If $A=\sin^{2}x+\cos^{4}x$, then for all real $x$
(1) $\frac{13}{16}\leq\mathrm{A}\leq 1$
(2) $1\leq A\leq 2$
(3) $\frac{3}{4}\leq\mathrm{A}\leq\frac{13}{16}$
(4) $\frac{3}{4}\leq\mathrm{A}\leq 1$

If $a \in R$ and the equation $- 3 ( x - [ x ] ) ^ { 2 } + 2 ( x - [ x ] ) + a ^ { 2 } = 0$ (where $[ x ]$ denotes the greatest integer $\leq x )$ has no integral solution, then all possible values of $a$ lie in the interval
(1) $( - 2 , - 1 )$
(2) $( - \infty , - 2 ) \cup ( 2 , \infty )$
(3) $( - 1,0 ) \cup ( 0,1 )$
(4) $( 1,2 )$

If non-zero real numbers $b$ and $c$ are such that $\min f ( x ) > \max g ( x )$, where $f ( x ) = x ^ { 2 } + 2 b x + 2 c ^ { 2 }$ and $g ( x ) = - x ^ { 2 } - 2 c x + b ^ { 2 } , ( x \in R )$; then $\left| \frac { c } { b } \right|$ lies in the interval
(1) $( \sqrt { 2 } , \infty )$
(2) $\left[ \frac { 1 } { 2 } , \frac { 1 } { \sqrt { 2 } } \right)$
(3) $\left( 0 , \frac { 1 } { 2 } \right)$
(4) $\left[ \frac { 1 } { \sqrt { 2 } } , \sqrt { 2 } \right]$

Q2 Consider the following three conditions (a), (b) and (c) on two real numbers $x$ and $y$:
(a) $x+y=5$ and $xy=3$,
(b) $x+y=5$ and $x^2+y^2=19$,
(c) $x^2+y^2=19$ and $xy=3$.
(1) Using the equality $x^2+y^2=(x+y)^2 - \square\mathbf{F}\, xy$, we see that
$$\text{condition (b) gives } xy = \mathbf{G},$$ $$\text{condition (c) gives } x+y = \mathbf{H} \text{ or } x+y = \mathbf{IJ}.$$
(2) For each of the following $\mathbf{K} \sim \mathbf{M}$, choose the most appropriate answer from among the choices (0)$\sim$(3) below.
(i) (a) is $\mathbf{K}$ for (b).
(ii) (b) is $\mathbf{L}$ for (c).
(iii) (c) is $\mathbf{M}$ for (a).
(0) a necessary and sufficient condition
(1) a sufficient condition but not a necessary condition
(2) a necessary condition but not a sufficient condition
(3) neither a necessary condition nor a sufficient condition

Q2 Consider the following three conditions (a), (b) and (c) on two real numbers $x$ and $y$:
(a) $x+y=5$ and $xy=3$,
(b) $x+y=5$ and $x^2+y^2=19$,
(c) $x^2+y^2=19$ and $xy=3$.
(1) Using the equality $x^2+y^2=(x+y)^2-\square\mathbf{F}\,xy$, we see that
condition (b) gives $xy=\mathbf{G}$,
condition (c) gives $x+y=\mathbf{H}$ or $x+y=\mathbf{IJ}$.
(2) For each of the following $\mathbf{K}\sim\mathbf{M}$, choose the most appropriate answer from among the choices (0)$\sim$(3) below.
(i) (a) is $\mathbf{K}$ for (b).
(ii) (b) is $\mathbf{L}$ for (c).
(iii) (c) is $\mathbf{M}$ for (a).
(0) a necessary and sufficient condition
(1) a sufficient condition but not a necessary condition
(2) a necessary condition but not a sufficient condition
(3) neither a necessary condition nor a sufficient condition

