Matrices and Determinants 2 Question 23
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25. The determinant $\begin{array}{cccc}a & b & a \alpha+b \ b & c & b \alpha+c \ a \alpha+b & b \alpha+c & 0\end{array}$ is equal to zero, then
======= ####25. The determinant $\begin{vmatrix}a & b & a \alpha+b \\ b & c & b \alpha+c \\ a \alpha+b & b \alpha+c & 0\end{vmatrix}$ is equal to zero, then
3e0f7ab6f6a50373c3f2dbda6ca2533482a77bed
$(1986,2 M)$
(a) $a, b, c$ are in $AP$
(b) $a, b, c$ are in GP
(c) $a, b, c$ are in $HP$
(d) $(x-\alpha)$ is a factor of $a x^{2}+2 b x+c$
Numerical Value
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Answer:
Correct Answer: 25. (b, d)
Solution:
- Given,
$\begin{vmatrix}a & b & a \alpha+b \\ b & c & b \alpha+c \\ a \alpha+b & b \alpha+c & 0\end{vmatrix}$=0
Applying $C _3 \rightarrow C _3-\left(\alpha C _1+C _2\right)$
$ \begin{vmatrix} a & b & 0 \\ b & c & 0 \\ a \alpha+b & b \alpha+c & -\left(a \alpha^{2}+2 b \alpha+c\right) \end{vmatrix} $=0
$\Rightarrow -(a \alpha^{2}+2 b \alpha+ct)(a c-b^{2}) =0 $
$\Rightarrow a \alpha^{2}+2 b \alpha+c=0 \text { or } b^{2}= a c $
$\Rightarrow x-\alpha$ is a factor of $a x^{2}+2 b x+c$ or $a, b, c$ are in GP.
