B2. Let

$$
f(x)=\ln x \quad \text { and } \quad g(x)=\frac{1}{3} x^{3}-x^{2}-6 x+4
$$

be two functions. Let $p$ be the tangent to the graph of $f$ at the point $S(a, b)$, and $q$ be the tangent to the graph of $g$ at the point $T(a, c)$. The tangent $p$ is perpendicular to the tangent $q$. Calculate the value of $a$.
B2.

Derivative of the function $f: f^{\prime}(x)=\frac{1}{x}$ 1 point

Derivative of the function $g: g^{\prime}(x)=x^{2}-2 x-6$

Considering $k_{1}=f^{\prime}(a)=\frac{1}{a}$ and $k_{2}=g^{\prime}(a)=a^{2}-2 a-6$ $1+1$ point
Considering $k_{1} k_{2}=-1$ or $\frac{a^{2}-2 a-6}{a}=-1$ ..... 1 point
Reformulating the equation to $a^{2}-a-6=0$ ..... 1 point
Writing the solutions of the quadratic equation $a_{1}=3$ and $a_{2}=-2$ ..... 1 point
Determining that the solution $a_{2}=-2$ is not valid ( $f(-2)$ does not exist) ..... 1 point