Let $a$ be a constant. Consider the quadratic function in $x$
$$y = 2 x ^ { 2 } + a x + 3 .$$
Suppose that the vertex of the graph of (1) is in the first (upper right-hand) quadrant.
(1) The range of values which $a$ can take is
$$\mathbf { A B } \sqrt { \mathbf { C } } < a < \mathbf { D } ,$$
and the least integer $a$ satisfying this inequality is $\mathbf{EF}$.
(2) Let $a = \mathrm { EF }$ in (1). Let
$$y = 2 x ^ { 2 } + p x + q$$
be the equation of the graph which is obtained by translating the graph of (1) by $- \frac { 1 } { n }$ in the $x$-direction and by $\frac { 6 } { n ^ { 2 } }$ in the $y$-direction. Then
$$p = \frac { \mathbf { G } } { n } - \mathbf { H } , \quad q = \frac { \mathbf { I } } { n ^ { 2 } } - \frac { \mathbf { J } } { n } + \mathbf { K } .$$
(3) The total number of natural numbers $n$ for which $p$ in (2) is an integer is $\mathbf { L }$. Among these $n$, consider the ones such that the value of $q$ is also an integer. Then $\mathbf { M }$ is the value of the $n$ for which $q$ takes the minimum value $\mathbf { N }$.

Suppose that $x$ and $y$ satisfy
$$3 x + y = 18 , \quad x \geqq 1 , \quad y \geqq 6$$
We are to find the maximum value and the minimum value of $x y$.
When we express $x y$ in terms of $x$, we have
$$x y = \mathbf { A B } ( x - \mathbf { C } ) ^ { 2 } + \mathbf { D E } .$$
Also, the range of values which $x$ can take is
$$\mathbf { F } \leq x \leqq \mathbf { G }$$
Hence, the value of $x y$ is maximized at $x = \mathbf { H }$ and its value there is $\mathbf { I J }$, and the value of $x y$ is minimized at $x = \mathbf { K }$ and its value there is $\mathbf { L M }$.

Consider two quadratic functions
$$\begin{aligned} & y = 2 x ^ { 2 } + 3 a x + 4 b \tag{1}\\ & y = b x ^ { 2 } + c x + d \tag{2} \end{aligned}$$
whose graphs are mutually symmetric with respect to the origin.
(1) From the symmetry with respect to the origin we see that
$$b = \mathbf { AB } , \quad c = \mathbf { C } a , \quad d = \mathbf { D } .$$
Hence (2) can be reduced to
$$y = \mathbf{AB} x ^ { 2 } + \mathbf{C} a x + \mathbf{D} . \tag{3}$$
(2) Let $0 < a < 1$, and consider the graph of (3).
When the range of values of $x$ is $0 \leqq x \leqq \frac { 3 } { 2 }$, the range of values of $y$ in (3) is
$$\frac { \mathbf { E } } { \mathbf { F } } a ^ { 2 } - \frac { \mathbf { G } } { \mathbf { H } } \leq y \leqq \frac { \mathbf { I } } { \mathbf{J}} a ^ { 2 } + \mathbf { K }$$
(3) For any value of $a$, the vertex of the graph of (3) is on the graph of the quadratic function
$$y = \mathbf { L } x ^ { 2 } + \mathbf { M } .$$

Let $a$ and $b$ be constants where $a > 0$. Translate the graph of the quadratic function
$$y = 4x^2 + 2ax + b$$
by $a$ in the $x$-direction and by $1 - 7a$ in the $y$-direction. If this graph passes through the point $(0, 4)$, we have
$$b = \mathbf{AB}\, a^2 + \mathbf{C}\, a + \mathbf{D},$$
and the quadratic function representing the graph resulting from these translations is
$$y = \mathbf{E}\, x^2 - \mathbf{F}\, ax + \mathbf{G}.$$
When the graph of quadratic function (1) is tangent to the $x$-axis, we have $a = \frac{\mathbf{H}}{\mathbf{I}}$, and the $x$-coordinate of the point of tangency is $x = \mathbf{J}$.

