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KCET 2020 Math Solved Paper

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Question : 57 of 60

Marks: +1, -0

If

a_{1}, a_{2}, a_{3}, \dots, a_{9}

are in A.P. then the value of

\begin{vmatrix} a_1 & a_2 & a_3 \\ a_4 & a_5 & a_6 \\ a_7 & a_8 & a_9 \end{vmatrix}

is

1
$\frac{9}{2}(a_1 + a_9)$
$a_1 + a_9$
$\log_e(\log_e e)$

Solution:

Let 𝑑 be the common difference of A.P.

Then,

\Delta = \begin{vmatrix} a_1 & a_2 & a_3 \\ a_4 & a_5 & a_6 \\ a_7 & a_8 & a_9 \end{vmatrix}

= \begin{vmatrix} a_1 & a_1+d & a_1+2d \\ a_1+3d & a_1+4d & a_1+5d \\ a_1+6d & a_1+7d & a_1+8d \end{vmatrix}

Applying

C_2 \rightarrow C_2 - C_1

and

C_3 \rightarrow C_3 - C_2

\Delta = \begin{vmatrix} a_1 & d & d \\ a_1+3d & d & d \\ a_1+6d & d & d \end{vmatrix}

=0 \qquad (\because C_2

and

C_3

are identical)

= \log_e(\log_e e) \qquad (\because \log_e 1 = 0)

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