Three vectors $\overrightarrow{P},\overrightarrow{Q}$ and $\overrightarrow{R}$ are shown in the figure. Let, $S$ be any point on the vector $\overrightarrow{R}$ . The distance between the points $P$ and $S$ is $b\left|\overrightarrow{R}\right|$. The general relation among vectors $\overrightarrow{P},\overrightarrow{Q}$ and $\overrightarrow{S}$ are

Two capacitors with capacitance values $C_1=\left(2000\pm 10\right) \mathrm{pF}$  and $C_2=\left(3000\pm 15\right) \mathrm{pF}$ are connected in series. The voltage applied across this combination is $V=\left(5.00\pm 0.02\right) \mathrm{V}$. The percentage error in the calculation of the energy stored in this combination of capacitors is __________.

$S_1$ and $S_2$ are two identical sound sources of frequency $656 \mathrm{Hz}$. The source $S_1$ is located at $O$ and $S_2$ moves anti-clockwise with a uniform speed $4\sqrt{2} \mathrm{m} {\mathrm{s}}^{-1}$ on a circular path around $O$, as shown in the figure. There are three points $P, Q$ and $R$ on this path such that $P$ and $R$ are diametrically opposite while $Q$ is equidistant from them. A sound detector is placed at point $P$. The source $S_1$ can move along direction $OP$.
[Given: The speed of sound in air is $324 \mathrm{m} {\mathrm{s}}^{-1}$]

When only $S_2$ is emitting sound and it is at $Q$, the frequency of sound measured by the detector in $\mathrm{Hz}$ is _____.

System shown in figure is in equilibrium and at rest. The spring and string are massless, now the string is cut. The acceleration of mass \(2 m\) and \(m\) just after the string is cut will be

An ideal gas is in thermodynamic equilibrium. The number of degrees of freedom of a molecule of the gas is $n$. The internal energy of one mole of the gas is $U_n$ and the speed of sound in the gas is $v_n$. At a fixed temperature and pressure, which of the following is the correct option?

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Modern trains are based on Maglev technology in which trains are magnetically leviated, which runs its EDS Maglev system.
There are coils on both sides of wheels. Due to motion of train, current induces in the coil of track which levitate it. This is in accordance with Lenz's law. If trains lower down then due to Lenz's law a repulsive force increases due to which train gets uplifted and if it goes much high then there is a net downward force disc to gravity. The advantage of Maglev train is that there is no friction between the train and the track, thereby reducing power consumption and enabling the train to attain very high speeds.
Disadvantage of Maglev train is that as it slows down the electromagnetic forces decreases and it becomes difficult to keep it leviated and as it moves forward according to Lenz law, there is an electromagnetic drag force.
Question:
What is the disadvantage of this system?

Consider an electron in the \(n=3\) orbit of a hydrogen-like atom with atomic number \(Z\). At absolute temperature \(T\), a neutron having thermal energy \(k_{\mathrm{B}} T\) has the same de Broglie wavelength as that of this electron. If this temperature is given by \(T=\frac{Z^2 h^2}{\alpha \pi^2 a_0^2 m_N k_B}\), (where \(h\) is the Planck's constant, \(k_B\) is the Boltzmann constant, \(m_{\mathrm{N}}\) is the mass of the neutron and \(a_0\) is the first Bohr radius of hydrogen atom) then the value of \(\alpha\) is ________

A point source \(\mathrm{S}\) emits unpolarized light uniformly in all directions. At two points \(\mathrm{A}\) and \(\mathrm{B}\), the ratio \(r=I_A / I_B\) of the intensities of light is 2. If a set of two polaroids having \(45^{\circ}\) angle between their pass-axes is placed just before point \(\mathrm{B}\), then the new value of \(r\) will be ______

Paragraph : In electromagnetic theory, the electric and magnetic phenomena are related to each other. Therefore, the dimensions of electric and magnetic quantities must also be related to each other. In the questions below, $\left[E\right] \mathrm{a}\mathrm{n}\mathrm{d} \left[B\right]$ stand for dimensions of electric and magnetic fields respectively, while $\left[{\epsilon }_0\right]$ and $\left[{\mu }_0\right]$ stand for dimensions of the permittivity and permeability of free space respectively. $\left[L\right]\mathrm{a}\mathrm{n}\mathrm{d} \left[T\right]$ are dimensions of length and time respectively. All the quantities are given in $SI$ units

Question :The relation between $\left[E\right]$ and $\left[B\right]$ is

A uniform circular disc of mass 1.5 kg and radius 0.5 m is initially at rest on a horizontal frictionless surface. Three forces of equal magnitude $\mathrm{F}\mathrm{}=\mathrm{}0.5\mathrm{}\mathrm{N}$ are applied simultaneously along the three sides of an equilateral triangle XYZ with its vertices on the perimeter of the disc (see figure). One second after applying the forces, the angular speed of the disc in rad ${\mathrm{s}}^{-1}$ is

In a photoelectric experiment, a parallel beam of monochromatic light with the power of $200\text{W}$ is incident on a perfectly absorbing cathode of work function $6.25 \mathrm{eV}$. The frequency of light is just above the threshold frequency so that the photoelectrons are emitted with negligible kinetic energy. Assume that the photoelectron emission efficiency is 100%. A potential difference of $500\text{V}$ is applied between the cathode and the anode. All the emitted electrons are incident normally on the anode and are absorbed. The anode experiences a force $F=n\times {10}^{-4}\mathrm{ N}$ due to the impact of the electrons. The value of $n$ is __________. Mass of the electron \(\text{m}_\text e =9 \times 10^{-31} kg\) and \(1.0 eV =1.6 \times\) \(10^{-19}\text{ J}.\)

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\(P_{14-16}\) : Paragraph for Questions Nos. 14 to 16 A fixed thermally conducting cylinder has a radius \(R\) and height \(L_0\). The cylinder is open at its bottom and has a small hole at its top. A piston of mass \(M\) is held at a distance \(L\) from the top surface as shown in the figure. The atmospheric pressure is \(p_0\).

Question :
The piston is now pulled out slowly and held at a distance \(2 L\) from the top. The pressure in the cylinder between its top and the piston will then be

The figure below shows the variation of specific heat capacity ($C$) of a solid as a function of temperature ($T$). The temperature is increased continuously from $0$ to $500 \mathrm{K}$ at a constant rate. Ignoring any volume change, the following statement (s) is (are) correct to a reasonable approximation.