A thermodynamic system is taken from an original state \(D\) to an intermediate state \(E\) by the linear process shown in the figure. Its volume is then reduced to the original volume from \(E\) to \(F\) by an isobaric process. The total work done by the gas from \(D\) to \(E\) to \(F\) will be \(......J\)

If velocity \([V],\) time \([T]\) and force \([F]\) are chosen as the base quantities, the dimensions of the mass will be

Given below are two statements: Statement \(I:\) For a planet, if the ratio of mass of the planet to its radius increases, the escape velocity from the planet also increases. Statement \(II:\) Escape velocity is independent of the radius of the planet. In the light of above statements, choose the most appropriate answer from the options given below on :

For a given series \(LCR\) circuit it is found that maximum current is drawn when value of variable capacitance is \(2.5 \mathrm{nF}\). If resistance of \(200 \Omega\) and \(100 \mathrm{mH}\) inductor is being used in the given circuit. The frequency of ac source is _______ \(\times 10^3 \mathrm{~Hz}\). (given \(\pi^2=10\) )

A flask contains hydrogen and oxygen in the ratio of \(2: 1\) by mass at temperature \(27^{\circ} C\). The ratio of average kinetic energy per molecule of hydrogen and oxygen respectively is :

Let the moment of inertia of a hollow cylinder of length \(30\, cm\) (inner radius \(10\, cm\) and outer radius \(20\, cm\)), about its axis be \(I\). The radius of a thin cylinder of the same mass such that its moment of inertia about its axis is also \(I\), is ......... \(cm\).

A wire of resistance R is bent into an equilateral triangle and an identical wire is bent into \(a\) square. The ratio of resistance between the two end points of an edge of the triangle to that of the square is:

The total number of \(4-\)digit numbers whose greatest common divisor with \(54\) is \(2\) , is \(..........\).

A test consists of \(6\) multiple choice questions, each having \(4\) alternative ans wers of which only one is correct. The number of ways, in which a candidate answers all six questions such that exactly four of the answers are correct, is

Muon \((\mu ^{-1})\) is negatively charged \(\left( {\left| q \right| = \left| e \right|} \right)\) with a mass \(m_{\mu}\, = 200\,m_e\), where \(m_e\) is the mass of the electron and \(e\) is the electronic charge. If \(\mu ^{-1}\) is bound to a proton to form a hydrogen like atom, identify the correct statements \((A)\) Radius of the muonic orbit is \(200\) times smaller than that of the electron \((B)\) the speed of the \(\mu ^{-1}\) in the \(n^{th}\) orbit is \(\frac{1}{{200}}\) times that of the election in the \(n^{th}\) orbit \((C)\) The Ionization energy of muonic atom is \(200\) times more than that of an hydrogen atom \((D)\) The momentum of the muon in the \(n^{th}\) orbit is \(200\) times more than that of the electron

A plano convex lens of refractive index \({\mu _1}\) and focal length \(f_1\) is kept in contact with another plano concave lens of refractive index \({\mu _2}\) and focal length \(f_2\). If the radius of curvature of their spherical faces is \(R\) each and \(f_1 = 2f_2\), then \({\mu _1}\) and \({\mu _2}\) are related as

\(\mathrm{N}\) moles of a polyatomic gas \((f=6)\) must be mixed with two moles of a monoatomic gas so that the mixture behaves as a diatomic gas. The value of \(\mathrm{N}\) is _______.

A plane electromagnetic wave propagating along y-direction can have the following pair of electric field \((\vec{E} )\) and magnetic field \((\overrightarrow{ B })\) components.