JEE Main 2026 · 5 Apr · Shift 1 · Mathematics Q4

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MathematicsSingle correct+4 / -1Question 4 of 75

Consider the system of linear equations in $x,y,z$ :
$x+2y+tz=0,\qquad 6x+y+5tz=0,\qquad 3x+t^2y+f(t)z=0,$
where $f:\mathbb R\to\mathbb R$ is a differentiable function. If this system has infinitely many solutions for all $t\in\mathbb R$ , then $f$

is a constant function

is strictly increasing on

\mathbb R

is strictly decreasing on

\mathbb R

has two critical points

Answer and worked solution

Official answer: B

Condition. A homogeneous system has infinitely many solutions when its determinant vanishes:
$\begin{vmatrix}1&2&t\\6&1&5t\\3&t^2&f\end{vmatrix}=(f-5t^3)-2(6f-15t)+t(6t^2-3)=-11f+t^3+27t=0.$

So $f(t)=\dfrac{t^3+27t}{11}$ , and $f'(t)=\dfrac{3t^2+27}{11}>0$ for all t: $f$ is strictly increasing.

(B)

Topics:Systems of linear equations Monotonicity

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JEE Main 2026 · 5 Apr · Shift 1, Mathematics question 4