Statement I If the accelerating potential in an X-ray tube is increased, the wavelengths of the characteristic \(X\)-rays do not change.
Statement II When an electron beam strikes the target in an X-ray tube, part of the kinetic energy is converted into X-ray energy.

A bullet is fired vertically upwards with velocity $\mathrm{\upsilon }$ from the surface of a spherical planet. When it reaches its maximum height, its acceleration due to the planet's gravity is $\frac{1}{4^{\mathrm{t}\mathrm{h}}}$ of its value at the surface of the planet. If the escape velocity from the planet is ${\mathrm{\upsilon }}_{\mathrm{e}\mathrm{s}\mathrm{c}}=\mathrm{\upsilon }\sqrt{\mathrm{N}}$ , then the value of N is (ignore energy loss due to atmosphere)

A spherical soap bubble inside an air chamber at pressure \(P_0=10^5 \mathrm{~Pa}\) has a certain radius so that the excess pressure inside the bubble is \(\Delta P=144 \mathrm{~Pa}\). Now, the chamber pressure is reduced to \(8 P_0 / 27\) so that the bubble radius and its excess pressure change. In this process, all the temperatures remain unchanged. Assume air to be an ideal gas and the excess pressure \(\Delta P\) in both the cases to be much smaller than the chamber pressure. The new excess pressure \(\Delta P\) in \(\mathrm{Pa}\) is ________ .

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A point charge \(Q\) is moving in a circular orbit of radius \(R\) in the \(x-y\) plane with an angular velocity \(\omega .\) This can be considered as equivalent to a loop carrying a steady current \(\frac{Q \omega}{2 \pi}\). A uniform magnetic field along the positive \(z\)-axis is now switched on, which increases at a constant rate from 0 to \(B\) in one second. Assume that the radius of the orbit remains constant. The application of the magnetic field induces an emf in the orbit. The induced emf is defined as the work done by an induced electric field in moving a unit positive charge around a closed loop. It is known that, for an orbiting charge, the magnetic dipole moment is proportional to the angular momentum with a proportionality constant \(\gamma\).

Question:

The change in the magnetic dipole moment associated with the orbit, at the end of the time interval of the magnetic field change, is

A container of fixed volume has a mixture of one mole of hydrogen and one mole of helium in equilibrium at temperature $\mathrm{T}$ . Assuming the gases are ideal, the correct statement(s) is (are)

The largest wavelength in the ultraviolet region of the hydrogen spectrum is \(122 \mathrm{~nm}\). The smallest wavelength in the infrared region of the hydrogen spectrum (to the nearest integer) is

Let us consider a system of units in which mass and angular momentum are dimensionless. If length has dimension of $\mathrm{L},$ which of the following statement (s) is/are correct?

Let $f :\left[0, 4\pi \right]\to \left[0, \pi \right]$ be defined by $f\left(x\right)={\cos }^{-1}\left(\cos x\right).$ The number of points $x \in \left[0, 4\pi \right]$ satisfying the equation $f\left(x\right)=\frac{10-x}{10}$ is_________

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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

Consider the following cell reaction,
\(2 \mathrm{Fe}(\mathrm{s})+\mathrm{O}_2(g)+4 \mathrm{H}^{+}(a q) \longrightarrow \)
\( 2 \mathrm{Fe}^{2+}(a q)+2 \mathrm{H}_2 \mathrm{O}(l), E^{\circ}=1.67 \mathrm{~V} \)
\( \text {At } {\left[\mathrm{Fe}^{2+}\right]=10^{-3} \mathrm{M}, \mathrm{P}\left(\mathrm{O}_2\right)=0.1 \mathrm{~atm}}\)
and \(\mathrm{pH}=3\), the cell potential at \(25^{\circ} \mathrm{C}\) is

Column I shows four situations of standard Young's double slit arrangement with the screen placed far away from the slits \(S_1\) and \(S_2\). In each of these cases \(S_1 P_0=S_2 P_0\), \(S_1 P_1-S_2 P_1=\frac{\lambda}{4}\) and \(S_1 P_2-S_2 P_2=\frac{\lambda}{3}\), where \(\lambda\) is the wavelength of the light used. In the cases \(\mathrm{B}, \mathrm{C}\) and \(\mathrm{D}\), a transparent sheet of refractive index \(\mu\) and thickness \(t\) is pasted on slit \(S_2\). The thickness of the sheets are different in different cases. The phase difference between the light waves reaching a point \(P\) on the screen from the two slits is denoted by \(\delta(P)\) and the intensity by \(I(P)\). Match each situation given in Column-I with the statement(s) in Column-II valid for that situation.

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 _____.

A linear octasaccharide (molar mass \(=1024 \mathrm{~g} \mathrm{~mol}^{-1}\) ) on complete hydrolysis produces three monosaccharides: ribose, 2-deoxyribose and glucose. The amount of 2-deoxyribose formed is \(58.26 \%(\mathrm{w} / \mathrm{w})\) of the total amount of the monosaccharides produced in the hydrolyzed products. The number of ribose unit(s) present in one molecule of octasaccharide is \(\qquad\)
Use : Molar mass \(\left(\right.\) in \(\left.\mathrm{g} \mathrm{mol}^{-1}\right)\) : ribose \(=150,2\)-deoxyribose \(=134\), glucose \(=180\); Atomic mass (in amu): \(\mathrm{H}=1, \mathrm{O}=16\)