Let S be the set of all values of for which the system of linear equations x + y + 3 z = 0 , - x + ( tan ) y + 7 z = 0 , x + y + ( tan ) z = 0 has a non-trivial solution. Then 120 is equal to [2023]

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Solution

Q.
Let $S$ be the set of all values of $θ \in [- π, π]$ for which the system of linear equations

$x + y + \sqrt{3} z = 0,$

$- x + (\tan θ) y + \sqrt{7} z = 0,$

$x + y + (\tan θ) z = 0$

has a non-trivial solution. Then $\frac{120}{π} \sum_{θ \in S} θ$ is equal to [2023]

1 30

2 10

3 40

4 20

Ans.
(4)

For non-trivial solution, we have $| \begin{matrix} 1 & 1 & \sqrt{3} \\ - 1 & \tan θ & \sqrt{7} \\ 1 & 1 & \tan θ \end{matrix} |$ $= 0$

$\Rightarrow 1 (\tan^{2} θ - \sqrt{7}) - 1 (- \tan θ - \sqrt{7}) + \sqrt{3} (- 1 - \tan θ) = 0$

$\Rightarrow \tan^{2} θ - \sqrt{7} + \tan θ + \sqrt{7} - \sqrt{3} - \sqrt{3} \tan θ = 0$

$\Rightarrow \tan^{2} θ + \tan θ - \sqrt{3} \tan θ - \sqrt{3} = 0$

$\Rightarrow \tan θ (\tan θ + 1) - \sqrt{3} (\tan θ + 1) = 0$

$\Rightarrow \tan θ = - 1 or \tan θ = \sqrt{3} \Rightarrow θ = \frac{- π}{4}, \frac{3 π}{4}, \frac{π}{3}, \frac{- 2 π}{3}$

So, $\frac{120}{π} \sum_{θ \in S} θ = \frac{120}{π} [\frac{- π}{4} + \frac{3 π}{4} + \frac{π}{3} - \frac{2 π}{3}] = \frac{120}{π} \times \frac{π}{6} = 20$

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