Consider the two functions $y = x ^ { 2 } + a x + a$ and $y = x + 1$.
(1) The number of points at which the graphs of the two functions meet depends on the relationship of $a$ with the numbers $\mathbf { Q }$ and $\mathbf { R }$ in the following way: (For $\mathbf { N } \sim \mathbf { P }$ choose which of (0) $\sim$ (2) gives the correct condition for the question.)
(i) The condition under which the graphs of the two functions intersect at two different points is $\mathbf { N }$.
(ii) The condition under which the graphs of the two functions are tangent at a point is $\mathbf{O}$.
(iii) The condition under which the graph of $y = x ^ { 2 } + a x + a$ is always above the graph of $y = x + 1$ is $\mathbf { P }$.
$$\begin{aligned} & \text { (0) } \mathrm { Q } < a < \mathrm { R } \\ & \text { (1) } a = \mathrm { Q } \text { or } a = \mathrm { R } \\ & \text { (2) } a < \mathrm { Q } \text { or } \mathrm { R } < a \end{aligned}$$
(2) Let us consider the case where the value of $a$ satisfies P. Let $g ( x )$ be the difference between the values of the two functions, so $g ( x ) = x ^ { 2 } + a x + a - ( x + 1 )$, and let $m$ be the minimum value of $g ( x )$. Then
$$m = - \frac { \mathbf { S } } { \mathbf { T } } \left( a ^ { 2 } - \mathbf { U } a + \mathbf { U } \right)$$
Hence $m$ takes the maximum at $a = \mathbf { W }$ and its value there is $m = \mathbf { W }$.

Consider the two functions $y = x ^ { 2 } + a x + a$ and $y = x + 1$.
(1) The number of points at which the graphs of the two functions meet depends on the relationship of $a$ with the numbers $\mathbf { Q }$ and $\mathbf { R }$ in the following way: (For $\mathbf { N } \sim \mathbf { P }$ choose which of (0) $\sim$ (2) gives the correct condition for the question.)
(i) The condition under which the graphs of the two functions intersect at two different points is $\mathbf { N }$.
(ii) The condition under which the graphs of the two functions are tangent at a point is $\mathbf{O}$.
(iii) The condition under which the graph of $y = x ^ { 2 } + a x + a$ is always above the graph of $y = x + 1$ is $\mathbf { P }$.
$$\begin{aligned} & \text { (0) } \mathrm { Q } < a < \mathrm { R } \\ & \text { (1) } a = \mathrm { Q } \text { or } a = \mathrm { R } \\ & \text { (2) } a < \mathrm { Q } \text { or } \mathrm { R } < a \end{aligned}$$
(2) Let us consider the case where the value of $a$ satisfies P. Let $g ( x )$ be the difference between the values of the two functions, so $g ( x ) = x ^ { 2 } + a x + a - ( x + 1 )$, and let $m$ be the minimum value of $g ( x )$. Then
$$m = - \frac { \mathbf { S } } { \mathbf { T } } \left( a ^ { 2 } - \mathbf { U } a + \mathbf { U } \right)$$
Hence $m$ takes the maximum at $a = \mathbf { W }$ and its value there is $m = \mathbf { W }$.

Let $a$ be a real number. Consider the quadratic expressions in $x$
$$\begin{aligned} & A = x^2 + ax + 1 \\ & B = x^2 + (a+3)x + 4 \end{aligned}$$
(1) The range of values taken by $a$ such that there exists a real number $x$ satisfying $A + B = 0$ is
$$a \leq -\sqrt{\mathbf{AB}} - \frac{\mathbf{C}}{\mathbf{D}} \text{ or } \sqrt{\mathbf{AB}} - \frac{\mathbf{C}}{\mathbf{D}} \leq a.$$
(2) The range of values taken by $a$ such that there exists a real number $x$ satisfying $AB = 0$ is
$$a \leq \mathbf{EF} \text{ or } \mathbf{G} \leq a.$$
(3) There exists a real number $x$ satisfying $A^2 + B^2 = 0$ only when $a = \mathbf{H}$. In this case $x = \mathbf{IJ}$.

