The sum of all possible values of , for which the system of equations: ; ; x + y + 3z = 0 has a non-trivial solution, is equal to: [2026]

Question

The sum of all possible values of $θ \in [0, 2 π]$ , for which the system of equations:

$x \cos 3 θ - 8 y - 12 z = 0$ ;

$x \cos 2 θ + 3 y + 3 z = 0$ ;

x + y + 3z = 0

has a non-trivial solution, is equal to: [2026]

Smart Achievers · Accepted Answer

(4)

Since the given system of equations has non-trivial solution,

$△ = | \begin{matrix} \cos 3 θ & - 8 & - 12 \\ \cos 2 θ & 3 & 3 \\ 1 & 1 & 3 \end{matrix} |$

Expanding along $C_{1}$

$\Rightarrow \cos 3 θ | \begin{matrix} 3 & 3 \\ 1 & 3 \end{matrix} | - \cos 2 θ | \begin{matrix} - 8 & - 12 \\ 1 & 3 \end{matrix} | + 1 | \begin{matrix} - 8 & - 12 \\ 3 & 3 \end{matrix} | = 0$

$\Rightarrow 6 \cos 3 θ - (- 12) \cos 2 θ + 12 = 0$

$\Rightarrow \cos 3 θ + 2 \cos 2 θ + 2 = 0$

Since, $\cos 2 θ = 2 \cos^{2} θ - 1 and \cos 3 θ = 4 \cos^{3} θ - 3 \cos θ$

$∴ (4 \cos^{3} θ - 3 \cos θ) + 4 \cos^{2} θ - 2 + 2 = 0$

$\Rightarrow 4 \cos^{3} θ + 4 \cos^{2} θ - 3 \cos θ = 0$

$\Rightarrow \cos θ (4 \cos^{2} θ + 4 \cos θ - 3) = 0$

$\Rightarrow \cos θ (2 \cos θ - 1) (2 \cos θ + 3) = 0$

$\Rightarrow \cos θ = 0, \cos θ = \frac{1}{2}$ and $\cos θ = \frac{- 3}{2}$ (no real soluion)

If $\cos θ = 0, θ = \frac{π}{2}, \frac{3 π}{2}$

If $\cos θ = \frac{1}{2}, θ = \frac{π}{3}, \frac{3 π}{3}$

$∴ Required sum = \frac{π}{2} + \frac{3 π}{2} + \frac{π}{3} + \frac{5 π}{3}$

$= 2 π + 2 π = 4 π$

Solution

Q.
The sum of all possible values of $θ \in [0, 2 π]$ , for which the system of equations:

$x \cos 3 θ - 8 y - 12 z = 0$ ;

$x \cos 2 θ + 3 y + 3 z = 0$ ;

x + y + 3z = 0

has a non-trivial solution, is equal to: [2026]

1 $π$

2 $2 π$

3 $3 π$

4 $4 π$

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Solution

Q. The sum of all possible values of θ∈[0,2π], for which the system of equations: xcos3θ–8y–12z=0; xcos2θ+3y+3z=0; x + y + 3z = 0 has a non-trivial solution, is equal to: [2026]

1 π

2 2π

3 3π