Three identical spheres each of mass \(M\) are placed at the corners of a right angled triangle with mutually perpendicular sides equal to \(3\,m\) each. Taking point of intersection of mutually perpendicular sides as origin, the magnitude of position vector of centre of mass of the system will be \(\sqrt{x} m\). The value of \(x\) is

A cubic block of mass \(m\) is sliding down on an inclined plane at \(60^{\circ}\) with an acceleration of \(\frac{\mathrm{g}}{2}\), the value of coefficient of kinetic friction is

The escape velocities of two planets \(A\) and \(B\) are in the ratio \(1: 2\). If the ratio of their radii respectively is \(1: 3\), then the ratio of acceleration due to gravity of planet \(A\) to the acceleration of gravity of planet \(B\) will be:

Consider four conducting materials copper, tungsten, mercury and aluminium with resistivity \(\rho_{ C }, \rho_{ T }, \rho_{ M }\) and \(\rho_{ A }\) espectively Then:

In the given figure the blocks \(A , B\) and C weigh \(4 kg, 6 kg\) and 8 kg respectively. The co-efficient of sliding friction between any two surfaces is 0.5 . The force \(\overrightarrow{ F }\) required to slide the block C with constant speed is _________ N . (Used \(g =10 m / s ^2\) )

Match List - I with List - II.

List - I	List - II
A. Boltzmann constant	I. \([M^{-1}L^3T^{-2}]\)
B. Stefan's constant	II. \([ML^2T^{-1}]\)
C. Planck's constant	III. \([ML^2T^{-2}K^{-1}]\)
D. Gravitational constant	IV. \([ML^0T^{-3}K^{-4}]\)

Choose the correct answer from the options given below :

A tuning fork of frequency \(480\, Hz\) is used in an experiment for measuring speed of sound \((\nu )\) in air by resonance tube method. Resonance is observed to occur at two successive lengths of the air column, \({\ell _1} = 30\,cm\) and \({\ell _2} = 70\,cm\). Then \(\nu\) is equal to ..... \(ms^{-1}\)

The maximum error in the measurement of resistance, current and time for which current flows in an electrical circuit are \(1 \%, 2 \%\) and \(3 \%\) respectively. The maximum percentage error in the detection of the dissipated heat will be

In a Screw Gauge, fifth division of the circular scale coincides with the reference line when the ratchet is closed. There are \(50\) divisions on the circular scale, and the main scale moves by \(0.5 \,{mm}\) on a complete rotation. For a particular observation the reading on the main scale is \(5\, {mm}\) and the \(20^{\text {th }}\) division of the circular scale coincides with reference line. Calculate the true reading. (in \({mm}\))

Match List \(I\) with List \(II\)

LIST \(I\)	LIST \(II\)
\(A\) Planck's constant \(( h )\)	\(I\) \(\left[ M ^1 L ^2 T ^{-2}\right]\)
\(B\) Stopping potential \(( Vs )\)	\(II\) \(\left[ M ^1 L ^1 T ^{-1}\right]\)
\(C\) Work function \((\phi)\)	\(III\) \(\left[ M ^1 L ^2 T ^{-1}\right]\)
\(D\) Momentum \(( p )\)	\(IV\) \(\left[ M ^1 L ^2 T ^{-3} A ^{-1}\right]\)