Let $A$, $B$ and $C$ be unknown constants. Consider the function $f(x)$ defined by
$$\begin{aligned} f(x) &= Ax^2 + Bx + C \text{ when } x \leq 0 \\ &= \ln(5x + 1) \text{ when } x > 0 \end{aligned}$$
Write the values of the constants $A$, $B$ and $C$ such that $f''(x)$, i.e., the double derivative of $f$, exists for all real $x$. If this is not possible, write ``not possible''. If some of the constants cannot be uniquely determined, write ``not unique'' for each such constant.

The function $$f(x) = \begin{cases} x^3 + ax & (x < 1) \\ bx^2 + x + 1 & (x \geq 1) \end{cases}$$ is differentiable at $x = 1$. What is the value of $a + b$? (Here, $a, b$ are constants.) [4 points]
(1) 5
(2) 6
(3) 7
(4) 8
(5) 9

Let the function $f ( x ) = \left( x ^ { 2 } + 1 \right) \left| x ^ { 2 } - a x + 2 \right| + \cos | x |$ be not differentiable at the two points $x = \alpha = 2$ and $x = \beta$. Then the distance of the point $( \alpha , \beta )$ from the line $12 x + 5 y + 10 = 0$ is equal to :
(1) 5
(2) 4
(3) 3
(4) 2