Find the number of values of $t$ for which the system of equations
$(a + 2 t) x + b y + c z = 0$
$b x + (c + 2 t) y + a z = 0$
$c x + a y + (b + 2 t) z = 0$
has non-trivial solutions.

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Solution

$S y s t e m o f e q u a t i o n s h a s a n o n - t r i v i a l s o l u t i o n i . e Δ = 0 l e t A b e t h e c o e f f i c i e n t m a t r i x, | A | = ∣ ∣ ∣ ∣ \begin{matrix} a + 2 t & b & c b & c + 2 t & a c & a & b + 2 t \end{matrix} ∣ ∣ ∣ ∣ = 0 c l e a r l y, d e t e r m i n a n t v a l u e i s a c u b i c e q u a t i o n i n t a n d h a s 3 r o o t s H e n c e n o o f v a l u e s o f t = 3$

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