If \(f(x)=\left|\begin{array}{ccc}a+\frac{\sin x}{x} & 1 & b \\ a & 1+\frac{\sin x}{x} & b \\ a & 1 …
If \(f(x)=\left|\begin{array}{ccc}a+\frac{\sin x}{x} & 1 & b \\ a & 1+\frac{\sin x}{x} & b \\ a & 1 & b+\frac{\sin x}{x}\end{array}\right|\)
if \(\lim _{x \rightarrow 0^{+}} f(x)=\mu+\alpha a+\beta b\) then find the value of \((\mu+\alpha+\beta)^2\)
16
\(\begin{aligned}
&\begin{aligned}
& \lim _{x \rightarrow 0^{+}} f(x)=\left|\begin{array}{ccc}
a+1 & 1 & b \\
a & 1+1 & b \\
a & 1 & b+1
\end{array}\right|=\mu+\alpha a+\beta b \\
& R_1 \rightarrow R_1-R_2 \\
& R_2 \rightarrow R_2-R_3 \\
& \left|\begin{array}{ccc}
1 & -1 & 0 \\
0 & 1 & -1 \\
a & 1 & b+1
\end{array}\right|=\mu+\alpha a+\beta b \\
& C_2 \rightarrow C_2+C_1 \\
& \left|\begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & -1 \\
a & a+1 & b+1
\end{array}\right|=\mu+\alpha a+\beta b \\
& a+b+2=\mu+\alpha a+\beta b \\
& \quad \text { So } \alpha=1, \mu=2, \beta=1
\end{aligned}\\
&\text { Now, } \quad(\mu+\alpha+\beta)^2=16
\end{aligned}\)
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