A series \(R-C\) combination is connected to an \(\mathrm{AC}\) voltage of angular frequency \(\omega=500 \mathrm{rad} / \mathrm{s}\). If the impedance of the \(R-C\) circuit is \(R \sqrt{1.25}\), the time constant (in millisecond) of the circuit is

Sometimes it is convenient to construct a system of units so that all quantities can be expressed in terms of only one physical quantity. In one such system, dimensions of different quantities are given in terms of a quantity $X$ as follows: $\left[\mathrm{position}\right]=\left[X^{\alpha }\right];$ $\left[\mathrm{speed}\right]=\left[X^{\beta }\right]$; $\left[\mathrm{acceleration}\right]=\left[X^p\right]$;$\left[\mathrm{linear}\mathrm{momentum}\right]=\left[X^q\right]$; $\left[\mathrm{force}\right]=\left[X^r\right]$. Then

Six charges are placed around a regular hexagon of side length $a$ as shown in the figure. Five of them have charge $q$, and the remaining one has charge $x$. The perpendicular from each charge to the nearest hexagon side passes through the center $O$ of the hexagon and is bisected by the side.

Which of the following statement(s) is(are) correct in SI units?

The electrostatic energy of Z protons uniformly distributed throughout a spherical nucleus of radius R is given by $E=\frac{3}{5}\frac{Z\left(Z-1\right)e^2}{4\pi {\epsilon }_{0}R}$ The measured masses of the neutron, ${}_1{}^{1}H, {}_8{}^{15}O$ are 1.008665 u, 1.007825 u, 15.000109 u and 15.003065 u respectively. Given that the radii of both the ${}_7{}^{15}N and {}_8{}^{15}O$ nuclei are same, 1u = 931.5 Me V/ $c^2$ (c is the speed of light) and $\frac{e^2}{\left(4\pi {ϵ}_0\right)}=1.44 MeV fm.$ Assuming that the difference between the binding energies of ${}_7{}^{15}N and {}_8{}^{15}O$ is purely due to the electrostatic energy, the radius of either of the nuclei is $\left(1 fm={10}^{-15}m\right)$

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When liquid medicine of density \(\rho\) is to be put in the eye, it is done with the help of a dropper. As the bulb on the top of the dropper is pressed, a drop forms at the opening of the dropper. We wish to estimate the size of the drop.
We first assume that the drop formed at the opening is spherical because that requires a minimum increase in its surface energy. To determine the size, we calculate the net vertical force due to the surface tension \(T\) when the radius of the drop is \(R\). When this force becomes smaller than the weight of the drop, the drop gets detached from the dropper.Question :
After the drop detaches, its surface energy is

While conducting the Young's double slit experiment, a student replaced the two slits with a large opaque plate in the x-y plane containing two small holes that act as two coherent point sources $\left(S_1,S_2\right)$ emitting light of wavelength 600 nm. The student mistakenly placed the screen parallel to the x-z plane (for z > 0) at a distance D = 3m from the mid-point of $\left(S_1,S_2\right)$ , as shown schematically in the figure. The distance between the sources d = 0.6003 mm. The origin O is at the intersection of the screen and the line joining $\left(S_1,S_2\right)$ . Which of the following is (are) true of the intensity pattern on the screen?

Consider an electron in the \(n=3\) orbit of a hydrogen-like atom with atomic number \(Z\). At absolute temperature \(T\), a neutron having thermal energy \(k_{\mathrm{B}} T\) has the same de Broglie wavelength as that of this electron. If this temperature is given by \(T=\frac{Z^2 h^2}{\alpha \pi^2 a_0^2 m_N k_B}\), (where \(h\) is the Planck's constant, \(k_B\) is the Boltzmann constant, \(m_{\mathrm{N}}\) is the mass of the neutron and \(a_0\) is the first Bohr radius of hydrogen atom) then the value of \(\alpha\) is ________

The tangent \(P T\) and the normal \(P N\) to the parabola \(y^2=4 a x\) at a point \(P\) on

A few electric field lines for a system of two charges \(Q_1\) and \(Q_2\) fixed at two different points on the \(x\)-axis are shown in the figure. These lines suggest that

If \(\vec{a}\) and \(\vec{b}\) are vectors such that \(|\vec{a}+\vec{b}|=\sqrt{29}\) and \(\vec{a} \times(2 \hat{i}+3 \hat{j}+4 \hat{k})=(2 \hat{i}+3 \hat{j}+4 \hat{k}) \times \vec{b}\), then a possible value of \((\vec{a}+\vec{b}) \cdot(-7 \hat{i}+2 \hat{j}+3 \hat{k})\) is

A curve passes through the point $\left(1,\frac{\pi }{6}\right)$. Let the slope of the curve at each point $\left(\text{x},\text{y}\right)$ be $\frac{\text{y}}{\text{x}}+\text{sec}\left(\frac{\text{y}}{\text{x}}\right),\; x > 0$. Then the equation of the curve is

The densities of two solid spheres $\mathrm{A}$ and $\mathrm{B}$ of the same radii $\mathrm{R}$ vary with radial distance $\mathrm{r}$ as ${\mathrm{\rho }}_{\mathrm{A}}\left(\mathrm{r}\right)=\mathrm{k}\left(\frac{\mathrm{r}}{\mathrm{R}}\right)$ and ${\mathrm{\rho }}_{\mathrm{B}}\left(\mathrm{r}\right)=\mathrm{k}{\left(\frac{\mathrm{r}}{\mathrm{R}}\right)}^5$ , respectively, where $\mathrm{k}$ is a constant. The moments of inertia of the individual spheres about axes passing through their centres are ${\mathrm{I}}_{\mathrm{A}}$ and ${\mathrm{I}}_{\mathrm{B}}$ , respectively. If $\frac{{\mathrm{I}}_{\mathrm{B}}}{{\mathrm{I}}_{\mathrm{A}}}=\frac{\mathrm{n}}{10}$ , the value of $\mathrm{n}$ is

The electrochemical cell shown below is a concentration cell. \(\mathrm{M} \mid \mathrm{M}^{2+}\) (saturated solution of a sparingly soluble salt, \(\left.\mathrm{MX}_{2}\right) \| \mathrm{M}^{2 \perp}\left(0.001 \mathrm{~mol} \mathrm{dm}^{-3}\right) \mid \mathrm{M}\).
The emf of the cell depends on the difference in concentrations of \(\mathrm{M}^{2+}\) ions at the two electrodes. The emf of the cell at \(298 \mathrm{~K}\) is \(0.059 \mathrm{~V}\).
Question:
The value of \(\Delta \mathrm{G}\left(\mathrm{kJ} \mathrm{mol}^{-1}\right)\) for the given cell is (take 1 \(\mathrm{F}=96500 \mathrm{Cmol}^{-1}\) )