Statement I: The perimeter of the triangle is 30 cm.
Statement II: The ratio between the base and the height of the triangle is 5:12.
Statement III: The area of the triangle is equal to the area of a rectangle of length 10 cm.

Let's analyze each statement one by one:

Statement I: The perimeter of the triangle is 30 cm.
The perimeter of a triangle is the sum of the lengths of its three sides. However, the perimeter alone does not provide enough information to determine the area of a right-angled triangle. For example, a right-angled triangle with sides of length 6 cm, 8 cm, and 10 cm has a perimeter of 24 cm, while a right-angled triangle with sides of length 9 cm, 12 cm, and 15 cm also has a perimeter of 36 cm. Therefore, statement I alone is not sufficient to determine the area of the triangle.

Statement II: The ratio between the base and the height of the triangle is 5:12.
In a right-angled triangle, the base and the height are the two sides that form the right angle. The area of a right-angled triangle is given by the formula: (base * height) / 2.
If the ratio between the base and the height is given as 5:12, we can assume that the base is 5x and the height is 12x, where x is a positive constant.
However, without knowing the actual lengths of the base and the height, we cannot determine the area of the triangle. Therefore, statement II alone is not sufficient to determine the area of the triangle.

Statement III: The area of the triangle is equal to the area of a rectangle of length 10 cm.
The area of a rectangle is given by the formula: length * width. If the area of the triangle is equal to the area of a rectangle of length 10 cm, it implies that the base of the triangle is 10 cm. However, we still do not have enough information to determine the height of the triangle. Therefore, statement III alone is not sufficient to determine the area of the triangle.

Combining Statements I and II:
Even when we combine statements I and II, we still do not have enough information to determine the area of the triangle. We know the perimeter and the ratio between the base and the height, but the actual lengths of the sides are not given. Therefore, combining statements I and II is not sufficient to determine the area of the triangle.

Combining Statements II and III:
By combining statements II and III, we know that the base of the triangle is 10 cm and the ratio between the base and the height is 5:12. From this information, we can determine that the height of the triangle is (12/5) * 10 = 24 cm. With the base and the height known, we can now calculate the area of the triangle using the formula: (base * height) / 2 = (10 * 24) / 2 = 120 cm².

Therefore, the correct answer is option A) I and II only, as combining these two statements will allow us to determine the area of the right-angled triangle.