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

$\begin{array}{l}\text{The slopes of both curves at }x=1\text{\hspace{0.17em} are }\\ m_1={\left[3x^2+a\right]}_{x=1}=a+3\\ m_2={\left[12x\right]}_{x=1}=12\\ \text{Since the curves touch each other at }x=1,\text{ their slopes are equal. }\\ \text{hence, }\\ a+3=12\\ a=9\\ \text{The ordinates must be same at }x=1\text{\hspace{0.17em} for both points on the curves, }\\ \text{because they touch each other}\\ 1+a-1=6+b\\ b+6=9\\ b=3\\ \text{Therefore, }a+b=12\end{array}$