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 $f : R \to R$ is a differentiable function. If this system has infinitely many solutions for all $t \in R$ , then f [2026]

Question

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 $f : R \to R$ is a differentiable function. If this system has infinitely many solutions for all $t \in R$ , then f [2026]

Smart Achievers · Accepted Answer

(2)

Given system of linear equations are

x + 2y + tz = 0

6x + y + 5tz = 0

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

Now, $△ = | \begin{matrix} 1 & 2 & t \\ 6 & 1 & 5 t \\ 3 & t^{2} & f (t) \end{matrix} | = 0$

$\Rightarrow f (t) + 30 t + 6 t^{3} - 3 t - 5 t^{3} - 12 f (t) = 0$

$\Rightarrow 11 f (t) = t^{3} + 27 t$

$\Rightarrow f (t) = \frac{t^{3} + 27 t}{11}$

$\Rightarrow f' (t) = \frac{3 t^{2} + 27}{11} > 0$

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