Determinants Matrices Question 163
Question: If the value of the determinant $ \begin{vmatrix} a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c \\ \end{vmatrix} $ is positive, then (a, b, c > 0)
Options:
A) $ abc>1 $
B) $ abc>-8 $
C) $ abc<-8 $
D) $ abc>-2 $
Show Answer
Answer:
Correct Answer: B
Solution:
- [b] we have, $ \Delta = \begin{vmatrix} a & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c \\ \end{vmatrix}=abc-(a+b+c)+2 $
$ \therefore \Delta >0\Rightarrow abc+2>a+b+c $
$ \Rightarrow abc+2>3{{(abc)}^{1/3}} $
$ [ \therefore A.M.>G.M.\Rightarrow \frac{a+b+c}{3}>{{(abc)}^{1/3}} ] $
$ \Rightarrow x^{3}+2>3x,wherex={{(abc)}^{1/3}} $
$ \Rightarrow x^{3}-3x+2>0 $
$ \Rightarrow {{(x-1)}^{2}}(x+2)>0 $
$ \Rightarrow x+2>0 $
$ \Rightarrow x>-2\Rightarrow {{(abc)}^{1/3}}>-2\Rightarrow abc>-8 $
BETA
