i need solution of part a$,$c and d by do it urgently and draw dig with hand

The schematic of a loaded, bipolar junction transistor $(3$C$2$BJT$1)$ inverter is shown below, where RC$=1$k$,$RB$=$RL$=10$k$$ and VCC$=5$V$.$ The datasheet for the $3$C$2$BJT$1$ is attached. Sketch a plot of the voltage transfer characteristic of the BJT inverter shown as Vi$$ is swept from $0$V to VCC$.$ Describe the transistor state in terms of cut$-$off, forward active or saturation at each point along with a suitable justification. $[5$ marks$]$ From basic circuit theory, derive an expression for ViHmin$$ and hence calculate ViHmin$$ for the above circuit, assuming that when Vi$=$ViHmin$$ the transistor is at the edge of saturation. $[7$ marks$]$ If the load resistance consists of N further stages as shown: Derive an expression for the fan$-$out of the single transistor inverter stage shown in terms of the minimum input voltage of the load stage, ViHmin$,$ and the other relevant circuit parameters. $[8$ marks$]$ Calculate the minimum value of RB$$ such that there is a fanout of F$=8$ if ViHmin$=0\mathrm{.}9$VCC$.[5$ marks$]$ Material properties of some common semiconducting crystals at standard temperature pressure are given below: BJT Datasheet Excerpt $(3$C$2$BJT$1)$ On Characteristics: