The largest value of the non-negative integer $a$ for which $\lim _{x \rightarrow 1}\left\{\frac{-a x+\sin (x-1)+a}{x+\sin (x-1)-1}\right\}^{\frac{1-x}{1-\sqrt{x}}}=\frac{1}{4}$ is

Let $\alpha(a)$ and $\beta(a)$ be the roots of the equation $(\sqrt[3]{1+a}-1) x^2+(\sqrt{1+a}-1) x+(\sqrt[6]{1+a}-1)=0$ where $a > -1$. Then $\lim _{a \rightarrow 0^{+}} \alpha(a)$ and $\lim _{a \rightarrow 0^{+}} \beta(a)$ are

Let $\alpha$ and $\beta$ be the roots of $ax^2 + bx + c = 0$,then $\lim_{x \to \alpha} \frac{1 - \cos(ax^2 + bx + c)}{(x - \alpha)^2}$ is equal to

For $t > -1$,let $\alpha_t$ and $\beta_t$ be the roots of the equation $\left((t+2)^{\frac{1}{7}}-1\right) x^2+\left((t+2)^{\frac{1}{6}}-1\right) x+\left((t+2)^{\frac{1}{21}}-1\right)=0$. If $\lim _{t \rightarrow -1^{+}} \alpha_t$ and $\lim _{t \rightarrow -1^{+}} \beta_t$ are the roots of the limiting equation,and $a+b$ is the sum of these roots,then $72(a+b)^2$ is equal to . . . . . . .

The values of $a$ and $b$ such that $\mathop {\lim }\limits_{x \to 0} \frac{{x(1 + a\cos x) - b\sin x}}{{{x^3}}} = 1$ are:

If $\lim _{x \rightarrow 0} \frac{2 a \sin x-\sin 2 x}{\tan ^{3} x}$ exists and is equal to $1$,then the value of $a$ is