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Early Conquests of Ranjit Singh

If a,b,c are ...

Question

If $a, b, c$ are in A.P, find value of $∣ ∣ ∣ ∣ \begin{matrix} 2 y + 4 & 5 y + 7 & 8 y + a 3 y + 5 & 6 y + 8 & 9 y + b 4 y + 6 & 7 y + 9 & 10 y + c \end{matrix} ∣ ∣ ∣ ∣$

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Solution

Simplification of given data
Given : $a, b, c$ are in A.P.

$\Rightarrow b - a = c - b$
$\Rightarrow b - a - c + b = 0$
$\Rightarrow 2 b - a - c = 0$ ...(i)

Solve given determinant

$△= ∣ ∣ ∣ ∣ \begin{matrix} 2 y + 4 & 5 y + 7 & 8 y + a 3 y + 5 & 6 y + 8 & 9 y + b 4 y + 6 & 7 y + 9 & 10 y + c \end{matrix} ∣ ∣ ∣ ∣$

Multiply & divide by $2$

$\Rightarrow△= \frac{2}{2} ∣ ∣ ∣ ∣ \begin{matrix} 2 y + 4 & 5 y + 7 & 8 y + a 3 y + 5 & 6 y + 8 & 9 y + b 4 y + 6 & 7 y + 9 & 10 y + c \end{matrix} ∣ ∣ ∣ ∣$

$\Rightarrow△= \frac{1}{2} ∣ ∣ ∣ ∣ ∣ \begin{matrix} 2 y + 4 & 5 y + 7 & 8 y + a 2 (3 y + 5) & 2 (6 y + 8) & 2 (9 y + b) 4 y + 6 & 7 y + 9 & 10 y + c \end{matrix} ∣ ∣ ∣ ∣ ∣$

$\Rightarrow△= \frac{1}{2} ∣ ∣ ∣ ∣ \begin{matrix} 2 y + 4 & 5 y + 7 & 8 y + a 6 y + 10 & 12 y + 16 & 18 y + 2 b 4 y + 6 & 7 y + 9 & 10 y + c \end{matrix} ∣ ∣ ∣ ∣$

Applying $R_{2} \to R_{2} - R_{1} - R_{3}$

$\Rightarrow△= \frac{1}{2} ∣ ∣ ∣ ∣ \begin{matrix} 2 y + 4 & 5 y + 7 & 8 y + a 0 & 0 & 2 b - a - c 4 y + 6 & 7 y + 9 & 10 y + c \end{matrix} ∣ ∣ ∣ ∣$

$\Rightarrow△= \frac{1}{2} ∣ ∣ ∣ ∣ \begin{matrix} 2 y + 4 & 5 y + 7 & 8 y + a 0 & 0 & 0 4 y + 6 & 7 y + 9 & 10 y + c \end{matrix} ∣ ∣ ∣ ∣$ ....[from (i)]

$\Rightarrow△= \frac{1}{2} \times 0$

$\Rightarrow△= 0$

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Early Conquests of Ranjit Singh

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