For some a, b, let f(x)= | ccc a+ x x & 1 & b a & 1+ x x & b a & … - Vidyadip

MCQ

For some $a, b$, let
$f(x)=\left|\begin{array}{ccc}a+\frac{\sin x}{x} & 1 & b \\ a & 1+\frac{\sin x}{x} & b \\ a & 1 & b+\frac{\sin x}{x}\end{array}\right|, \quad x \neq 0$,
$\lim _{x \rightarrow 0} f(x)=\lambda+\mu a+v b$. Then $(\lambda+\mu+v)^{2}$ is equal to:

A
25
B
9
C
36
✓
16

✓

Answer

Correct option: D.

16

(D)
Sol. $\quad \lim _{x \rightarrow 0} f(x)=\left|\begin{array}{ccc}a+1 & 1 & b \\ a & 1+1 & b \\ a & 1 & b+1\end{array}\right|$
$=(a+1)(2(b+1)-b)+1(a b-a(b+1))+b a$
$=(a+1)(b+2)-a+a b$
$=b+a+2=\lambda+\mu a+\nu b$
$\lambda=2, \mu=1, v=1 \Rightarrow(\lambda+\mu+v)^{2}=16$

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