A tiny spherical oil drop carrying a net charge \(q\) is balanced in still air with a vertical uniform electric field of strength \(\frac{81 \pi}{7} \times 10^5 \mathrm{Vm}^{-1}\). When the field is switched off, the drop is observed to fall with terminal velocity \(2 \times 10^{-3} \mathrm{~ms}^{-1}\). Given \(g=9.8 \mathrm{~ms}^{-2}\), viscosity of the air \(=1.8 \times 10^{-5} \mathrm{Ns} \mathrm{m}^{-2}\) and the density of oil \(=900 \mathrm{~kg} \mathrm{~m}^{-3}\), the magnitude of \(q\) is

The density of a solid ball is to be determined in an experiment. The diameter of the ball is measured with a screw gauge, whose pitch is \(0.5 \mathrm{~mm}\) and there are 50 divisions on the circular scale. The reading on the main scale is \(2.5 \mathrm{~mm}\) and that on the circular scale is 20 divisions. If the measured mass of the ball has a relative error of \(2 \%\), the relative percentage error in the density is

A string of length $1 \mathrm{m}$ and mass $2\times {10}^{-5} \mathrm{kg}$ is under tension $T$. When the string vibrates, two successive harmonics are found to occur at frequencies $750 \mathrm{Hz}$ and $1000 \mathrm{Hz}$. The value of tension $T$ is _____ newton.

Consider a neutral conducting sphere. A positive point charge is placed outside the sphere. The net charge on the sphere is then

A spherical metal shell \(A\) of radius \(R_A\) and a solid metal sphere \(B\) of radius \(R_B < \left(R_A\right)\) are kept far apart and each is given charge \(+Q\). Now, they are connected by a thin metal wire. Then

A uniform rod of length \(L\) and mass \(M\) is pivoted at the centre. Its two ends are attached to two springs of equal spring constants \(k\). The springs are fixed to rigid supports as shown in the figure, and rod is free to oscillate in the horizontal plane. The rod is gently pushed through a small angle \(\theta\) in one direction and released. The frequency of oscillation is

If the wavelength of the \(n^{\text {th }}\) line of Lyman series is equal to the de-Broglie wavelength of electron in initial orbit of a hydrogen like element \((Z=11)\). Find the value of \(n\).

Consider a concave mirror and a convex lens (refractive index = $1.5$) of focal length $10$ $\mathrm{cm}$ each, separated by a distance of $50$ $\mathrm{cm}$ in air (refractive index = $1$), as shown in the figure. An object is placed at a distance of $15$ $\mathrm{cm}$ from the mirror. Its erect image, formed by this combination, has magnification ${\mathrm{M}}_1$ . When the set-up is kept in a medium of refractive index $\frac{7}{6}$ , the magnification becomes ${\mathrm{M}}_2$. The magnitude $\left|\frac{{\mathrm{M}}_2}{{\mathrm{M}}_1}\right|$ is

Let \(\vec{p}=2 \hat{i}+\hat{j}+3 \hat{k}\) and \(\vec{q}=\hat{i}-\hat{j}+\hat{k}\). If for some real numbers \(\alpha, \beta\), and \(\gamma\), we have \(15 \hat{i}+10 \hat{j}+6 \hat{k}=\alpha(2 \vec{p}+\vec{q})+\beta(\vec{p}-2 \vec{q})+\gamma(\vec{p} \times \vec{q})\) then the value of \(\gamma\) is

The correct combination of names for isomeric alcohols with molecular formula ${\mathrm{C}}_4{\mathrm{H}}_{10}\mathrm{O}$ is/are

Most materials have the refractive index, \(n>1\). So, when a light ray from air enters a naturally occurring material, then by Snell's

law, \(\frac{\sin \theta_{1}}{\sin \theta_{2}}=\frac{n_{2}}{n_{1}}\), it is understood that the refracted ray bends towards the normal. But it never emerges on the same side of the normal as the incident ray. According to electromagnetism, the refractive index of the medium is given by the relation, \(\mathrm{n}=c / v=\pm \sqrt{\varepsilon_{r} \mu_{r}}\), where \(c\) is the speed of electromagnetic waves in vacuum, \(v\) its speed in the medium, \(\varepsilon_{r}\) and \(\mu_{r}\) are the relative permittivity and permeability of the medium respectively. In normal materials, both \(\varepsilon_{r}\) and \(\mu_{r}\), are positive, implying positive \(n\) for the medium. When both \(\varepsilon_{r}\) and \(\mu_{r}\) are negative, one must choose the negative root of \(n\). Such negative refractive index materials can now be artificially prepared and are called metamaterials. They exhibit significantly different optical behavior, without violating any physical laws. Since \(n\) is negative, it results in a change in the direction of propagation of the refracted light. However, similar to normal materials, the frequency of light remains unchanged upon refraction even in meta-materials.
Question : Choose the correct statement.

A current carrying wire heats a metal rod. The wire provides a constant power $\left(\mathrm{P}\right)$ to the rod. The metal rod is enclosed in an insulated container. It is observed that the temperature $\left(\mathrm{T}\right)$ in the metal rod changes with time $\left(\mathrm{t}\right)$ as:
$\mathrm{T}\left(\mathrm{t}\right)={\mathrm{T}}_0\left(1+\mathrm{\beta }{\mathrm{t}}^{1/4}\right)$
where $\mathrm{\beta }$ is a constant with appropriate dimension while ${\mathrm{T}}_0$ is a constant with dimension of temperature. The heat capacity of the metal is:

For every twice differentiable function $f:\mathbb{R}\to \left[-2, 2\right] \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} {\left(f\left(0\right)\right)}^2+{\left(f^{\prime }\left(0\right)\right)}^2=85$ , which of the following statement(s) is (are) TRUE?