Let $a$ and $b$ be constants where $a > 0$. Translate the graph of the quadratic function
$$y = 4x^2 + 2ax + b$$
by $a$ in the $x$-direction and by $1 - 7a$ in the $y$-direction. If this graph passes through the point $(0, 4)$, we have
$$b = \mathbf{AB}\, a^2 + \mathbf{C}\, a + \mathbf{D}$$
and the quadratic function representing the graph resulting from these translations is
$$y = \mathbf{E}\, x^2 - \mathbf{F}\, ax + \mathbf{G}.$$
When the graph of quadratic function (1) is tangent to the $x$-axis, we have $a = \frac{\mathbf{H}}{\mathbf{I}}$, and the $x$-coordinate of the point of tangency is $x = \mathbf{J}$.

Let $a \neq 0$. Let $G$ be a curve which is symmetric with respect to the origin $(0,0)$ to the graph of the quadratic function in $x$
$$y = ax^2 - 4x - 4a. \tag{1}$$
(1) The coordinates of the vertex of the graph of (1) are
$$\left( \frac{\mathbf{A}}{a}, -\frac{\mathbf{B}}{a} - 4a \right).$$
(2) Among the following choices, the quadratic function whose graph is $G$ is $\square$ C. (0) $y = ax^2 + 4x + 4a$
(1) $y = ax^2 + 4x - 4a$
(2) $y = ax^2 - 4x + 4a$
(3) $y = -ax^2 + 4x + 4a$
(4) $y = -ax^2 - 4x + 4a$
(5) $y = -ax^2 - 4x - 4a$
(3) The curve $G$ intersects the graph of the quadratic function (1) at the two points
$$(\mathrm{DE},\ \mathrm{F}) \text{ and } (\mathrm{G},\ \mathrm{HI}).$$
(4) Let $a = 2$. Then over the interval $\mathrm{DE} \leqq x \leqq \square$, the maximum and the minimum values of the quadratic function whose graph is $G$ are $\square$ JK and $\square$ LM, respectively.

Let $x$ and $y$ be real numbers which satisfy
$$3x^2 + 2xy + 3y^2 = 32. \tag{1}$$
Then we are to find the ranges of the values of $x + y$ and $xy$.
First, we set
$$a = x + y. \tag{2}$$
By eliminating $y$ from (1) and (2), we obtain the quadratic equation in $x$
$$\mathbf{A}x^2 - \mathbf{B}ax + \mathbf{C}a^2 - 32 = 0.$$
Since $x$ is a real number, we have
$$\mathbf{DE} \leqq a \leqq \mathbf{F}.$$
Next, we set
$$b = xy. \tag{4}$$
From (1), (2) and (4) we obtain
$$b = \frac{\mathbf{G}}{\mathbf{H}}a^2 - \mathbf{I}.$$
Hence from (3) and (5) we have
$$\mathbf{JK} \leqq b \leqq \mathbf{L}.$$

Suppose we have a quadratic function $y = a x ^ { 2 } + b x + c$ in $x$ which satisfies the following conditions 【＊】：
【＊】 When $x = - 1$ ，then $y = - 8$ and when $x = 3$ ，then $y = 16$ ．Further，in the interval $- 1 \leqq x \leqq 3$ ，the value of $y$ increases with the increase of the value of $x$ ．
We are to find the conditions which $a , b$ and $c$ must satisfy．
From【＊】，it follows that $b$ and $c$ can be expressed in terms of $a$ as
$$\begin{aligned} & b = \mathbf { A B } a + \mathbf { A } \\ & c = \mathbf { D E } a - \mathbf { F } . \end{aligned}$$
Hence，the axis of symmetry of the graph of this quadratic function has the equation
$$x = \mathbf { G } - \frac { \mathbf { H } } { a } .$$
Thus $a , b$ and $c$ must satisfy the relationships（1）and（2），and furthermore
$$0 < a \leqq \frac { \mathbf { I } } { \mathbf { J } } \quad \text { or } \frac { \mathbf { K L } } { \mathbf { M } } \leqq a < 0 .$$

