An open-ended U-tube of uniform cross-sectional area contains water (density ${10}^3\mathrm{kg}{\mathrm{m}}^{-3}$ ). Initially the water level stands at $0.29\mathrm{m}$ from the bottom in each arm. Kerosene oil (a water-immiscible liquid) of density $800\mathrm{kg}{\mathrm{m}}^{-3}$ is added to the left arm until its length is $0.1\mathrm{m}$, as shown in the schematic figure below. The ratio $\left(\frac{h_1}{h_2}\right)$ of the heights of the liquid in the two arms is

A person sitting inside an elevator performs a weighing experiment with an object of mass 50 kg . Suppose that the variation of the height \(y\) (in m ) of the elevator, from the ground, with time \(t\) (in s ) is given by \(y=8\left[1+\sin \left(\frac{2 \pi t}{T}\right)\right]\), where \(T=40 \pi \mathrm{~s}\). Taking acceleration due to gravity, \(g=10 \mathrm{~m} / \mathrm{s}^2\), the maximum variation of the object's weight (in N ) as observed in the experiment is _______ .

In the determination of Young's modulus \(\left(Y=\frac{4 M L g}{\pi l d^{2}}\right)\) by using Searle's method, a wire of length \(L=2 \mathrm{~m}\) and diameter \(d=0.5 \mathrm{~mm}\) is used. For a load \(M=2.5 \mathrm{~kg}\), an extension \(l=0.25 \mathrm{~mm}\) in the length of the wire is observed. Quantities \(d\) and \(l\) are measured using a screw gauge and a micrometer, respectively. They have the same pitch of \(0.5 \mathrm{~mm}\). The number of divisions on their circular scale is 100 . The contributions to the maximum probable error of the \(Y\) measurement

A cylindrical furnace has height $\left(H\right)$ and diameter $\left(D\right)$ both $1 \mathrm{m}$. It is maintained at temperature $360 \mathrm{K}$. The air gets heated inside the furnace at constant pressure $P_a$ and its temperature becomes $T=360 \mathrm{K}$. The hot air with density $\rho$ rises up a vertical chimney of diameter $d=0.1 \mathrm{m}$ and height $h=9 \mathrm{m}$ above the furnace and exits the chimney (see the figure). As a result, atmospheric air of density ${\rho }_a=1.2 \mathrm{kg} {\mathrm{m}}^{-3}$, pressure $P_a$ and temperature $T_a=300 \mathrm{K}$ enters the furnace. Assume air as an ideal gas, neglect the variations in $\rho$ and $T$ inside the chimney and the furnace. Also ignore the viscous effects.
[Given: The acceleration due to gravity $g=10 \mathrm{m} {\mathrm{s}}^{-2}$ and $\pi =3.14$]

Considering the air flow to be streamline, the steady mass flow rate of air exiting the chimney is _____ $\mathrm{gm} {\mathrm{s}}^{-1}$.

A metal surface is illuminated by light of two different wavelengths 248 nm and 310 nm. The maximum speeds of the photoelectrons corresponding to these wavelengths are u₁ and u₂, respectively. If the ratio $u_1:u_2=2:1$ , the work function of the metal is nearly

A ray of light, travelling in the direction $\frac{1}{2}\left(\hat{\mathrm{i}}+\sqrt{3} \hat{\mathrm{j}}\right)$, is incident on a plane mirror. After reflection, it travels along the direction $\frac{1}{2}\left(\hat{\mathrm{i}}-\sqrt{3}\widehat{ \mathrm{j}}\right)$. The angle of incidence is

A disk of radius \(\frac{a}{4}\) having a uniformly distributed charge \(6 \mathrm{C}\) is placed in the \(x-y\) plane with its centre at \(\left(\frac{-a}{2}, 0,0\right)\). A rod of length a carrying a uniformly distributed charge \(8 \mathrm{C}\) is placed on the \(x\)-axis from \(x=\frac{a}{4}\) to \(x=\frac{5 a}{4}\). Two point charges \(-7 \mathrm{C}\) and \(3 \mathrm{C}\) are placed at \(\left(\frac{a}{4}, \frac{-a}{4}, 0\right)\) and \(\left(\frac{-3 a}{4}, \frac{3 a}{4}, 0\right)\), respectively. Consider a cubical surface formed by six surfaces \(x=\pm \frac{a}{2}, y=\pm \frac{a}{2}, z=\pm \frac{a}{2}\).
The electric flux through this cubical surface is

Consider the following statements related to colloids.
(I) Lyophobic colloids are not formed by simple mixing of dispersed phase and dispersion medium.
(II) For emulsions, both the dispersed phase and the dispersion medium are liquid.
(III) Micelles are produced by dissolving a surfactant in any solvent at any temperature.
(IV) Tyndall effect can be observed from a colloidal solution with dispersed phase having the same refractive index as that of the dispersion medium.
The option with the correct set of statements is

Match each of the compounds is Column I with its characteristic reaction(s) is Column II.

If the normal of the parabola ${\mathrm{y}}^2=4\mathrm{x}$ drawn at the end points of its latus rectum are tangents to the circle ${\left(\mathrm{x}-3\right)}^2+{\left(\mathrm{y}+2\right)}^2={\mathrm{r}}^2,$ then the value of ${\mathrm{r}}^2$ is

Let $f: \left(0, \infty \right)\to R$ be a differentiable function such that $f^{\text{'}}\left(x\right)=2-\frac{f\left(x\right)}{x}\forall x\in (0, \infty )$ and $f\left(1\right)\ne 1$. Then,

A beaker of radius $r$ is filled with water$\left(\text{refractive index}\frac{4}{3}\right)$ up to a height $H$ as shown in the figure on the left. The beaker is kept on a horizontal table rotating with angular speed $\omega$. This makes the water surface curved so that the difference in the height of water level at the centre and at the circumference of the beaker is $h\left(h\ll H\text{,}h\ll r\right)$, as shown in the figure on the right. Take this surface to be approximately spherical with a radius of curvature $R.$ Which of the following is/are correct? ($g$ is the acceleration due to gravity)

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 value of $\alpha$ is _____.