Mathematics · Determinants

JEE Main 2026 — 5 April, Morning Shift — Question 27

Consider the system of linear equations in $x, y, z: x+2y+tz=0, 6x+y+5tz=0, 3x+t²y+f(t)z=0,$ where $f: R→R$ is differentiable. If this system has infinitely many solutions for all $t∈R,$ then $f:$

Option A:
is a constant function
Option B:
is strictly increasing on R
Correct
Option C:
is strictly decreasing on R
Option D:
has two critical points

Answer: B

Step-by-step solution

$D=\left|\begin{array}{ccc}1 & 2 & t \\ 6 & 1 & 5 t\\ 3 & t^{2} & f(t)\end{array}\right|=0$ $\Rightarrow 1\left(\mathrm{f}(\mathrm{t})-5 \mathrm{t}^{3}\right)-2(6 \mathrm{f}(\mathrm{t})-15 \mathrm{t})+\mathrm{t}\left(6 \mathrm{t}^{2}-3\right)=0$ $f(t)=\frac{t^{3}+12 t}{11}$ $\mathrm{f}^{\prime}(\mathrm{t})=\frac{1}{11}\left(3 \mathrm{t}^{2}+12\right)>0 \forall \mathrm{t} \in \mathrm{R}$ $\Rightarrow \mathrm{f}(\mathrm{t})$ is strictly increasing on $R$

Answer key and solution verified before publishing.

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Exam: JEE Main 2026
Paper: 5 April, Morning Shift
Subject: Mathematics
Chapter: Determinants
Topic: System of Linear Equations using Determinants