If x : y = 3 : 2, then the ratio (2x² + 3y²) : (3x² − 2y²) is equal to:

1. 5 : 3

2. 6 : 5

3. 12 : 5

4. 30 : 19

Given,

x : y = 3 : 2

Let x = 3k and y = 2k for some k.

For ratio (2x² + 3y²) : (3x² − 2y²),

Substituting value of x and y in the antecedent,

⇒ 2x² + 3y²

⇒ 2(3k)² + 3(2k)²

⇒ 2 × 9k² + 3 × 4k²

⇒ 18k² + 12k²

⇒ 30k².

Substituting value of x and y in the consequent,

⇒ 3x² - 2y²

⇒ 3(3k)² - 2(2k)²

⇒ 3 × 9k² - 2 × 4k²

⇒ 27k² - 8k²

⇒ 19k².

Therefore the required ratio is:

$\Rightarrow \dfrac{2x^2 + 3y^2}{3x^2 - 2y^2} \\[1em] \Rightarrow \dfrac{30k^2}{19k^2} \\[1em] \Rightarrow \dfrac{30}{19} \\[1em] \Rightarrow 30 : 19.$

Hence, option 4 is the correct option.