The numbers 52, 53, 54, 54, (2x + 1), 55, 55, 56, 57 have been arranged in an ascending order and their median is 55.

Arranging the given data in ascending order:
52, 53, 54, 54, (2x + 1), 55, 55, 56, 57

Number of terms = 9 (odd)

$\therefore$ Median $=\left(\frac{n+1}{2}\right)^{\text {th }}$ term

$\Rightarrow 55=\left(\frac{9+1}{2}\right)^{\text {th }}$ term

$\Rightarrow 55=(5)^{\text {th }}$ term

$\Rightarrow 55=2 x+1$

$\Rightarrow 2 x=55-1$

$\Rightarrow 2 x=54$

$\Rightarrow x=27$

Hence, the value of x is 27.

Arranging the given data in ascending order:
52, 53, 54, 54, 55, 55, 55, 56, 57

Here, 55 occurs maximum number of times.
∴ Mode = 55

Hence, the mode of the data is 55.