Here we can use the formula

$$\begin{gathered} \text{Average (A)=}\frac{\mathrm{Sum} \mathrm{of} \mathrm{observations} \left(\mathrm{S}\right)}{\mathrm{No}. \mathrm{of} \mathrm{observations} \left(\mathrm{N}\right)} \\ \text{⇒ S=A×N} \end{gathered}$$

According to the question,

Sum of weight of 34 students (A=42, N=34)

⇒S=42 X 34 =1428

Later, Wieght of the teacher included, The mean rises by 400 g. (so A=42.4, N=35)

So, sum of weight of 35 students

⇒S=42.4 X 35 =1484

Now,

The teacher's age =(sum of weight of 34 students)-(Sum of weight of 35 students)

   =1484-1428

   = 56

Hence, Option B is correct.