Question

What will be the value of f(x) if \(\begin{vmatrix}1 & ab & (\frac{1}{a} + \frac{1}{b}) \\1 & bc & (\frac{1}{b} + \frac{1}{c}) \\1 & ca & (\frac{1}{c} + \frac{1}{a})\end {vmatrix}\)?

(a) -1

(b) 0

(c) 1

(d) Can’t be predicted

I had been asked this question in examination.

Question is taken from Determinant in division Determinants of Mathematics – Class 12

Answer 1

Right choice is (c) 1

Best explanation: We have,\(\begin{vmatrix}1 & ab & (\frac{1}{a} + \frac{1}{b}) \\1 & bc & (\frac{1}{b} + \frac{1}{c}) \\1 & ca & (\frac{1}{c} + \frac{1}{a})\end {vmatrix}\)

= (1/abc)\(\begin{vmatrix}1 & ab & \frac{b + a}{ab} * abc \\1 & bc & \frac{b + c}{bc} * abc \\1 & ca & \frac{c + a}{ac} * abc \end {vmatrix}\)

= (1/abc)\(\begin{vmatrix}1 & ab & bc + ac \\1 & bc & ac + ab \\1 & ca & ab + bc \end {vmatrix}\)

Operating, C3 = C3 + C2

= (1/abc)\(\begin{vmatrix}1 & ab & ab + bc + ac \\1 & bc & ab + bc + ac \\1 & ca & ab + bc + ac \end {vmatrix}\)

= ((ab + bc + ac)/abc)\(\begin{vmatrix}1 & ab & 1 \\1 & bc & 1 \\1 & ca & 1 \end {vmatrix}\)

= 0