Question

If x is real number such that $\left| \begin{matrix} x + 1 & x + 2 & x + \alpha \\ x + 2 & x + 3 & x + \beta \\ x + 3 & x + 4 & x + \gamma \end{matrix} \right|$ = 0 .

Then a, b, g are in

• A. A.P
• B. G.P
• C. H.P.
• D. None of these

Answer 1

Answer: (A)

A.P

Answer 2

(i) R₃ ® R₃ – R₂

(ii) R₂ ® R₂ – R₁

$\left| \begin{matrix} x + 1 & x + 2 & x + \alpha \\ 1 & 1 & \beta - \alpha \\ 1 & 1 & \gamma - \beta \end{matrix} \right|$

$\left| \begin{matrix} x + 1 & x + 2 & x + \alpha \\ 0 & 0 & 2\beta - \alpha - \gamma \\ 1 & 1 & \gamma - \beta \end{matrix} \right|$ = 0;R₂ ® R₂ – R₃

expanding by 2^nd row –(2b – a – g) (–1) = 0

2b = a + g

a, b, g are in AP.