If a 1 b 1 c 1 - a 2 b 2 c 2 and a 3 b 3 c 3 are three-digit even natural numbers and - - c 1 a 1 b 1 c 2 a 2 b 2 c 3 a 3 b 3- then- - is

The correct option is A divisible by 2 but not necessarily by 4 Δ = c _ 1 amp; a _ 1 amp; b _ 1 c _ 2 amp; a _ 2 amp; b _ 2 c _ ...

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Solving Simultaneous Linear Equation Using Cramer's Rule

If a 1 b 1 ...

Question

If $a_{1} b_{1} c_{1}, a_{2} b_{2} c_{2}$ and $a_{3} b_{3} c_{3}$ are three-digit even natural numbers and $Δ = ∣ ∣ ∣ ∣ \begin{matrix} c_{1} & a_{1} & b_{1} c_{2} & a_{2} & b_{2} c_{3} & a_{3} & b_{3} \end{matrix} ∣ ∣ ∣ ∣$ , then, $Δ$ is

divisible by 2 but not necessarily by 4

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divisible by but not necessarily by 8

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divisible by 8

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none of these

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Solution

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a_{1}, b_{1}, c_{1}

a_{2} = g c d (b_{1}, c_{1}), b_{2} = g c d (c_{1}, a_{1}), c_{2} = g c d (a_{1}, b_{1}),

a_{3} = l c m (b_{2}, c_{2}), b_{3} = l c m (c_{2}, a_{2}), c_{3} = l c m (a_{2}, b_{2}) .

g c d (b_{3}, c_{3}) = a_{2}

(a_{1}, a_{2}, a_{3})

(b_{1}, b_{2}, b_{3})

(c_{1}, c_{2}, c_{3})

x (a_{1}, a_{2}, a_{3}) + y (b_{1}, b_{2}, b_{3}) + z (c_{1}, c_{2}, c_{3}) = 0

Δ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} l & a^{2} & a^{3} l & b^{2} & b^{3} l & c^{2} & c^{3} \end{matrix} ∣ ∣ ∣ ∣ ∣

Δ

| a_{1} | > | a_{2} | + | a_{3} |, | b_{2} | > | b_{1} | + | b_{3} |

| c_{3} | > | c_{1} | + | c_{2} |

∣ ∣ ∣ ∣ \begin{matrix} a_{1} & a_{2} & a_{3} b_{1} & b_{2} & b_{3} c_{1} & c_{2} & c_{3} \end{matrix} ∣ ∣ ∣ ∣ = 0

a_{1} x^{2} + b_{1} x + c_{1} = 0, a_{2} x^{2} + b_{2} x + c_{2} = 0

a_{3} x^{2} + b_{3} x + c_{3} = 0

\frac{c_{1} a_{2} + c_{2} a_{1}}{c_{1} a_{2} - c_{2} a_{1}} + \frac{a_{1} a_{2} b_{3}}{a_{3} (a_{1} b_{2} - a_{2} b_{1})} = 0

{(\frac{a_{1} b_{2} - a_{2} b_{1}}{a_{1} c_{2} - a_{2} c_{1}})}^{2} = \frac{a_{1} a_{2} c_{3}}{c_{1} c_{2} a_{3}}

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