List of practice Questions

Consider that an electron is revolving in an excited state of Hydrogen atom with velocity \[ \sqrt{25.6}\times10^5 \ \text{ms}^{-1}. \] The radius of the orbit is \(x\times10^{-9}\) m. The value of \(x\) is : [Take mass of electron \(=9\times10^{-31}\) kg, charge of electron \(=-1.6\times10^{-19}\) C and \[ \frac{1}{4\pi\varepsilon_0}=9\times10^9 \ \text{Nm}^2\text{C}^{-2} \]

The mean free path of molecules in an ideal gas A is half that of another ideal gas B. The diameter of the spherical molecules of gas A is twice the diameter of the molecules of gas B. If number densities of the gases A and B are \(n_A\) and \(n_B\), respectively, then

A cylindrical cork of uniform density \(\rho_1\) floats in a liquid of density \(\rho_1\). If the cork is depressed slightly and released, it oscillates harmonically with time period \(T\). If the same cork floats in another liquid of density \(\rho_2\), then the similar oscillation has time period \(2T\). The value of \(\dfrac{\rho_2}{\rho_1}\) is:

For sound waves, if the number of nodes for the 5th harmonic of an open-ended pipe is \(n\) and that for the 9th harmonic of the same pipe with one of its ends closed is \(m\), the ratio \(n/m\) is:

Consider the nuclear reaction \[ ^{238}\mathrm{U} \rightarrow ^{234}\mathrm{Th} + ^{4}\mathrm{He} \] Take masses of \(^{238}\mathrm{U}\), \(^{234}\mathrm{Th}\), and \(^{4}\mathrm{He}\) as \[ 238.050\,u,\qquad 234.043\,u,\qquad 4.003\,u \] respectively. The \(Q\)-value for the reaction, in keV, is: \[ 1u = 931.5\ \mathrm{MeV}/c^2 \]

A car travels on a circular racetrack of radius \(50\,\text{m}\), which is banked at an angle \(\theta\). If the car travels at a speed \(10\,\text{m s}^{-1}\), then the wear and tear on its tyres is minimum. Taking \(g=10\,\text{m s}^{-2}\), the value of \(\theta\) is:

A frictionless circular wire of unit radius is fixed on a horizontal plane. Two point particles of unit mass start moving simultaneously from point \(A\) \((\theta=\pi/2)\) with identical uniform angular speeds in opposite directions and meet again at point \(B\). During this time, which graph correctly represents the magnitude of total linear momentum \(P\) of the system as a function of time?

Three identical p-n junction diodes \(D_1\), \(D_2\) and \(D_3\) are connected across a battery as shown in the figure. If the widths of the depletion regions of \(D_1\), \(D_2\) and \(D_3\) are \(W_1\), \(W_2\) and \(W_3\), respectively, then the correct option is:

The lens combination as shown consists of two thin lenses \(L_1\) and \(L_2\) of focal lengths \(+10\ \text{cm}\) and \(-10\ \text{cm}\), respectively. The object is placed \(30\ \text{cm}\) to the left of \(L_1\), and the distance between the two lenses is \(3\ \text{cm}\). The position of the image formed is:

A particle of mass \(M\) moves along the horizontal \(x\)-axis from \(x=0\) to \(x=L\). The coefficient of kinetic friction varies as \[ \mu_k(x)=\frac{\mu_0}{L}x \] where \(\mu_0\) and \(L\) are constants. If the total work done by friction during the motion is \[ -\frac{\mu_0 MgL}{n} \] where \(g\) is the acceleration due to gravity, find \(n\).

Consider three media \(P\), \(Q\) and \(R\) with refractive indices \[ n_P=1,\qquad n_Q=1.25,\qquad n_R=1.5 \] respectively. Medium \(Q\) has a thickness of \(5\,\text{cm}\) and is placed between media \(P\) and \(R\) as shown. An object \(O\) is placed at the centre of medium \(Q\). If viewed from medium \(P\) near the normal direction, the apparent depth of \(O\) is \(h_1\). For the same object viewed from medium \(R\), the apparent depth is \(h_2\). Find \[ |h_1-h_2|. \]

Consider a fixed uniformly charged insulating sphere with radius \(R\) and total charge \(+Q\). A point charge \(-q\) (\(q \ll Q\)) with mass \(m\) is released from rest at a distance of \(3R\) from the centre of the charged sphere. When the point charge reaches the surface of the sphere, its speed is:

