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

If $\lim_{x \to 2} \frac{\sin(x^3 - 5x^2 + ax + b)}{(\sqrt{x-1} - 1)\log_e(x-1)} = m$, then $a+b+m$ is equal to:

If $f(x) = \begin{cases} 3ax - 2b, & x > 1 \\ ax + b + 1, & x < 1 \end{cases}$ and $\lim_{x \rightarrow 1} f(x)$ exists,then the relation between $a$ and $b$ is

If $\lim _{x \rightarrow 3} \left( \frac{x^n - 3^n}{x - 3} \right) = 108$ and $n \in N$,then the value of $n$ is

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 . . . . . . .

If $\lim _{x \rightarrow 5} \frac{x^{k}-5^{k}}{x-5}=500$,then the value of $k$,where $k \in N$ is