The energy of a system as a function of time $\mathrm{t}$ is given as $\mathrm{E}\left(\mathrm{t}\right)={\mathrm{A}}^2\left(-\mathrm{\alpha }\mathrm{t}\right)$ , where $\alpha =0.2\mathrm{ }{\mathrm{s}}^{-1}$ . The measurement of $\mathrm{A}$ has an error of 1.25%. If the error in the measurement of time is 1.50%, the percentage error in the value of $\mathrm{E}\left(\mathrm{t}\right)$ at $\mathrm{t}=5$ s is

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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 :
If the radius of the opening of the dropper is \(r\), the vertical force due to the surface tension on the drop of radius \(R\) (assuming \(r< < R\) ) is

Two soap bubbles \(A\) and \(B\) are kept in a closed chamber where the air is maintained at pressure \(8 \mathrm{Nm}^{-2}\). The radii of bubbles \(A\) and \(B\) are \(2 \mathrm{~cm}\), respectively. Surface tension of the soap-water used to make bubbles is \(0.04\) \(\mathrm{Nm}^{-1}\). Find the ratio \(\frac{n_B}{n_A}\), where \(n_A\) and \(n_B\) are the number of moles of air in bubbles \(A\) and \(B\), respectively. [Neglect the effect of gravity]

A lamina is made by removing a small disc of diameter \(2 R\) from a bigger disc of uniform mass density and radius \(2 R\), as shown in the figure. The moment of inertia of this lamina about axes passing though \(O\) and \(P\) is \(I_{O}\) and \(I_{P}\) respectively. Both these axes are perpendicular to the plane of the lamina. The ratio \(I_{P} / I_{O}\) to the nearest integer is

Under the influence of the coulomb field of charge \(+Q\), a charge \(-q\) is moving around it in an elliptical orbit. Find out the correct statement(s).

A table tennis ball has radius \((3 / 2) \times 10^{-2} \mathrm{~m}\) and mass \((22 / 7) \times 10^{-3} \mathrm{~kg}\). It is slowly pushed down into a swimming pool to a depth of \(d=0.7 \mathrm{~m}\) below the water surface and then released from rest. It emerges from the water surface at speed \(v\), without getting wet, and rises up to a height \(H\). Which of the following option(s) is(are) correct?
[Given: \(\pi=22 / 7, g=10 \mathrm{~m} \mathrm{~s}^{-2}\), density of water \(=1 \times 10^3 \mathrm{~kg} \mathrm{~m}^{-3}\), viscosity of water \(=1 \times 10^{-3} \mathrm{~Pa}\)-s.]

A uniform circular disc of mass $\text{50 kg}$ and radius $\text{0.4 m}$ is rotating with an angular velocity of ${\text{10 rad s}}^{-1}$ about its own axis, which is vertical. Two uniform circular rings, each of mass $\text{6.25 kg}$ and radius $\text{0.2 m}$, are gently placed symmetrically on the disc in such a manner that they are touching each other along the axis of the disc and are horizontal. Assume that the friction is large enough such that the rings are at rest relative to the disc and the system rotates about the original axis. The new angular velocity $\left({in\; rad\; s}^{-1}\right)$ of the system is

Three numbers are chosen at random, one after another with replacement, from the set \(S=\{1,2,3, \ldots, 100\}\). Let \(p_{1}\) be the probability that the maximum of chosen numbers is at least 81 and \(p_{2}\) be the probability that the minimum of chosen numbers is at most 40 .
The value of $\frac{625}{4}{p}_1$ is

Which ordering of compounds is according to the decreasing order of the oxidation state of nitrogen?

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The reactions of \(\mathrm{Cl}_{2}\) gas with cold-dilute and hot-concentrated \(\mathrm{NaOH}\) in water give sodium salts of two (different) oxoacids of chlorine, \(\boldsymbol{P}\) and \(\boldsymbol{Q}\), respectively. The \(\mathrm{Cl}_{2}\) gas reacts with \(\mathrm{SO}_{2}\) gas, in presence of charcoal, to give a product \(\boldsymbol{R}\). \(\boldsymbol{R}\) reacts with white phosphorus to give a compound \(\boldsymbol{S}\). On hydrolysis, \(\boldsymbol{S}\) gives an oxoacid of phosphorus, \(\boldsymbol{T}\).
Question:
\(P\) and \(Q\), respectively, are the sodium salts of

The spin only magnetic moment value (in Bohr magneton units) of \(\mathrm{Cr}(\mathrm{CO})_6\) is

The tangent to the curve \(y=e^x\) drawn at the point \(\left(c, e^c\right)\) intersects the line joining the points \(\left(c-1, e^{c-1}\right)\) and \(\left(c+1, e^{c+1}\right)\)

Total number of cis $\mathrm{N}-\mathrm{M}\mathrm{n}-\mathrm{C}\mathrm{l}$ bond angles (that is, $\mathrm{M}\mathrm{n}-\mathrm{N}$ and $\mathrm{M}\mathrm{n}-\mathrm{C}\mathrm{l}$ bonds in cis positions) present in a molecule of cis- $\left[\mathrm{M}\mathrm{n}{\left(\mathrm{e}\mathrm{n}\right)}_2{\mathrm{C}\mathrm{l}}_2\right]$ complex is _________ $\left(en={\mathrm{N}\mathrm{H}}_2{\mathrm{C}\mathrm{H}}_2{\mathrm{C}\mathrm{H}}_2{\mathrm{N}\mathrm{H}}_2\right)$