$$ \text { The value of } \lim\limits_{x \rightarrow 0}\left(\frac{x^2 \sin ^2 x}{x^2-\sin ^2 x}\right) \text { is : } $$

Let $f(x)=\lim \limits_{y \rightarrow 0} \frac{(1-\cos (x y)) \tan (x y)}{y^3}$. Then the number of solutions of the equation $f(x)=\sin x$, $x \in \mathbf{R}$ is :

Let $f(x)$ and $g(x)$ be twice differentiable functions satisfying $f^{\prime \prime}(x)=g^{\prime \prime}(x)$ for all $x \in \mathbf{R}, f^{\prime}(1)=2 g^{\prime}(1)=4$ and $g(2)=3 f(2)=9$. Then $f(25)-g(25)$ is equal to :

The product of all possible values of $\alpha$, for which

$\lim \limits_{x \rightarrow 0}\left(\frac{1-\cos (\alpha x) \cos ((\alpha+1) x) \cos ((\alpha+2) x)}{\sin ^2((\alpha+1) x)}\right)=2$, is :