The following table presents the part of the electromagnetic spectrum and their corresponding major applications. Match the following and choose the correct option. \[ \begin{array}{|c|c|c|c|} \hline \text{Part of Spectrum} & & \text{Applications} & \\ \hline P & \text{Microwave} & I & \text{For purifying water} \\ \hline Q & \text{UV rays} & II & \text{For warming food} \\ \hline R & \text{Gamma rays} & III & \text{For AM and FM communication systems} \\ \hline S & \text{Radio waves} & IV & \text{Cancer cells treatment} \\ \hline \end{array} \]

An ideal gas is made of polyatomic molecules. Each molecule has three translational, three rotational and \(f\) number of vibrational modes. If the ratio of heat capacities \[ \frac{C_P}{C_V}=\frac{8}{7} \] then the value of \(f\) is:

A unit positive point charge is slowly moved through an infinitely thin tube inside a uniformly charged dielectric sphere of radius \(R\) and volume charge density \(\rho\). The initial and final positions of the charge are \(B\) and \(A\), located at distances \(3R\) and \(2R\) respectively from the centre. If the magnitude of work done on the charge is \[ \frac{\rho R^2}{n\varepsilon_0} \] then find \(n\).

A current \(I_0\) flows through a metallic circular loop of radius \(r\) as shown. The resistance of arc \(ABC\) is half that of arc \(ADC\). Find the magnetic field at the centre \(O\).

Bob \(B\) of mass \(m\) at rest is hanging vertically from the ceiling by a massless string of length \(10\,\text{m}\), as shown in the figure. Point mass \(A\) of mass \(m\) travelling horizontally with speed \(10\,\text{m s}^{-1}\) collides with the bob \(B\) elastically. The bob \(B\) rises to a height \(h\) after the collision. Taking acceleration due to gravity \(g=10\,\text{m s}^{-2}\) and neglecting the size of the bob, the value of \(h\) is:

An electromagnetic wave travelling in a lossless dielectric medium having a dielectric constant, \[ \varepsilon_r = 9, \] has the electric field \[ E_x=E_0\sin(kz-2\pi\times10^6 t)\ \text{V m}^{-1} \] where \(E_0\) is the amplitude and \(k\) is the wave vector. Among the following options, the incorrect choice is:

An AC voltage \[ V=220\sin(2\times10^{3}t)\ \text{Volt} \] is applied to a series LCR circuit. Then the current amplitude in the circuit is: Given: \[ L=10\,\text{mH},\quad C=25\,\mu\text{F},\quad R=100\,\Omega \]

Two identical inductors are connected in two different configurations \(P\) and \(Q\), where a time varying current \(I(t)\) is flowing, as shown in the figure. If the induced emf between points \(a\) and \(b\) for configuration \(P\) is \(E_P\) and that for configuration \(Q\) is \(E_Q\), then the ratio \[ \frac{E_P}{E_Q} \] is:

Three identical capacitors \(P\), \(Q\) and \(S\), each of capacitance \(C\), are connected to a battery of voltage \(V\), as shown in the figure. If the potential energy stored in the capacitor \(P\) and total energy stored in the system are \(U_P\) and \(U_T\), respectively, then the ratio \[ \frac{U_P}{U_T} \] is:

A conducting loop of finite resistance lies on the \(x-y\) plane. There is a constant magnetic field in the \(y\)-direction. The area of the loop varies with time \(t\) as \[ A=A_0(1+\sin t) \] The figure that correctly indicates the qualitative behaviour of the power dissipated in the loop as a function of time is:

In an adiabatic expansion, the temperature of one mole of an ideal monoatomic gas \((\gamma=\frac{5}{3})\) decreases from \(60\,\text{K}\) to \(50\,\text{K}\). The work done by the gas in the process is: (Take the universal gas constant as \(R=8.3\,\text{J mol}^{-1}\text{K}^{-1}\))

Consider a particle moving along a straight line, whose position as a function of time is given by \[ s(t)=\alpha t^2-\beta t+\gamma \] where \(\alpha=1\,\text{m s}^{-2}\), \(\beta=6\,\text{m s}^{-1}\) and \(\gamma=5\,\text{m}\). The average speed of the particle, in \(\text{m s}^{-1}\), from \(t=0\) to \(t=6\,\text{s}\) is:

Consider two circuits, (A) and (B), each having two resistors. One of them has a positive temperature coefficient of resistance, \(+\alpha\), while the other one has a negative temperature coefficient of resistance, \(-\alpha\), as shown in the figure. The current through these circuits are denoted by \(I_A\) and \(I_B\). At initial temperature, the resistance of the two resistors is \(R_0\). As the temperature is increased, the correct option that describes the variation of current in these circuits is: