\(ABC\) is a plane lamina of the shape of an equilateral triagnle. \(D,\) \(E\) are mid points of \(AB\), \(AC\) and \(G\) is the centroid of the lamina. Moment of inertia of the lamina about an axis passing through \(G\) and perpendicular to the plane \(ABC\) is \(I _{0} .\) If part \(ADE\) is removed, the moment of inertia of the remaining part about the same axis is \(\frac{ NI _{0}}{16}\) where \(N\) is an integer. Value of \(N\) is

A particle executes \(S.H.M.,\) the graph of velocity as a function of displacement is :-

A rod of length \(50\,cm\) is pivoted at one end. It is raised such that if makes an angle of \(30^o\) fro the horizontal as shown and released from rest. Its angular speed when it passes through the horizontal (in \(rad\,s^{-1}\) ) will be \((g = 10\,ms^{-2})\)

A copper wire is stretched to make it \(0.5\%\) longer. The percentage change in its electric resistance if its volume remains unchanged is ................ \(\%\)

Two blocks of masses \(10 \; kg\) and \(30 \; kg\) are placed on the same straight line with coordinates \((0,0) \; cm\) and \(( x , 0) \; cm\) respectively. The block of \(10 \; kg\) is moved on the same line through a distance of \(6 \; cm\) towards the other block. The distance through which the block of \(30 \; kg\) must be moved to keep the position of centre of mass of the system unchanged is

A current of \(2\, mA\) was passed through an unknown resistor which dissipated a power of \(4.4\, W\). Dissipated power when an ideal power supply of \(11\, V\) is connected across it is

A bar magnet with a magnetic moment \(5.0\,Am ^2\) is placed in parallel position relative to a magnetic field of \(0.4\,T\). The amount of required work done in turning the magnet form parallel to antiparallel position relative to the field direction is \(.........\,J\)

Let the domain of the function
\(f(x)=\cos ^{-1}\left(\frac{4 x+5}{3 x-7}\right)\) be \([\alpha, \beta]\) and the domain of \(\mathrm{g}(\mathrm{x})=\log _2\left(2-6 \log _{27}(2 \mathrm{x}+5)\right)\) be \((\gamma, \delta)\).
Then \(|7(\alpha+\beta)+4(\gamma+\delta)|\) is equal to ________

If the mean of the following probability distribution of a random variable \(X\);

\(X\)	\(0\)	\(2\)	\(4\)	\(6\)	\(8\)
\(P(X)\)	\(a\)	\(2a\)	\(a+b\)	\(2b\)	\(3b\)

is \(\frac{46}{9}\) , then the variance of the distribution is

In a compound microscope the focal length of objective lens is \(1.2\, cm\) and focal length of eye piece is \(3.0\, cm\). When object is kept at \(1.25\, cm\) in front of objective, final image is formed at infinity. Magnifying power of the compound microscope should be

If the curves \(x=y^{4}\) and \(x y=k\) cut at right angles, then \((4 k )^{6}\) is equal to ..... .

The source that illuminates the double - slit in 'double - slit interference experiment' emits two distinct monochromatic waves of wavelength \(500\,nm\) and \(600\, nm\), each of them producing its own pattern on the screen. At the central point of the pattern when path difference is zero, maxima of both the patterns coincide and the resulting interference pattern is most distinct at the region of zero path difference. But as one moves out of this central region, the two fringe systems are gradually out of step such that maximum due to on wavelength coincides with the minimum due to the other and the combined fringe system becomes completely indistinct. This may happen when path difference in \(nm\) is

A system consists of two identical spheres each of mass \(1.5\, {kg}\) and radius \(50\, {cm}\) at the end of light rod. The distance between the centres of the two spheres is \(5\, {m}\). What will be the moment of inertia of the system about an axis perpendicular to the rod passing through its midpoint ? (in \({kgm}^{2}\))