Suppose x-y-z are positive integers - 1 if - - det of - 1 logxy logxz- logyx 1 logyz- sinx-y -cosx-y sin2 z - then - is I Independent of x II Independent of y III Independent of zThe which of the above statement is - are correct

The correct option is D all the three I,II,III Δ = 1 amp; log_ x y amp; log_ x z log_ y x amp; 1 amp; log_ y z sin(x+y) amp; co ...

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Byju's Answer

Standard XII

Mathematics

Definition of a Determinant

Suppose x,y,z...

Question

Suppose x,y,z are positive integers $(\neq 1)$ if $Δ$ = det of $⎡ ⎢ ⎣ \begin{matrix} 1 & l o g_{x} y & l o g_{x} z l o g_{y} x & 1 & l o g_{y} z s i n (x + y) & - c o s (x + y) & s i n^{2} z \end{matrix} ⎤ ⎥ ⎦$ then $Δ$ is I) Independent of x II) Independent of y III) Independent of z
The which of the above statement is / are correct

only I and II

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only II and III

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only I and III

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all the three I,II,III

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Solution

The correct option is D all the three I,II,III
$Δ = ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & l o g_{x} y & l o g_{x} z l o g_{y} x & 1 & l o g_{y} z s i n (x + y) & - c o s (x + y) & s i n^{2} z \end{matrix} ∣ ∣ ∣ ∣ ∣$
$Δ = ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} 1 & \frac{{log}_{e} y}{{log}_{e} x} & \frac{{log}_{e} z}{{log}_{e} x} \frac{{log}_{e} x}{{log}_{e} y} & 1 & \frac{{log}_{e} z}{{log}_{e} y} s i n (x + y) & - c o s (x + y) & s i n^{2} z \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣$
$Δ = \frac{1}{{log}_{e} x {log}_{e} y} ∣ ∣ ∣ ∣ ∣ \begin{matrix} {log}_{e} x & {log}_{e} y & {log}_{e} z {log}_{e} x & {log}_{e} y & {log}_{e} z s i n (x + y) & - c o s (x + y) & s i n^{2} z \end{matrix} ∣ ∣ ∣ ∣ ∣$
$= 0 ∵ R_{1} a n d R_{2}$ are identical

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Suppose x,y,z are positive integers (≠1) if Δ = det of ⎡⎢⎣1logxylogxzlogyx1logyzsin(x+y)−cos(x+y)sin2z⎤⎥⎦ then Δ is I) Independent of x II) Independent of y III) Independent of zThe which of the above statement is / are correct