Answer: Option C
$$\eqalign{
& {\text{Principal = Rs. 2000}} \cr
& {\text{Amount = Rs. 2420}} \cr
& {\text{Rate = 10% }} \cr
& {\text{By using formula,}} \cr
& \Rightarrow 2420 = 2000{\left( {1 + \frac{{10}}{{100}}} \right)^n} \cr
& \Rightarrow \frac{{2420}}{{2000}} = {\left( {1 + \frac{{10}}{{100}}} \right)^n} \cr
& \Rightarrow \frac{{121}}{{100}} = {\left( {\frac{{11}}{{10}}} \right)^n} \cr
& \Rightarrow {\left( {\frac{{11}}{{10}}} \right)^2} = {\left( {\frac{{11}}{{10}}} \right)^n} \cr
& \Rightarrow n = 2 \cr
& {\text{Hence,}} \cr
& {\text{required time = 2 years}} \cr} $$
Alternative
Note : In such type of questions to save your valuable time follow the given below method.
 
 
Principal
  :  
Amount
Ratio
  →  
2000
  :  
2420
 
 
100
  :  
121
$${\text{Rate = 10% = }}\frac{1}{{10}}$$
 
 
Principal
  :  
Amount
1st year
  →  
10
  :  
11
2nd year
  →  
10
  :  
11
Ratio
  →  
100
  :  
121
Note : Now after 2nd year both the principal and amount will be in the same ratio.
Hence required time = 2 years