For some $\mathrm{a}, \mathrm{b}$, let $f(x)=\left|\begin{array}{ccc}\mathrm{a}+\frac{\sin x}{x} & 1 & \mathrm{~b} \ \mathrm{a} & 1+\frac{\sin x}{x} & \mathrm{~b} \ \mathrm{a} & 1 & \mathrm{~b}+\frac{\sin x}{x}\end{array} ight|$, $x
eq 0, \lim _{x ightarrow 0} f(x)=\lambda+\mu…

Question

For some $\mathrm{a}, \mathrm{b}$, let $f(x)=\left|\begin{array}{ccc}\mathrm{a}+\frac{\sin x}{x} & 1 & \mathrm{~b} \ \mathrm{a} & 1+\frac{\sin x}{x} & \mathrm{~b} \ \mathrm{a} & 1 & \mathrm{~b}+\frac{\sin x}{x}\end{array}ight|$, $x 
eq 0, \lim _{x ightarrow 0} f(x)=\lambda+\mu \mathrm{a}+
u \mathrm{b}$. Then $(\lambda+\mu+
u)^2$ is equal to :

Aicharya · Accepted Answer

$\lim _{x ightarrow 0}\left|\begin{array}{ccc} a+\frac{\sin x}{x} & 1 & b \ a & 1+\frac{\sin x}{x} & b \ a & 1 & b+\frac{\sin x}{x} \end{array}ight|=\lambda+\mu a+v b$ At $\lim x ightarrow 0$, $\begin{aligned} & f(x)=\left|\begin{array}{ccc} a+1 & 1 & b \ a & 1+1 & b \ a & 1 & b+1 \end{array}ight|=\lambda+\mu a+v b \ & R_1 ightarrow R_1-R_2 \ & R_2 ightarrow R_2-R_3 \ & \left|\begin{array}{ccc} 1 & -1 & 0 \ 0 & 1 & -1 \ a & 1 & b+1 \end{array}ight|=\lambda+\mu a+v b \ & \mathrm{C}_2 ightarrow \mathrm{C}_1-\mathrm{C}_2 \ & \left|\begin{array}{ccc} 1 & 0 & 0 \ 0 & 1 & -1 \ a & a+1 & b+1 \end{array}ight|=\lambda+\mu a+v b \ & a+b+2=\lambda+\mu a+v b \ & \lambda=2, \mu=1, \quad v=1 \ & (\lambda+\mu+v)=(2+1+1)^2=16 \end{aligned}$

For some $\mathrm{a}, \mathrm{b}$, let $f(x)=\left|\begin{array}{ccc}\mathrm{a}+\frac{\sin x}{x} & 1 &…

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