Question

What will be the value of f(x) if \(\begin{vmatrix}1 & 1 & 1 \\x & y & z \\x^3 & y^3 & z^3 \end {vmatrix}\)?

(a) -1

(b) 0

(c) 1

(d) 2

I got this question during an online interview.

I want to ask this question from Determinant topic in section Determinants of Mathematics – Class 12

Answer 1

Correct choice is (b) 0

For explanation: Given, \(\begin{vmatrix}1 & 1 & 1 \\x & y & z \\x^3 & y^3 & z^3  \end {vmatrix}\)

Operating, C1 = C1 – C2 and C2 = C2 – C3

= \(\begin{vmatrix}1 & 1 & 1 \\x – y & y – z & y \\x^3 – y^3 & y^3 – z^3 & z^3  \end {vmatrix}\)

Expanding by the 1^st row,

= (x – y)(y^3 – z^3) – (y – z)(x^3 – y^3)

= (x – y)(y – z)[(y^2 + yz + z^2) – (x^2 + xy + y^2)]

= (x – y)(y – z)(z – x)(x + y + z)

As, x + y + z = 0

= 0