If 0- and the systems of equations x- sin- y- cos- z y- z- cos- x z- sin- x-yhas a nontrivial solution- find 4 -

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Standard XII

Mathematics

Perpendicular Distance of a Point from a Line

If 0≤θ≤π an...

Question

If $0 \leq θ \leq π$ and the systems of equations
$x = (sin θ) y + (cos θ) z$
$y = z + (cos θ) x$
$z = (sin θ) x + y$
has a nontrivial solution, find $4 θ / π$

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Solution

$R e w r i t e t h e s y s t e m o f e q u a t i o n s g i v e n s y s t e m h a s a n o n - t r i v i a l s o l u t i o n i . e Δ = 0 l e t A b e t h e c o e f f i c i e n t m a t r i x, | A | = ∣ ∣ ∣ ∣ \begin{matrix} - 1 & sin θ & cos θ cos θ & - 1 & 1 sin θ & 1 & - 1 \end{matrix} ∣ ∣ ∣ ∣ = 0 \Rightarrow {(sin θ + cos θ)}^{2} = 0 sin θ + cos θ = 0 tan θ = - 1 (g i v e n 0 \leq θ \leq π) \Rightarrow θ = \frac{3 π}{4} H e n c e \frac{4 θ}{π} = 3$

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Q. Assertion :The system of linear equations

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If 0≤θ≤π and the systems of equationsx=(sinθ)y+(cosθ)zy=z+(cosθ)xz=(sinθ)x+yhas a nontrivial solution, find 4θ/π

Rewritethesystemofequationsgivensystemhasanon−trivialsolutioni.eΔ=0letAbethecoefficientmatrix,|A|=∣∣ ∣∣−1sinθcosθcosθ−11sinθ1−1∣∣ ∣∣=0⇒(sinθ+cosθ)2=0sinθ+cosθ=0tanθ=−1(given0≤θ≤π)⇒θ=3π4Hence4θπ=3

If $0 \leq θ \leq π$ and the systems of equations
$x = (sin θ) y + (cos θ) z$
$y = z + (cos θ) x$
$z = (sin θ) x + y$
has a nontrivial solution, find $4 θ / π$