"If x² - 5x + 4 is a factor of x⁴ - px² + q, then what are the values of p and q respectively?"

Given:

If x² - 5x + 4 is a factor of x⁴ - px² + q.

Formula used:

If (x² - 5x + 4) is a factor of (x⁴ - px² + q), then the roots of x² - 5x + 4 (i.e., 1 and 4) must satisfy x⁴ - px² + q = 0.

Calculation:

Roots of x² - 5x + 4 are x = 1 and x = 4.

Substitute x = 1 in x⁴ - px² + q = 0

⇒ 1⁴ - p(1²) + q = 0

⇒ 1 - p + q = 0

⇒ q = p - 1 ----- (1)

Substitute x = 4 in x⁴ - px² + q = 0

⇒ 4⁴ - p(4²) + q = 0

⇒ 256 - 16p + q = 0

⇒ q = 16p - 256 ------ (2)

Equate q values from 1 and 2

⇒ p - 1 = 16p - 256

⇒ 255 = 15p

⇒ p = 17

Substitute p = 17 into q = p - 1

⇒ q = 17 - 1

⇒ q = 16

∴ The values of p and q are 17 and 16 respectively.

The correct answer is option (1).