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