How can I design the BJT based inverter question given in description?
It's a bit of a tricky problem. You seem to understand that you can consider the base and collector circuits independently, so that's a good start.

The phrase "the input connected to the 1.5 V supply through a resistance equal to RC" means that the base current flows through the series combination of the previous stage's Rc plus this stage's Rb. That's the only way you can meet the forced beta constraint. Here's the circuit:







simulate this circuit &ndash; Schematic created using CircuitLab

The total power dissipation is specified as 1 mW, and the supply voltage is 1.5 V. This means that the total current consumption of both circuits can only be 0.6667 mA.

The "forced &beta;" is specified as 10, which means that 10&times; the base current flows through the collector. This means that Ib must be 0.0606 mA and Ic must be 0.6061 mA.

This should be enough of a hint to let you complete the problem.