A) Are in A.P.
B) Are in G. P.
C) Are in H. P.
D) Satisfy \[a+2b+3c=0\]
Correct Answer: C
Solution :
\[\left| \,\begin{matrix} 1 & 2a & a \\ 1 & 3b & b \\ 1 & 4c & c \\ \end{matrix}\, \right|\,=0\,\], \[[{{C}_{2}}\to {{C}_{2}}-2{{C}_{3}}]\] Þ\[\left| \,\begin{matrix} 1 & 0 & a \\ 1 & b & b \\ 1 & 2c & c \\ \end{matrix}\, \right|=0\], \[[{{R}_{3}}\to {{R}_{3}}-{{R}_{2}},\,{{R}_{2}}\to {{R}_{2}}-{{R}_{1}}]\] Þ \[\left| \,\begin{matrix} 1 & 0 & a \\ 0 & b & b-a \\ 0 & 2c-b & c-b \\ \end{matrix}\, \right|\,=0\] ; \[b(c-b)-(b-a)\,(2c-b)=0\] On simplification, \[\frac{2}{b}=\frac{1}{a}+\frac{1}{c}\] \ a, b, c are in Harmonic progression.
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