Let us consider the three quadratic functions
$$f ( x ) = - x ^ { 2 } - 2 x + 1 , \quad g ( x ) = - x ^ { 2 } + 4 x , \quad h ( x ) = 2 x ^ { 2 } + a x + b$$
(1) When we denote the discriminant of the quadratic equation $h ( x ) - f ( x ) = 0$ by $D _ { 1 }$ and the discriminant of the quadratic equation $h ( x ) - g ( x ) = 0$ by $D _ { 2 }$, we have
$$D _ { 1 } = \mathbf { N } , \quad D _ { 2 } = \mathbf { O }$$
(for N and O, choose the correct answers from among choices (0) $\sim$ (5) below). (0) $a ^ { 2 } + 4 a - 3 b + 7$
(1) $a ^ { 2 } - 8 a - 12 b + 16$
(2) $a ^ { 2 } + 4 a - 12 b + 16$
(3) $a ^ { 2 } + 8 a + 12 b + 16$
(4) $a ^ { 2 } - 4 a + 12 b + 16$
(5) $a ^ { 2 } - 8 a - 3 b + 7$
(2) The values of $a$ and $b$ such that both of the two equations $f ( x ) = h ( x )$ and $g ( x ) = h ( x )$ have only one real solution are
$$a = \mathbf { P } , \quad b = \frac { \mathbf { Q } } { \mathbf{4} } .$$
In this case, the solution of $f ( x ) = h ( x )$ is $x = - \frac { \mathbf { S } } { \mathbf{T} }$ and the solution of $g ( x ) = h ( x )$ is $x = \frac { \mathbf { U } } { \mathbf{4} }$.
(3) Let $b = 3$. Then the range of the values of $a$ such that both $f ( x ) < h ( x )$ and $g ( x ) < h ( x )$ hold for any $x$ is $\square$ W (for $\square$ W, choose the correct answer from among choices (0) $\sim$ (5) below). (0) $- 2 - 2 \sqrt { 6 } < a < 10$
(1) $a < - 2 - 2 \sqrt { 6 } , 10 < a$
(2) $a < - 1 - \sqrt { 6 } , 10 < a$
(3) $- 2 < a < - 1 + \sqrt { 6 }$
(4) $- 2 < a < - 2 + 2 \sqrt { 6 }$
(5) $- 1 - \sqrt { 6 } < a < 10$

Q2 Let $a$ be a real number. For the two quadratic functions in $x$
$$\begin{aligned} & f ( x ) = x ^ { 2 } + 2 a x + a ^ { 2 } - a , \\ & g ( x ) = 4 - x ^ { 2 } , \end{aligned}$$
answer the following questions.
(1) The range of the values of $a$ such that the equation $f ( x ) = g ( x )$ has two different solutions is
$$- \mathbf { K } < a < \mathbf { L } .$$
(2) In the case of (1), the parabolas $y = f ( x )$ and $y = g ( x )$ intersect at two points. We are to find the range of the values of $a$ such that both of the $y$ coordinates of these points of intersection are positive.
First, let $h ( x ) = f ( x ) - g ( x )$. Since the solutions of the equation $f ( x ) = g ( x )$ are the $x$ coordinates of the points of intersection of parabolas $y = f ( x )$ and $y = g ( x )$, the solutions of $h ( x ) = 0$ have to be between $- \mathbf { M }$ and $\mathbf { N }$. Accordingly, we have
$$\begin{aligned} & h \left( - \mathbf { M } \right) = a ^ { 2 } - \mathbf { O } a + \mathbf { P } > 0 , \quad \cdots \cdots \cdots (1)\\ & h ( \mathbf { N } ) = a ^ { 2 } + \mathbf { Q } a + \mathbf { R } > 0 . \quad \cdots \cdots \cdots (2) \end{aligned}$$
Also, from the position of the axis of the parabola $y = h ( x )$ we have that
$$- \mathbf { S } < a < \mathbf { T } . \quad \cdots \cdots \cdots (3)$$
Therefore, from (1), (2), (3) and (4) we obtain
$$- \mathbf { U } < a < \mathbf{V}.$$