Question

For positive numbers x, y, z the numerical value of the determinant `|(1, log_x y, log_x z),(log_y x, 3, log_y z),(log_z x, log_z y, 5)|` is

Options

• 0

• log x log y log z

• 1

• 8

Answer 1

8

Explanation:

Replace log_ba by `loga/logb`

∴ Δ = `1/(log x log y log z) xx |(logx, logy, logz),(log x, 3logy, log z),(log x, log y, 5 log z)|`

Take log x, log y, log z common from C₁, C₂, C₃ respectively.

Δ = `|(1, 1, 1),(1, 3, 1),(1, 1, 5)| = |(1, 0, 0),(1, 2, 0),(1, 0, 4)| = 1 xx 2 xx 4` = 8