The density of \(\beta\)-Fe is 7.6 g/cm\(^3\). It crystallizes in a cubic lattice with \( a = 290 \) pm.
What is the value of \( Z \)? (\( Fe = 56 \) g/mol, \( N_A = 6.022 \times 10^{23} \) mol\(^{-1}\))
Arrange the following in the increasing order of number of unpaired electrons present in the central metal ion:
I. \([MnCl_6]^{4-}\)
II. \([FeF_6]^{3-}\)
III. \([Mn(CN)_6]^{3-}\)
IV. \([Fe(CN)_6]^{3-}\)
A small block of mass \(m\) slides down from the top of a frictionless inclined surface, while the inclined plane is moving towards left with constant acceleration \(a_0\). The angle between the inclined plane and ground is \(\theta\) and its base length is \(L\). Assuming that initially the small block is at the top of the inclined plane, the time it takes to reach the lowest point of the inclined plane is _______. 
A cylindrical tube \(AB\) of length \(l\), closed at both ends, contains an ideal gas of \(1\) mol having molecular weight \(M\). The tube is rotated in a horizontal plane with constant angular velocity \(\omega\) about an axis perpendicular to \(AB\) and passing through the edge at end \(A\), as shown in the figure. If \(P_A\) and \(P_B\) are the pressures at \(A\) and \(B\) respectively, then (consider the temperature to be same at all points in the tube) 