Density of water at \(4^{\circ} C\) and \(20^{\circ} C\) are \(1000 kg / m ^3\) and respectively. The increase in internal energy of 4 kg water when it is heated from \(4{ }^{\circ} C\) to \(20^{\circ} C\) is _________ J.
(Specific heat capacity of water \(=4.2 KJ / kg\). and 1 atmospheric pressure \(=10^5 Pa\) )

For a given gas at \(1\,atm\) pressure, \(rms\) speed of the molecules is \(200\,m/s\) at \(127\,^oC.\) At \(2\,atm\) pressure and at \(227\,^oC,\) the \(rms\) speed of the molecules will be

Given below are two statements: Statement-\(I\): Figure shows the variation of stopping potential with frequency \((v)\) for the two photosensitive materials \(M_1\) and \(M_2\). The slope gives value of \(\frac{h}{e}\), where \(h\) is Planck's constant, \(e\) is the charge of electron. Statement-\(II\): \(M_2\) will emit photoelectrons of greater kinetic energy for the incident radiation having same frequency. In the light of the above statements, choose the most appropriate answer from the options given below.

A body of mass \(2\,kg\) is initially at rest. It starts moving unidirectionally under the influence of a source of constant 10.Power P. Its displacement in \(4\,s\) is \(\frac{1}{3} \alpha^2 \sqrt{ P } m\). The value of \(\alpha\) will be \(.............\)

The parameter that remains the same for molecules of all gases at a given temperature is _______.

A particle starts from origin at \(t=0\) with a velocity \(5 \hat{i} \mathrm{~m} / \mathrm{s}\) and moves in \(x-y\) plane under action of a force which produces a constant acceleration of \((3 \hat{i}+2 \hat{j}) \mathrm{m} / \mathrm{s}^2\). If the \(x\)-coordinate of the particle at that instant is \(84 \mathrm{~m}\), then the speed of the particle at this time is \(\sqrt{\alpha} \mathrm{m} / \mathrm{s}\). The value of \(\alpha\) is___________.

A block of mass \(m\) is placed on a surface having vertical cross section given by \(y=x^2 / 4\). If coefficient of friction is \(0.5\) , the maximum height above the ground at which block can be placed without slipping is _______.

The electric potential as a function of \(x, y\) is given by \(V = 5(x^2 - y^2)\) V. The electric field at a point \((2, 3)\) m is __________ V/m.

Solid sphere \(A\) is rotating about an axis \(PQ\). If the radius of the sphere is \(5\,cm\) then its radius of gyration about \(P Q\) will be \(\sqrt{x} \;cm\). The value of \(x\) is \(................\).

Two coils \(P\) and \(Q\) are separated by some distance. When a current of \(3\, A\) flows through coil \(P\) a magnetic flux of \(10^{-3}\, Wb\) passes through \(Q\). No current is passed through \(Q\). When no current passes through \(P\) and a current of \(2\, A\) passes through \(Q\), the flux through \(P\) is

The ratio of de Broglie wavelength of a deutron with kinetic energy \(E\) to that of an alpha particle with kinetic energy \(2 E\), is \(n: 1\). The value of \(n\) is ___________.
(Assume mass of proton = mass of neutron) :

A particle moves in \(\mathrm{x}-\mathrm{y}\) plane under the influence of a force \(\vec{F}\) such that its linear momentum is \(\vec{P}(t)=\hat{i} \cos (k t)-\hat{j} \sin (k t)\). If \(k\) is constant, the angle between \(\overrightarrow{\mathrm{F}}\) and \(\overrightarrow{\mathrm{P}}\) will be _______.