Consider the system of linear equations in $x,y,z$ :$x + 2y + tz = 0$,$6x + y + 5tz = 0$,$3x + t^{2}

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Consider the system of linear equations in $x,y,z$ :

$x + 2y + tz = 0$,

$6x + y + 5tz = 0$,

$3x + t^{2}y + f(t)z = 0$,

Where $f:R \rightarrow R$ is a differentiable function. If this system has infinitely many solutions for all $t \in R$, then $f$

Accepted Answer

Sol. [Key Concept: Infinitely many solutions for all $t$ force $D(t) = 0$; solving gives $f(t)$, and $f'(t) > 0$ shows it is strictly increasing.]$$\ D = \left| \begin{matrix} 1 & 2 & t \ 6 & 1 & 5t \ 3 & t^{2} & f(t) \end{matrix} ight| = 0$$$$ {\Rightarrow 1\left( f(t) - 5t^{3} ight) - 2(6f(t) - 15t) + t\left( 6t^{2} - 3 ight) = 0}$$$$ {f(t) = \frac{t^{3} + 12t}{11}}$$$$ {f^{'}(t) = \frac{1}{11}\left( 3t^{2} + 12 ight) > 0\ \forall t \in R}$$$$ \Rightarrow f(t)$$ is strictly increasing on R

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