jee-main 2025 Q18

jee-main · India · session1_24jan_shift2 Polynomial Division & Manipulation
For some $\mathrm{a}, \mathrm{b}$, let $f(x) = \left| \begin{array}{ccc} \mathrm{a} + \frac{\sin x}{x} & 1 & \mathrm{b} \\ \mathrm{a} & 1 + \frac{\sin x}{x} & \mathrm{b} \\ \mathrm{a} & 1 & \mathrm{b} + \frac{\sin x}{x} \end{array} \right|$, $x \neq 0$, $\lim_{x \rightarrow 0} f(x) = \lambda + \mu\mathrm{a} + \nu\mathrm{b}$. Then $(\lambda + \mu + \nu)^{2}$ is equal to:
(1) 16
(2) 25
(3) 9
(4) 36
For some $\mathrm{a}, \mathrm{b}$, let $f(x) = \left| \begin{array}{ccc} \mathrm{a} + \frac{\sin x}{x} & 1 & \mathrm{b} \\ \mathrm{a} & 1 + \frac{\sin x}{x} & \mathrm{b} \\ \mathrm{a} & 1 & \mathrm{b} + \frac{\sin x}{x} \end{array} \right|$, $x \neq 0$, $\lim_{x \rightarrow 0} f(x) = \lambda + \mu\mathrm{a} + \nu\mathrm{b}$. Then $(\lambda + \mu + \nu)^{2}$ is equal to:\\
(1) 16\\
(2) 25\\
(3) 9\\
(4) 36