If a,b and care non zero numbers, then Δ=∣∣ ∣ ∣

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If a,b and care non zero numbers, then

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

is equal to

Options:

A . abc

B . a2b2c2

C . ab+bc+ca

D . None of these

Answer: Option D
:
D
Multiplying

R_{1}

by

a, R_{2}

by b and

R_{3}

by c, we have

Δ = \frac{1}{a b c} ∣ ∣∣∣ ∣ \begin{matrix} a b^{2} c^{2} & a b c & a b + a c a^{2} b c^{2} & a b c & b c + a b a^{2} b^{2} c & a b c & a c + b c \end{matrix} ∣ ∣∣∣ ∣ = \frac{a^{2} b^{2} c^{2}}{a b c} ∣ ∣∣ ∣ \begin{matrix} b c & 1 & a b + a c a c & 1 & b c + a b a b & 1 & a c + b c \end{matrix} ∣ ∣∣ ∣ = a b c ∣ ∣∣∣ ∣ \begin{matrix} b c & 1 & \sum a b a c & 1 & \sum a b a b & 1 & \sum a b \end{matrix} ∣ ∣∣∣ ∣ {b y C_{3} \to C_{3} + C_{1}} = a b c . \sum a b ∣ ∣∣ ∣ \begin{matrix} b c & 1 & 1 c a & 1 & 1 a b & 1 & 1 \end{matrix} ∣ ∣∣ ∣ = 0, [S i n c e C_{2} \equiv C_{3}]

.
Trick : Put a=1, b=2, c=3 and check it.

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