A stone tied to \(180 cm\) long string at its end is making 28 revolutions in horizontal circle in every minute. The magnitude of acceleration of stone is \(\frac{1936}{ x }\,ms ^{-2}\). The value of \(x.........\left(\text { Take } \pi=\frac{22}{7}\right)\)

de-Broglie wavelength of an electron accelerated by a voltage of \(50\,\,V\) is close to ............. \(\mathop {\rm{A}}\limits^o \)  \((\left| e \right| = 1.6 \times {10^{ - 19}}\,\,C,\,\,{m_e} = 9.1 \times {10^{ - 31}}\,kg,\,\,h = 6.6 \times {10^{ - 34}}\,\,Js)\)

Heat energy of \(735\,J\) is given to a diatomic gas allowing the gas to expand at constant pressure. Each gas molecule rotates around an internal axis but do not oscillate. The increase in the internal energy of the gas will be \(..........\,J\)

A cord of negligible mass is wound around the rim of a wheel supported by spokes with negligible mass. The mass of wheel is 10 kg and radius is 10 cm and it can freely rotate without any friction. Initially the wheel is at rest. If a steady pull of 20 N is applied on the cord, the angular velocity of the wheel, after the cord is unwound by 1 m , would be :

An electron in the ground state of the hydrogen atom has the orbital radius of \(5.3 \times 10^{-11} \mathrm{~m}\) while that for the electron in third excited state is \(8.48 \times 10^{-10} \mathrm{~m}\). The ratio of the de Broglie wavelengths of electron in the excited state to that in the ground state is _______.

A rod of linear mass density ' \(\lambda\) ' and length ' \(L\) ' is bent to form a ring of radius 'R'. Moment of inertia of ring about any of its diameter is :

Heat energy of \(184\,kJ\) is given to ice of mass \(600\,g\) at \(-12^{\circ}\,C\), Specific heat of ice is \(2222.3\,J\,kg ^{-1^{\circ}}\,C ^{-1}\) and latent heat of ice in \(336\,kJ / kg ^{-1}\) \((A)\) Final temperature of system will be \(0^{\circ} C\). \((B)\) Final temperature of the system will be greater than \(0^{\circ} C\). \((C)\) The final system will have a mixture of ice and water in the ratio of \(5: 1\). \((D)\) The final system will have a mixture of ice and water in the ratio of \(1: 5\). \((E)\) The final system will have water only. Choose the correct answer from the options given below:

The equations of two waves are given by \(y_{1}=5 \sin 2 \pi(x-v t) \,c m\,\) \(y_{2}=3 \sin 2 \pi(x-v t+1.5) \,c m\) These waves are simultaneously passing through a string. The amplitude of the resulting wave is.........\(cm\)

A thin circular disc of mass \(M\) and radius \(R\) is rotating in a horizontal plane about an axis passing through its centre and perpendicular to its plane with angular velocity \(\omega\). If another disc of same dimensions but of mass \(\mathrm{M} / 2\) is placed gently on the first disc co-axially, then the new angular velocity of the system is _______.

\(\int \frac{\left(x^{2}+1\right) e^{x}}{(x+1)^{2}} d x=f(x) e^{x}+C\), Where \(C\) is a constant, then \(\frac{d^{3} f}{d x^{3}}\) at \(x =1\) is equal to

If non-zero real numbers \(b\) and \(c\) are such that \(min \,f\left( x \right) > \max \,g\left( x \right)\), where \(f\left( x \right) = {x^2} + 2bx + 2{c^2}\)  and \(g\left( x \right) = {-x^2} - 2cx + {b^2}\)\(\left( {x \in R} \right)\); then \(\left| {\frac{c}{b}} \right|\) lies in the interval

A particle is moving with constant speed in a circular path. When the particle turns by an angle \(90^{\circ}\), the ratio of instantaneous velocity to its average velocity is \(\pi: x \sqrt{2}\). The value of \(x\) will be \(.........\)

Let \(y(x)\) be the solution of the differential equation \(\left( {xlogx} \right)\frac{{dy}}{{dx}} + y = 2xlogx,\left( {x \ge 1} \right)\) . Then \(y(e) \) is equal to :      \([y(1)=0]\)