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3 and 4 .Determinants and Matrices
hard
Let $A_1, A_2, A_3$ be the three A.P. with the same common difference $d$ and having their first terms as $A , A +1, A +2$, respectively. Let $a , b , c$ be the $7^{\text {th }}, 9^{\text {th }}, 17^{\text {th }}$ terms of $A_1, A_2, A_3$, respectively such that $\left|\begin{array}{lll} a & 7 & 1 \\ 2 b & 17 & 1 \\ c & 17 & 1\end{array}\right|+70=0$ If $a=29$, then the sum of first $20$ terms of an $AP$ whose first term is $c - a - b$ and common difference is $\frac{ d }{12}$, is equal to $........$.
A
$494$
B
$495$
C
$496$
D
$498$
(JEE MAIN-2023)
Solution
$\left|\begin{array}{lll}A+6 d & 7 & 1 \\ 2(A+1+8 d) & 17 & 1 \\ A+2+16 d & 17 & 1\end{array}\right|+70=0$
$\Rightarrow A=-7 \text { and } d =6$
$\therefore c – a – b =20$
$S _{20}=495$
Standard 12
Mathematics
