will be

Two independent harmonic oscillators of equal mass are oscillating about the origin with angular frequencies ${\mathrm{\omega }}_1$ and ${\mathrm{\omega }}_2$ and have total energies ${\mathrm{E}}_1$ and ${\mathrm{E}}_2$ , respectively. The variations of their momenta $\mathrm{p}$ with positions $\mathrm{x}$ are shown in the figures. If $\frac{\mathrm{a}}{\mathrm{b}}={\mathrm{n}}^2$ and $\frac{\mathrm{a}}{\mathrm{R}}=\mathrm{n}$ , then the correct equation(s) is (are)

Column I shows four situations of standard Young's double slit arrangement with the screen placed far away from the slits \(S_1\) and \(S_2\). In each of these cases \(S_1 P_0=S_2 P_0\), \(S_1 P_1-S_2 P_1=\frac{\lambda}{4}\) and \(S_1 P_2-S_2 P_2=\frac{\lambda}{3}\), where \(\lambda\) is the wavelength of the light used. In the cases \(\mathrm{B}, \mathrm{C}\) and \(\mathrm{D}\), a transparent sheet of refractive index \(\mu\) and thickness \(t\) is pasted on slit \(S_2\). The thickness of the sheets are different in different cases. The phase difference between the light waves reaching a point \(P\) on the screen from the two slits is denoted by \(\delta(P)\) and the intensity by \(I(P)\). Match each situation given in Column-I with the statement(s) in Column-II valid for that situation.

Statement I The total translational kinetic energy of all the molecules of a given mass of an ideal gas is \(1.5\) times the product of its pressure and its volume.
Statement II The molecules of a gas collide with each other and the velocities of the molecules change due to the collision.

A positive point charge of \(10^{-8} \mathrm{C}\) is kept at a distance of 20 cm from the center of a neutral conducting sphere of radius 10 cm . The sphere is then grounded and the charge on the sphere is measured. The grounding is then removed and subsequently the point charge is moved by a distance of 10 cm further away from the center of the sphere along the radial direction. Taking \(\frac{1}{4 \pi \epsilon_0}=9 \times 10^9 \mathrm{Nm}^2 / \mathrm{C}^2\) (where \(\epsilon_0\) is the permittivity of free space), which of the following
statements is/are correct: :

Incandescent bulbs are designed by keeping in mind that the resistance of their filament increases with the increase in temperature. If at room temperature, \(100 \mathrm{~W}, 60 \mathrm{~W}\) and \(40 \mathrm{~W}\) bulbs have filament resistances \(R_{100}, R_{60}\) and \(R_{40}\), respectively, the relation between these resistances is

A planar structure of length $\mathrm{L}$ and width $\mathrm{W}$ is made of two different optical media of refractive indices ${\mathrm{n}}_1=1.5$ and ${\mathrm{n}}_2=1.44$ as shown in figure. If $\mathrm{L}>>\mathrm{W},$ a ray entering from end $\mathrm{A}\mathrm{B}$ will emerge from end $\mathrm{C}\mathrm{D}$ only if the total internal reflection condition is met inside the structure. For $\mathrm{L}=9.6\mathrm{}\mathrm{m},$ if the incident angle $\mathrm{\theta }$ is varied, the maximum time taken by a ray to exit the plane $\mathrm{C}\mathrm{D}$ is $\mathrm{t}\times {10}^{-9}\mathrm{s},$ where $\mathrm{t}$ is ____.
[Speed of light $\mathrm{c}=3\times {10}^8\mathrm{m}/\mathrm{s}]$

For $x\in R$, the number of real roots of the equation $3x^2-4\left|x^2-1\right|+x-1=0$ is

The number of neutrons emitted when \({ }_{92}^{235} \mathrm{U}\) undergoes controlled nuclear fission to \({ }_{54}^{142} \mathrm{Xe}\) and \({ }_{38}^{90} \mathrm{Sr}\) is

Which of the following statement(s) is(are) true?

Let \(\alpha(a)\) and \(\beta(a)\) be the roots of the equation \((\sqrt[3]{1+a}-1) x^{2}+(\sqrt{1+a}-1) x+(\sqrt[6]{1+a}-1)=0\) where \(a\) \(>-1\). Then \(\lim _{a \rightarrow 0^{+}} \alpha(a)\) and \(\lim _{x \rightarrow 0^{+}} \beta(a)\) are

Hydrogen bonding plays a central role in the following phenomena: