A soft plastic bottle, filled with water of density \(1 \mathrm{gm} / \mathrm{cc}\), carries an inverted glass test-tube with some air (ideal gas) trapped as shown in the figure. The test-tube has a mass of \(5 \mathrm{gm}\), and it is made of a thick glass of density \(2.5 \mathrm{gm} / \mathrm{cc}\). Initially the bottle is sealed at atmospheric pressure \(p_{0}=10^{5} \mathrm{~Pa}\) so that the volume of the trapped air is \(v_{0}=3.3 \mathrm{cc}\). When the bottle is squeezed from outside at constant temperature, the pressure inside rises and the volume of the trapped air reduces. It is found that the test tube begins to sink at pressure \(p_{0}+\Delta p\) without changing its orientation. At this pressure, the volume of the trapped air is \(v_{0}-\Delta v\).
Let \(\Delta v=X\) cc and \(\Delta p=Y \times 10^{3} \mathrm{~Pa}\).

The value of $Y$ is _____.

A cube of unit volume contains \(35 \times 10^7\) photons of frequency \(10^{15} \mathrm{~Hz}\). If the energy of all the photons is viewed as the average energy being contained in the electromagnetic waves within the same volume, then the amplitude of the magnetic field is \(\alpha \times 10^{-9} \mathrm{~T}\). Taking permeability of free space \(\mu_0=4 \pi \times 10^{-7} \mathrm{Tm} / \mathrm{A}\), Planck's constant \(h=6 \times 10^{-34} \mathrm{Js}\) and \(\pi=\frac{22}{7}\), the value of \(\alpha\) is_______

Paragraph:
When a particle is restricted to move along \(x\)-axis between \(x=0\) and \(x=a\), where \(a\) is of nanometer dimension, its energy can take only certain specific values. The allowed energies of the particle moving in such a restricted region, correspond to the formation of standing waves with nodes at its ends \(x=0\) and \(x=a\). The wavelength of this standing wave is related to the liner momentum \(p\) of the particle according to the de Broglie relation. The energy of the particle of mass \(m\) is related to its linear momentum as \(E=\frac{p^2}{2 m}\). Thus, the energy of the particle can be denoted by a quantum number \(n\) taking values \(1,2,3, \ldots(n=1\), called the ground state) corresponding to the number of loops in the standing wave.
Use the model described above to answer the following three questions for a particle moving in the line \(x=0\) to \(x=a\) [Take \(h=6.6 \times 10^{-34} \mathrm{Js}\) and \(e=1.6 \times 10^{-19} \mathrm{C}\) ]
Question:
If the mass of the particle is \(m=1.0 \times 10^{-30} \mathrm{~kg}\) and \(a=6.6 \mathrm{~nm}\), the energy of the particle in its ground state is closest to

In a circuit shown in the figure, the capacitor $C$ is initially uncharged and the key $K$ is open. In this condition, a current of $1 \mathrm{A}$ flows through the $1 \Omega$ resistor. The key is closed at time $t=t_0$. Which of the following statement(s) is(are) correct?
[Given : $e^{–1}=0.36$]

Two spherical stars $\mathrm{A}$ and $\mathrm{B}$ emit blackbody radiation. The radius of $\mathrm{A}$ is 400 times that of $\mathrm{B}$ and $\mathrm{A}$ emits ${10}^4$ times the power emitted from $\mathrm{B}$ . The ratio $\left(\frac{{\mathrm{\lambda }}_{\mathrm{A}}}{{\mathrm{\lambda }}_{\mathrm{B}}}\right)$ of their wavelengths ${\mathrm{\lambda }}_{\mathrm{A}}$ and ${\mathrm{\lambda }}_{\mathrm{B}}$ at which the peaks occur in their respective radiation curves is

A source, approaching with speed $u$ towards the open end of a stationary pipe of length $L$, is emitting a sound of frequency $f_s$. The farther end of the pipe is closed. The speed of sound in air is $v$ and $f_0$ is the fundamental frequency of the pipe. For which of the following combination(s) of $u$ and $f_s$, will the sound reaching the pipe lead to a resonance?

A circuit is connected as shown in the figure with the switch \(S\) open. When the switch is closed, the total amount of charge that flows from \(Y\) to \(X\) is

The density of a solid ball is to be determined in an experiment. The diameter of the ball is measured with a screw gauge, whose pitch is \(0.5 \mathrm{~mm}\) and there are 50 divisions on the circular scale. The reading on the main scale is \(2.5 \mathrm{~mm}\) and that on the circular scale is 20 divisions. If the measured mass of the ball has a relative error of \(2 \%\), the relative percentage error in the density is

A block of mass \(0.18 \mathrm{~kg}\) is attached to a spring of force constant \(2 \mathrm{~N} / \mathrm{m}\). The coefficient of friction between the block and the floor is 0.1. Initially, the block is at rest and the spring is unstretched. An impulse is given to the block as shown in the figure. The block slides a distance of \(0.06 \mathrm{~m}\) and comes to rest for the first time. The initial velocity of the block in \(\mathrm{m} / \mathrm{s}\) is \(v=\frac{N}{10}\). Then, \(N\) is

Bleaching powder and bleach solution are produced on a large scale and used in several household products. The effectiveness of bleach solution is often measured by iodometry.
Question :
Bleaching powder contains a salt of an oxoacid as one of its components. The anhydride of that oxoacid 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)

The image of an object, formed by a plano - convex lens at a distance of 8 m behind the lens, is real and is one-third the size of the object. The wavelength of light inside the lens is  $\frac{2}{3}$ times the wavelength in free space. The radius of the curved surface of the lens is

Let \(f, g\) and \(h\) be real-valued functions defined on the interval \([0,1]\) by \(f(x)=e^{x^2}+e^{-x^2}, \quad g(x)=x e^{x^2}+e^{-x^2}\) and \(h(x)=x^2 e^{x^2}+e^{-x^2}\). If \(a, b\) and \(c\) denote respectively, the absolute maximum of \(f, g\) and \(h\) on \([0,1]\), then