The value of the determinant - -log x log y log z log2x log2y log2z log3x log3y log3z is

The correct option is A 0 Δ = log x amp; log y amp; log z log2x amp; log2y amp; log2z log3x amp; log3y amp; log3z C ...

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Standard XII

Mathematics

Equation of a Line Passing through 2 Points

The value of ...

Question

The value of the determinant $Δ = ∣ ∣ ∣ ∣ \begin{matrix} l o g x & l o g y & l o g z l o g 2 x & l o g 2 y & l o g 2 z l o g 3 x & l o g 3 y & l o g 3 z \end{matrix} ∣ ∣ ∣ ∣$ is

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l o g (x y z)

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l o g (6 x y z)

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6 l o g (x y z)

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Solution

The correct option is A 0
$Δ = ∣ ∣ ∣ ∣ \begin{matrix} l o g x & l o g y & l o g z l o g 2 x & l o g 2 y & l o g 2 z l o g 3 x & l o g 3 y & l o g 3 z \end{matrix} ∣ ∣ ∣ ∣$
$C_{1} \to C_{1} - C_{2}, C_{2} \to C_{2} - C_{3}$

$Δ = ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} l o g \frac{x}{y} & l o g \frac{y}{z} & l o g z l o g \frac{x}{y} & l o g \frac{y}{z} & l o g 2 z l o g \frac{x}{y} & l o g \frac{y}{z} & l o g 3 z \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣$
$Δ = l o g \frac{x}{y} l o g \frac{y}{z} ∣ ∣ ∣ ∣ \begin{matrix} 1 & 1 & l o g z 1 & 1 & l o g 2 z 1 & 1 & l o g 3 z \end{matrix} ∣ ∣ ∣ ∣$
$= 0$

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The value of the determinant
$Δ = ∣ ∣ ∣ ∣ \begin{matrix} l o g x & l o g y & l o g z l o g 2 x & l o g 2 y & l o g 2 z l o g 3 x & l o g 3 y & l o g 3 z \end{matrix} ∣ ∣ ∣ ∣$ is

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Δ = ∣ ∣ ∣ ∣ \begin{matrix} 3 & - 2 & sin 3 θ - 7 & 8 & cos 2 θ - 11 & 14 & 2 \end{matrix} ∣ ∣ ∣ ∣ = 0

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Q. If

A_{1}, B_{1}, C_{1}, . .

are, respectively, the cofactors of the elements

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

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

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, then the value of

∣ ∣ ∣ \begin{matrix} B_{2} & C_{2} B_{3} & C_{3} \end{matrix} ∣ ∣ ∣

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Q. If

A_{1}, B_{1}, C_{1}, . . . .

are respectively, the cofactors of the elements

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

of the determinant

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

, then the value of

∣ ∣ ∣ \begin{matrix} B_{2} & C_{2} B_{3} & C_{3} \end{matrix} ∣ ∣ ∣

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Q. The value of the determinant

∆ = |\begin{matrix} \sec^{2} θ & \tan^{2} θ & 1 \\ \tan^{2} θ & \sec^{2} θ & - 1 \\ 22 & 20 & 2 \end{matrix}|

is _____________.

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The value of the determinant Δ=∣∣ ∣∣logxlogylogzlog2xlog2ylog2zlog3xlog3ylog3z∣∣ ∣∣ is

The correct option is A 0Δ=∣∣ ∣∣logxlogylogzlog2xlog2ylog2zlog3xlog3ylog3z∣∣ ∣∣C1→C1−C2,C2→C2−C3Δ=∣∣ ∣ ∣ ∣∣logxylogyzlogzlogxylogyzlog2zlogxylogyzlog3z∣∣ ∣ ∣ ∣∣Δ=logxylogyz∣∣ ∣∣11logz11log2z11log3z∣∣ ∣∣=0

The value of the determinant $Δ = ∣ ∣ ∣ ∣ \begin{matrix} l o g x & l o g y & l o g z l o g 2 x & l o g 2 y & l o g 2 z l o g 3 x & l o g 3 y & l o g 3 z \end{matrix} ∣ ∣ ∣ ∣$ is