$\text{ The two curves }y=x^3+ax-1\text{ and }y=6x^2+b\text{ touch each other at a point having abscissca }1.\text{ Then }a+b=$

$\begin{array}{l}\text{ Both the curves satisfied by the point }\left(1,y_1\right)\\ \text{ So, }1+a-1=6+b\Rightarrow a-b=6\\ \text{ Also }{\left.\frac{\text{dy}}{\text{dx}}\right|}_{\text{curve }1}={\left.\frac{\text{dy}}{\text{dx}}\right|}_{\text{curve }2}\text{ at x}=1\\ {\left[3{\text{x}}^2+\text{a}\right]}_{x=1}={\left[12x\right]}_{x=1}\Rightarrow \text{a}=9\\ \text{a}=9\text{ and b}=3\end{array}$