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Question

The determinant ∣ ∣xp+yxyyp+zyz0xp+yyp+z∣ ∣=0 if

A
x, y, z are in A.P.
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B
x, y, z are in G.P.
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C
x, y, z are in H.P.
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D
xy, yz, zx are in A.P.
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Solution

The correct option is B x, y, z are in G.P.
Given that,
∣ ∣xp+yxyyp+zyz0xp+yyp+z∣ ∣=0
Operating C1C1pC2C3, we get
∣ ∣ ∣0xy0yz(xp2+2py+z)xp+yyp+z∣ ∣ ∣=0or, (xzy2)(xp2+2py+z)=0or, xzy2=0or, y2=xzx,y,z are in G.P.

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