A uniform cylinder of mass \(M\) and radius \(R\) is to be pulled over a step of height \(a (a < R\)) by applying a force \(F\) at its centre \('O'\) perpendicular to the plane through the axes of the cylinder on the edge of the step ( see figure). The minimum value of \(F\) required is 

Statement \(I:\) If three forces \(\vec{F}_{1}, \vec{F}_{2}\) and \(\vec{F}_{3}\) are represented by three sides of a triangle and \(\overrightarrow{{F}}_{1}+\overrightarrow{{F}}_{2}=-\overrightarrow{{F}}_{3}\), then these three forces are concurrent forces and satisfy the condition for equilibrium. Statement \(II:\) A triangle made up of three forces \(\overrightarrow{{F}}_{1}, \overrightarrow{{F}}_{2}\) and \(\overrightarrow{{F}}_{3}\) as its sides taken in the same order, satisfy the condition for translatory equilibrium. In the light of the above statements, choose the most appropriate answer from the options given below:

In a meter bridge experiment the balance point in obtained if the gaps are closed by \(2\,\Omega\) and \(3\,\Omega\). A shunt of \(x\,\Omega\) is added to \(3\,\Omega\) resistor to shift the balancing point by \(22.5\,cm\). The value of \(X\) is \(................\)

A travelling microscope has \(20\) divisions per \(cm\) on the main scale while its Vernier scale has total \(50\) divisions and \(25\) Vernier scale divisions are equal to \(24\) main scale divisions, what is the least count of the travelling microscope \(..........\,cm\)

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 bowl filled with very hot soup cools from \(98^{\circ}\,C\) to \(86^{\circ}\,C\) in \(2\) minutes when the room temperature is \(22^{\circ}\,C\). \(..........\,minutes\) long it will take to cool from \(75^{\circ}\,C\) to \(69^{\circ}\,C\) ?

A \(10\,m\) long horizontal wire extends from North East to South West. It is falling with a speed of \(5.0\,ms^{-1},\) at right angles to the horizontal component of the earth’s magnetic field, of \(0.3\times 10^{-4}\,Wb/m^2\). The value of the induced \(emf\) in wire is 

If \(A\) and \(B\) are two events such that \(P ( A )=\frac{1}{3}, P ( B )=\frac{1}{5} \) and \(P ( A \cup B )=\frac{1}{2}\), then \(P \left( A \mid B ^{\prime}\right)+ P \left( B \mid A ^{\prime}\right)\) is equal to

If the projection of the vector \(\hat{\mathrm{i}}+2 \hat{\mathrm{j}}+\hat{\mathrm{k}}\) on the sum of the two vectors \(2 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}-5 \hat{\mathrm{k}}\) and \(-\lambda \hat{i}+2 \hat{j}+3 \hat{k}\) is \(1,\) then \(\lambda\) is equal to ..... .

The electric field of a plane electromagnetic wave propagating along the \(x\) direction in vacuum is \(\overrightarrow{ E }= E _{0} \hat{ j } \cos (\omega t - kx )\). The magnetic field \(\overrightarrow{ B },\) at the moment \(t =0\) is :

Two gases-argon (atomic radius \(0.07 \;\mathrm{nm}\),atomic weight \(40\) ) and xenon (atomic radius \(0.1\; \mathrm{nm},\) atomic weight \(140\) ) have the same number density and are at the same temperature. The raito of their respective mean free times is closest to

Let \(\vec \alpha \, = \,3\hat i\, + \hat j\) and \(\vec \beta \, = \,2\hat i\, - \hat j + 3\hat k.\) If \(\vec \beta \, = \,{\vec \beta _1} - {\vec \beta _2},\) where \({\vec \beta _1}\) is parallel to \(\vec \alpha \) and \(\vec \beta_2 \) is perpendicular to \(\vec \alpha ,\)  then \({\vec \beta _1} \times {\vec \beta _2}\) is equal to

Let the lines \((2-i) z=(2+i) \bar{z}\) and \((2+i) z+(i-2) \bar{z}-4 i=0,\) (here \(\left.i^{2}=-1\right)\) be normal to a circle \(C\). If the line \(iz +\overline{ z }+1+ i =0\) is tangent to this circle \(C\), then its radius is