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

$x+2 y+t z=0$,

$6 x+y+5 t z=0$,

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

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

[JEE Main 2026, 5 Apr (Shift 1)]

Question

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

$x+2 y+t z=0$,

$6 x+y+5 t z=0$,

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

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

[JEE Main 2026, 5 Apr (Shift 1)]

YESPYQ · Accepted Answer

Correct answer: b) is strictly increasing on $\mathbb{R}$. $D=\left|\begin{array}{ccc}1 & 2 & t \ 6 & 1 & 5 t \ 3 & t^2 & f(t)\end{array}\right|=0$ $\Rightarrow 1\left(\mathrm{f}(\mathrm{t})-5 \mathrm{t}^3\right)-2(6 \mathrm{f}(\mathrm{t})-15 \mathrm{t})+\mathrm{t}\left(6 \mathrm{t}^2-3\right)=0$ $f(t)=\frac{t^3+27 t}{11}$ $f^{\prime}(\mathrm{t})=\frac{1}{11}\left(3 \mathrm{t}^2+27\right)>0 \ \forall \mathrm{t} \in \mathrm{R}$ $\Rightarrow f(t)$ is strictly increasing on $\mathbb{R}$

Consider the system of linear equations in \(x, y, z:\) \(x+2 y+t z=0\), \(6 x+y+5 t z=0\), \(3 x+t^2 y+f(t) z=0\), wher…

Practice more JEE Maths PYQs