1. Tardigrade
  2. Question
  3. Mathematics
  4. If α+β+γ=2 π, then the system of equations x+( cos γ) y+( cos β) z=0 ( cos γ) x+y+( cos α) z=0 ( cos β) x+( cos α) y+z=0 has :

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. If $\alpha+\beta+\gamma=2 \pi$, then the system of equations
$x+(\cos \gamma) y+(\cos \beta) z=0 $
$(\cos \gamma) x+y+(\cos \alpha) z=0$
$(\cos \beta) x+(\cos \alpha) y+z=0$ has :

JEE MainJEE Main 2021Determinants

Solution:

$\alpha+\beta+\gamma=2 \pi$
$\begin{vmatrix}1 & \cos \gamma & \cos \beta \\ \cos \gamma & 1 & \cos \alpha \\ \cos \beta & \cos \alpha & 1\end{vmatrix}$
$=1+2 \cos \alpha \cdot \cos \beta \cdot \cos \gamma-\cos ^{2} \alpha-\cos ^{2} \beta-\cos ^{2} \gamma$
$=\sin ^{2} \gamma-\cos ^{2} \alpha-\cos ^{2} \beta+(\cos (\alpha+\beta)+\cos (\alpha-$
$\beta)) \cos \gamma$
$=\sin ^{2} \gamma-\cos ^{2} \alpha-\cos ^{2} \beta+\cos ^{2} \gamma+\cos (\alpha-\beta) \cos \gamma$
$=\sin ^{2} \alpha-\cos ^{2} \beta+\cos (\alpha-\beta) \cos (\alpha+\beta)$
$=\sin ^{2} \alpha-\cos ^{2} \beta+\cos ^{2} \alpha-\sin ^{2} \beta=0$