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Question

if
Δ=∣ ∣ ∣ ∣ ∣ ∣ ∣1z1z(x+y)z2(y+z)x21x1xy(y+z)x2z(x+2y+z)xz(x+y)xz2∣ ∣ ∣ ∣ ∣ ∣ ∣
then

A
Δ is independent of x
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B
Δ is independent of y
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C
Δ is independent of z
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D
Δ=0
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Solution

The correct options are
A Δ is independent of y
B Δ is independent of x
C Δ=0
D Δ is independent of z
Multiplying C1 by x,C2 by y and C3 by z we obtain
Δ=1xyz∣ ∣ ∣ ∣ ∣ ∣xzyzx+yzy+zxyxdzxy(y+z)xzy(x+2y+z)xzy(x+y)xz∣ ∣ ∣ ∣ ∣ ∣

Applying C1C1+C2+C3 we get

Δ=1xyz∣ ∣ ∣ ∣ ∣ ∣0yz(x+y)z0yxzx0y(x+2y+z)zxy(x+y)xz∣ ∣ ∣ ∣ ∣ ∣=0
This shows that Δ is independent of x,y,z

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