An incandescent bulb has a thin filament of tungsten that is heated to high temperature by passing an electric current. The hot filament emits black-body radiation. The filament is observed to break up at random locations after a sufficiently long time of operation due to non-uniform evaporation of tungsten from the filament. If the bulb is powered at constant voltage, Which of the following statement(s) is (are) true?

The  filament will be a conductor of non uniform crossection
A) For any element of length dx in equilibrium

   $$\begin{gathered} I^{2}R=P_{radiated} \\ \Rightarrow \frac{I^2\rho \left(dx\right)}{\pi r^2}=\sigma \left(2\pi r\right)dx{T}^4 \end{gathered}$$

  $$\begin{gathered} \Rightarrow T^4=\frac{I^2\rho }{2{\pi }^2\sigma r^3} \\ \Rightarrow T\propto \frac{1}{r^{3/4}} \end{gathered}$$
T will be non uniform as r is non uniform
B)  $$\begin{gathered} R=\frac{\rho \left(dx\right)}{\pi r^2} \\ \Rightarrow R\propto \frac{1}{r^2} \end{gathered}$$
Due to evaporation, r of every element decreases with time $\Rightarrow$  ‘R’ increases with time
C) $\lambda T=$  constant

 $\frac{T}{f}$  =constant 

$\Rightarrow$ $T\alpha f$   

from (A) ,  $T\alpha \frac{1}{r^{3/4}}$
$\therefore f\alpha \frac{1}{r^{3/4}}$  ‘r’  decrease with time $\Rightarrow \text{'}f\text{'}$  increases with time i.e. it emits bond of higher frequencies before it breaks  up if I is taken as constant. 
D)  $P=\frac{V^2}{R}$ Due to evaporation, ‘R’ increases with  time as radius of cross section decreases for every element  $\Rightarrow \text{'}P\text{'}$ decrease with time