To solve this problem, we can use the concept of average and the formula for calculating average.

Let's assume that the batsman's average before the 21st match is 'x'.

We know that the batsman scores 87 runs in the 21st match, and his average runs per match increases by 2 after this match. So, after the 21st match, his average becomes 'x + 2'.

We can now use the formula for average:

Average = (Total runs) / (Number of matches)

To find the total runs before the 21st match, we can multiply the average 'x' by the number of matches before the 21st match. Let's assume this number is 'n'.

Total runs before the 21st match = x * n

To find the total runs after the 21st match, we can add the runs scored in the 21st match (87) to the total runs before the 21st match.

Total runs after the 21st match = Total runs before the 21st match + 87

According to the given information, the average after the 21st match is 'x + 2'. So, we can now set up an equation:

(x * n + 87) / (n + 1) = x + 2

Now, let's solve this equation to find the value of 'x'.

(x * n + 87) = (x + 2) * (n + 1)

xn + 87 = xn + x + 2n + 2

Combining like terms, we get:

x + 2n + 2 = 87

x + 2n = 85

Since we don't know the values of 'x' and 'n', we cannot solve this equation directly. However, we can use the answer choices given to us and substitute the value of 'x' into the equation to check if it satisfies the equation.

Let's try option 'A' which states that the average before the 21st match is 45.

If x = 45, the equation becomes:

45 + 2n = 85

2n = 40

n = 20

Substituting these values into the total runs equation:

Total runs before the 21st match = 45 * 20 = 900

Total runs after the 21st match = 900 + 87 = 987

Average after the 21st match = 987 / 21 = 47

Since the average after the 21st match is 47, which is an increase of 2 from the assumed average of 45, option 'A' is the correct answer.

Therefore, the average of the batsman before the 21st match was 45 runs per match.