(i) Radius of cone, r =7 cm

Slant height of cone, l = 25 cm

Height of cone, h = \(\sqrt{l^2 - r^2}\)

h = \(\sqrt{25^2 - 7^2}\)

h = \(\sqrt{625 - 49}\)

h = 24

Height of the cone is 24 cm

Volume of cone, V = (1/3) πr²h (formula)

V = (1/3) × (22/7) × 7² × 24

= (154 × 8)

= 1232

So, the volume of the vessel is 1232 cm³

Therefore, capacity of the conical vessel = (1232/1000) liters (because 1L = 1000 cm³)

= 1.232 Liters.

(ii) Height of cone, h = 12 cm

Slant height of cone, l = 13 cm

Height of cone, r = \(\sqrt{l^2 - h^2}\)

r = \(\sqrt{13^2 - 12^2}\)

h = \(\sqrt{169 - 144}\)

r = 5

Hence, the radius of cone is 5 cm.

Now, Volume of cone, V = (1/3)πr²h

V = (1/3) × (22/7) × 52 × 12 cm³

= \(\frac{2200}{7}\)

Volume of cone is \(\frac{2200}{7}\) cm³

Now, Capacity of the conical vessel

= \(\frac{2200}{7000}\) litres (1L = 1000 cm³)

= \(\frac{11}{35}\) litres