For each of $\mathbf { A } \sim \mathbf { M }$ in the following statements, choose the correct answer from among (0) ~ (9) at the bottom of this page. We are to solve the following simultaneous inequalities $$\left\{ \begin{aligned}
x ^ { 2 } - 2 x < 3 & \cdots \cdots \cdots (1)\\
a x ^ { 2 } - a x - x + 1 > 0 , & \cdots \cdots \cdots (2)
\end{aligned} \right.$$ where $0 < a < 1$. When we solve (1), we have $$\mathbf { A } < x < \mathbf { B } .$$ Next, when we transform (2), we have $$( a x - \mathbf { C } ) ( x - \mathbf { D } ) > 0 .$$ Hence, noting that $0 < a < 1$, we see that the solution of (2) is $$x < \mathbf { E } \text { or } \mathbf { F } < x .$$ Thus, when $0 < a \leqq \mathbf { G }$, the solution of the simultaneous inequalities is $$\mathbf { H } < x < \mathbf { I }$$ and when $\mathbf { G } < a < 1$, the solution is $$\mathbf { J } < x < \mathbf { K } \text { or } \mathbf { L } < x < \mathbf { M }$$ where $\mathbf { K } < \mathbf { M }$. (0) 0 (1) 1 (2) 2 (3) 3 (4) $- 1$ (5) $\frac { 1 } { 2 }$ (6) $\frac { 1 } { 3 }$ (7) $\frac { 1 } { a }$ (8) $\frac { 2 } { a }$ (9) $\frac { 3 } { a }$
For each of $\mathbf { A } \sim \mathbf { M }$ in the following statements, choose the correct answer from among (0) ~ (9) at the bottom of this page.
We are to solve the following simultaneous inequalities
$$\left\{ \begin{aligned}
x ^ { 2 } - 2 x < 3 & \cdots \cdots \cdots (1)\\
a x ^ { 2 } - a x - x + 1 > 0 , & \cdots \cdots \cdots (2)
\end{aligned} \right.$$
where $0 < a < 1$.
When we solve (1), we have
$$\mathbf { A } < x < \mathbf { B } .$$
Next, when we transform (2), we have
$$( a x - \mathbf { C } ) ( x - \mathbf { D } ) > 0 .$$
Hence, noting that $0 < a < 1$, we see that the solution of (2) is
$$x < \mathbf { E } \text { or } \mathbf { F } < x .$$
Thus, when $0 < a \leqq \mathbf { G }$, the solution of the simultaneous inequalities is
$$\mathbf { H } < x < \mathbf { I }$$
and when $\mathbf { G } < a < 1$, the solution is
$$\mathbf { J } < x < \mathbf { K } \text { or } \mathbf { L } < x < \mathbf { M }$$
where $\mathbf { K } < \mathbf { M }$.
(0) 0\\
(1) 1\\
(2) 2\\
(3) 3\\
(4) $- 1$\\
(5) $\frac { 1 } { 2 }$\\
(6) $\frac { 1 } { 3 }$\\
(7) $\frac { 1 } { a }$\\
(8) $\frac { 2 } { a }$\\
(9) $\frac { 3 } { a }$