Question

If det $\left[\begin{array}{ccc}b& c& bL+c\\ c& d& cL+d\\ bL+c& cL+d& cL-a{L}^3\end{array}\right]=0$
I) b,c,d are in A.P.
II) b,c,d are in G.P.
III) b,c,d are in H.P.
IV) L is a root of $a{x}^3+b{x}^2+cx+d=0$

• A. only I, IV
• B. only II, IV
• C. only III, IV
• D. only IV

Answer 1

Answer: (B)

only II, IV

$\left|\begin{array}{ccc}b& c& bL+c\\ c& d& cL+d\\ bL+c& cL+d& cL-a{L}^3\end{array}\right|=0$
$C_3\to C_3{-(LC}_1+C_2)$
$\left|\begin{array}{ccc}b& c& 0\\ c& d& 0\\ bL+c& cL+d& -a{L}^3-b{L}^2-cL-d\end{array}\right|=0$
$(-a{L}^3-b{L}^2-cL-d)(bd-c^2)=0$
$\Rightarrow bd=c^2$ or $a{L}^3+b{L}^2+cL+d=0$
Hence, (i) b,c,d are in G.P.
(ii) L is the root of $a{x}^3+b{x}^2+cx+d=0$