#### Provide solution for RD Sharma maths class 12 chapter Determinants exercise multiple choise question 30

Correct option (a)

Hint:

Solve the determinant by applying row operation then use limit.

Given:

Given that,

$f(x)=\left|\begin{array}{ccc} \cos x & x & 1 \\ 2 \sin x & x & 2 x \\ \sin x & x & x \end{array}\right|$

We have to find

$\lim_{x\rightarrow 0}\frac{f(x)}{x^{2}}$

Solution:

Here, we have

$f(x)=\left|\begin{array}{ccc} \cos x & x & 1 \\ 2 \sin x & x & 2 x \\ \sin x & x & x \end{array}\right|$

Applying R2 → R2 - R3

$f(x)=\left|\begin{array}{ccc} \cos x & x & 1 \\ \sin x & 0 & x \\ \sin x-cos\, x & 0 & x-1 \end{array}\right|$

Expanding along column 2

\begin{aligned} &f(x)=-x[\sin \sin x \cdot(x-1)-x(\sin \sin x-\cos \cos x)] \\ &f(x)=-x[x x-\sin \sin x-x x+x x] \\ &f(x)=-x[x \cos x-\sin x] \\ &f(x)=x[\sin x-x \cos x] \end{aligned}

Again,

\begin{aligned} \lim _{x \rightarrow 0} \frac{f(x)}{x^{2}} &=\lim _{x \rightarrow 0} \frac{x(\sin x-x \cos x)}{x^{2}} \\ &=\lim _{x \rightarrow 0}\left(\frac{\sin x}{x}-\cos x\right) \\ \lim _{x \rightarrow 0} \frac{f(x)}{x^{2}} &=\lim _{x \rightarrow 0} \frac{\sin x}{x}-\lim _{x \rightarrow 0} \cos x \\ &=1-1=0 \end{aligned}

$Hence, \: \: \lim_{x\rightarrow 0}\frac{f(x)}{x^{2}} =0$