To determine the half-life of a radioactive element, a student plots a graph of \(\ln \left|\frac{d N(t)}{d t}\right|\) versust. Here \(\frac{d N(t)}{d t}\) is the rate of radioactive decay at time \(t\). If the number of radioactive nuclei of this element decreases by a factor of \(p\) after \(4.16 \mathrm{yr}\), the value of \(p\) is

Two spheres $P$ and $Q$ of equal radii have densities ${\rho }_1$ and ${\rho }_2$ , respectively. The spheres are connected by a massless string and placed in liquids $L_1$ and $L_2$ of densities ${\sigma }_1$ and ${\mathrm{\sigma }}_2$ and viscosities ${\eta }_1$ and ${\eta }_2$ , respectively. They float in equilibrium with the sphere $P$ in $L_1$ and sphere $Q$ in $L_2$ and the string being taut (see figure). If sphere $P$ alone in $L_2$ has terminal velocity ${\overrightarrow{V}}_P$ and $Q$ alone in $L_1$ has terminal velocity ${\overrightarrow{V}}_Q$ , then

A length - scale $\left(\mathcal{l}\right)$ depends on the permittivity $\left(\mathrm{\epsilon }\right)$ of a dielectric material, Boltzmann constant $k_B$ , the absolute temperature T, the number per unit volume (n) of certain charged particles, and the charge (q) carried by each of the particles, Which of the following expressions (s) for $\mathcal{l}$ is (are) dimensionally correct?

A nuclear power plant supplying electrical power to a village uses a radioactive material of half life $\mathrm{T}$ years as the fuel. The amount of fuel at the beginning is such that the total power requirement of the village is 12.5% of the electrical power available from the plant at that time. If the plant is able to meet the total power needs of the village for a maximum period of $\mathrm{n}\mathrm{T}$ years, then the value of $\mathrm{n}$ is

A student performed the experiment to measure the speed of sound in air using resonance air-column method. Two resonances in the air-column were obtained by lowering the water level. The resonance with the shorter air-column is the first resonance and that with the longer air column is the second resonance. Then,

Two non-reactive monoatomic ideal gases have their atomic masses in the ratio 2 : 3. The ratio of their partial pressures, when enclosed in a vessel kept at a constant temperature, is 4 : 3. The ratio of their densities is 

In terms of potential difference $V$ , electic current $I$ , permittivity ${\epsilon }_0$ , permeability ${\mu }_0$ and speed of light $c$ , the dimensionally correct equation(s) is (are)

A thin and uniform rod of mass $\mathrm{M}$ and length $\mathrm{L}$ is held vertical on a floor with large friction. The rod is released from rest so that it falls by rotating about its contact-point with the floor without slipping. Which of the following statement(s) is/are correct, when the rod makes an angle ${60}^{\mathrm{o}}$ with vertical?
[ $\mathrm{g}$ is the acceleration due to gravity]

In thermodynamics, the $\mathrm{P}-\mathrm{V}$ work done is given by $\mathrm{w}=-\int {\mathrm{dVP}}_{\mathrm{ext}}$. For a system undergoing a particular process, the work done is,
$\mathrm{w}=-\int \mathrm{dV}\left(\frac{\mathrm{RT}}{\mathrm{V}-\mathrm{b}}-\frac{\mathrm{a}}{{\mathrm{V}}^2}\right)$. This equation is applicable to a

A conducting square loop of side \(L\), mass \(M\) and resistance \(R\) is moving in the \(X Y\) plane with its edges parallel to the \(X\) and \(Y\) axes. The region \(y \geq 0\) has a uniform magnetic field, \(\vec{B}=B_0 k\). The magnetic field is zero everywhere else. At time \(t=0\), the loop starts to enter the magnetic field with an initial velocity \(v_0 \hat{\jmath} \mathrm{~m} / \mathrm{s}\), as shown in the figure. Considering the quantity \(K=\frac{B_0^2 L^2}{R M}\) in appropriate units, ignoring self-inductance of the loop and gravity, which of the following statements is/are correct:

Let P be a matrix of order $3\times 3$ such that all the entries in P are from the set $\left\{-1, \mathrm{0,} 1\right\}$ . Then, the maximum possible value of the determinant of P is _______.

A small block is connected to one end of a massless spring of un-stretched length \(4.9 \mathrm{~m}\). The other end of the spring (see the figure) is fixed. The system lies on a horizontal frictionless surface. The block is stretched by \(0.2 \mathrm{~m}\) and released from rest at \(t=0\). It then executes simple harmonic motion with angular frequency \(\omega=\pi / 3 \mathrm{rad} / \mathrm{s}\). Simultaneously at \(t=0\), a small pebble is projected with speed \(v\) form point \(P\) at an angle of \(45^{\circ}\) as shown in the figure. Point \(P\) is at a horizontal distance of \(10 \mathrm{~m}\) from \(O\). If the pebble hits the block at \(t=1 \mathrm{~s}\), the value of \(v\) is (take \(g\) \(=10 \mathrm{~m} / \mathrm{s}^{2}\) )

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A spray gun is shown in the figure where a piston pushes air out of a nozzle. A thin tube of uniform cross section is connected to the nozzle. The other end of the tube is in a small liquid container. As the piston pushes air through the nozzle, the liquid from the container rises into the nozzle and is sprayed out. For the spray gun shown, the radii of the piston and the nozzle are \(20 \mathrm{~mm}\) and \(1 \mathrm{~mm}\), respectively. The upper end of the container is open to the atmosphere.

Question :
If the piston is pushed at a speed of \(5 \mathrm{~mms}^{-1}\), the air comes out of the nozzle with a speed of