Two large circular discs separated by a distance of $0.01\mathrm{m}$ are connected to a battery via a switch as shown in the figure. Charged oil drops of density $900\mathrm{kg}{\mathrm{m}}^{-3}$ are released through a tiny hole at the centre of the top disc. Once some oil drops achieve terminal velocity, the switch is closed to apply a voltage of $200\mathrm{V}$ across the discs. As a result, an oil drop of radius $8\times {10}^{-7}\mathrm{m}$ stops moving vertically and floats between the discs. The number of electrons present in this oil drop is __________. (neglect the buoyancy force, take acceleration due to gravity $=10\mathrm{m}{\mathrm{s}}^{-2}$ and charge on an electron (e) $=1.6\times {10}^{-19}\mathrm{C}$)

Assume that the nuclear binding energy per nucleon \((B / A)\) versus mass number \((A)\) is as shown in the figure. Use this plot to choose the correct choice (s) given below.

Two non-conducting spheres of radii $R_1$ and $R_2$ and carrying uniform volume charge densities $+\rho$ and $-\rho$, respectively, are placed such that they partially overlap, as shown in the figure. At all points in the overlapping region,

A ball is thrown from the location \(\left(x_0, y_0\right)=(0,0)\) of a horizontal playground with an initial speed \(v_0\) at an angle \(\theta_0\) from the \(+x\)-direction. The ball is to be hit by a stone, which is thrown at the same time from the location \(\left(x_1, y_1\right)=(L, 0)\). The stone is thrown at an angle \(\left(180-\theta_1\right)\) from the \(+x\)-direction with a suitable initial speed. For a fixed \(v_0\), when \(\left(\theta_0, \theta_1\right)=\left(45^{\circ}, 45^{\circ}\right)\), the stone hits the ball after time \(T_1\), and when \(\left(\theta_0, \theta_1\right)=\left(60^{\circ}, 30^{\circ}\right)\), it hits the ball after time \(T_2\). In such a case, \(\left(T_1 / T_2\right)^2\) is _______ .

A planet of radius $\mathrm{R}=\frac{1}{10}\times$ (radius of Earth) has the same mass density as Earth. Scientists dig a well of depth $\frac{\mathrm{R}}{5}$ on it and lower a wire of the same length and of linear mass density ${10}^{-3}\mathrm{ }\mathrm{k}\mathrm{g}{\mathrm{m}}^{-1}$ into it. If the wire is not touching anywhere, the force applied at the top of the wire by a person holding it in place is (take the radius of Earth $=6\times {10}^6\mathrm{ }\mathrm{m}$ and the acceleration due to gravity of Earth is $10\mathrm{ }\mathrm{m}{\mathrm{s}}^{-2}$ )

In a radioactive decay chain, ${\mathrm{ }}_{90}^{232}\mathrm{Th}$ nucleus decays to ${\mathrm{ }}_{82}^{212}\mathrm{Pb}$ nucleus. Let $N_{\alpha } \text{and} N_{\beta }$ be the number of $\alpha \text{and} {\beta }^{-}$ particles, respectively, emitted in this decay process. Which of the following statements is (are) true?

A source of constant voltage $\mathrm{V}$ is connected to a resistance$\mathrm{R}$ and two ideal inductors ${\mathrm{L}}_1$ and ${\mathrm{L}}_2$ through a switch $\mathrm{S}$ as shown. There is no mutual inductance between the two inductors. The switch $\mathrm{S}$ is initially open. At $\mathrm{t}=0$ , the switch is closed and current begins to flow. Which of the following options is/are correct?

The value of \(\frac{(5050) \int_0^1\left(1-x^{50}\right)^{100} d x}{\int_0^1\left(1-x^{50}\right)^{101} d x}\)

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\(\mathbf{P}_{17 \text { - } 19}\) : Paragraph for Questions Nos. 17 to 19 Two discs \(A\) and \(B\) are mounted coaxially on a vertical axle. The discs have moments of inertia I and 2I, respectively about the common axis. Disc \(A\) is imparted an initial angular velocity \(2 \omega\) using the entire potential energy of a spring compressed by a distance \(x_1\). Disc \(B\) is imparted an angular velocity \(\omega\) by a spring having the same spring constant and compressed by a distance \(x_2\). Both the discs rotate in the clockwise direction.
Question :
When disc \(B\) is brought in contact with disc \(A\), they acquire a common angular velocity in time \(t\). The average frictional torque on one disc by the other during this period is

Let \(f(\theta)=\sin \left[\tan ^{-1}\left(\frac{\sin \theta}{\sqrt{\cos 2 \theta}}\right)\right]\), where \(-\frac{\pi}{4} < \theta < \frac{\pi}{4}\). Then, the value of \(\frac{d}{d(\tan \theta)}(f(\theta))\) is

Consider the region \(R=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x \geq 0\right.\) and \(\left.y^{2} \leq 4-x\right\}\). Let \(\mathcal{F}\) be the family of all circles that are contained in \(R\) and have centers on the \(x\)-axis. Let \(C\) be the circle that has largest radius among the circles in \(\mathcal{F}\). Let \((\alpha, \beta)\) be a point where the circle \(C\) meets the curve \(y^{2}=4-x\).
The radius of the circles $C$ is____.

Let \(f: I R \rightarrow I R\) be defined as \(f(x)=|x|+\left|x^{2}-1\right|\). The total number of points at which \(f\) attains either a local maximum or a local minimum is

Among the following, the number of reaction(s) that produce(s) benzaldehyde is