Given:
- The player tosses two fair coins.
- He wins Rs.5 if two heads appear.
- He wins Rs.2 if one head appears.
- He wins Rs.1 if no head occurs.

To Find:
The player's expected amount of winning.

Solution:
To calculate the expected amount of winning, we need to find the probability of each outcome and multiply it by the corresponding amount of winning. Then, we sum up these values to find the expected amount.

Let's calculate the probabilities of each outcome first:

1. Probability of two heads appearing:
- When tossing two fair coins, the probability of getting a head on each coin is 1/2.
- So, the probability of two heads appearing is (1/2) * (1/2) = 1/4.

2. Probability of one head appearing:
- There are two ways to get one head: (H, T) or (T, H), where H represents a head and T represents a tail.
- The probability of getting a head on the first coin is 1/2, and the probability of getting a tail on the second coin is also 1/2.
- So, the probability of one head appearing is 2 * (1/2) * (1/2) = 1/2.

3. Probability of no head occurring:
- The only way to get no head is to get tails on both coins.
- The probability of getting a tail on each coin is 1/2.
- So, the probability of no head occurring is (1/2) * (1/2) = 1/4.

Now, let's calculate the expected amount of winning:

Expected amount of winning = (Probability of two heads appearing * Amount of winning for two heads) + (Probability of one head appearing * Amount of winning for one head) + (Probability of no head occurring * Amount of winning for no head)
= (1/4 * Rs.5) + (1/2 * Rs.2) + (1/4 * Rs.1)
= Rs.5/4 + Rs.4/2 + Rs.1/4
= Rs.5/4 + Rs.8/4 + Rs.1/4
= (5 + 8 + 1)/4
= 14/4
= Rs.3.5

Therefore, the player's expected amount of winning is Rs.3.5, which corresponds to option 'A'.