Suppose we have a quadratic function $y = a x ^ { 2 } + b x + c$ in $x$ which satisfies the following conditions 【＊】：
【＊】 When $x = - 1$ ，then $y = - 8$ and when $x = 3$ ，then $y = 16$ ．Further，in the interval $- 1 \leqq x \leqq 3$ ，the value of $y$ increases with the increase of the value of $x$ ．
We are to find the conditions which $a , b$ and $c$ must satisfy．
From【＊】，it follows that $b$ and $c$ can be expressed in terms of $a$ as
$$\begin{aligned} & b = \mathbf { A B } a + \mathbf { C } \\ & c = \mathbf { D E } a - \mathbf { F } . \end{aligned}$$
Hence，the axis of symmetry of the graph of this quadratic function has the equation
$$x = \mathbf { G } - \frac { \mathbf { H } } { a } .$$
Thus $a , b$ and $c$ must satisfy the relationships（1）and（2），and furthermore
$$0 < a \leqq \frac { \mathbf { I } } { \mathbf { J } } \quad \text { or } \frac { \mathbf { K L } } { \mathbf { M } } \leqq a < 0 .$$

Consider the quadratic function
$$y = - x ^ { 2 } - a x + 3 .$$
(1) If $a > 0$ and the maximum value of function (1) is 7 , then $a = \square$. In this case, the equation of the axis of symmetry of the graph is $x = \mathbf { B C }$, and the $x$-coordinates of the points of intersection of this graph and the $x$-axis are $\mathbf { D E } \pm \sqrt { \mathbf { F } }.$
(2) If the curve obtained by translating the graph of function (1) by 2 in the $x$-direction and by $-3$ in the $y$-direction passes through $( - 3 , - 5 )$, then $a =$ $\square$ G.

Consider the quadratic function
$$y = - x ^ { 2 } - a x + 3 .$$
(1) If $a > 0$ and the maximum value of function (1) is 7 , then $a = \square$. In this case, the equation of the axis of symmetry of the graph is $x = \mathbf { B C }$, and the $x$-coordinates of the points of intersection of this graph and the $x$-axis are $\mathbf { D E } \pm \sqrt { \mathbf { F } }.$
(2) If the curve obtained by translating the graph of function (1) by 2 in the $x$-direction and by $-3$ in the $y$-direction passes through $( - 3 , - 5 )$, then $a =$ $\square$ G.

Q1 A quadratic function $y = ax^2 + bx + \frac{3}{a}$ satisfies the following two conditions:
(i) $y$ is maximized at $x = 3$,
(ii) the value of $y$ at $x = 1$ is 2.
We are to find the values of $a$ and $b$.
Using conditions (i) and (ii), we obtain the following relationships between $a$ and $b$:
$$\left\{ \begin{aligned} b & = \mathbf{AB}a \\ \mathbf{C} & = a + b + \frac{\mathbf{D}}{a}. \end{aligned} \right.$$
From these two equalities, we have the equation
$$\mathbf{E}a^2 + \mathbf{F}a - \mathbf{G} = 0$$
and hence
$$a = \mathbf{HI}, \quad b = \mathbf{J}.$$
Thus the maximum value of this function is $\mathbf{K}$.

Consider a quadratic function in $x$
$$y = ax^2 + bx + c$$
such that the graph of function (1) passes through the two points $(-1, -1)$ and $(2, 2)$.
(1) When we express $b$ and $c$ in terms of $a$, we have
$$b = \mathbf{A} - a, \quad c = \mathbf{BC}a.$$
(2) Suppose that one of the points of intersection of the graph of function (1) and the $x$-axis is within the interval $0 < x \leqq 1$. Then the range of values of $a$ is [see figure].
(3) When the value of $a$ varies within interval (2), the range of values of $a + bc$ is
$$\frac{\mathbf{GH}}{\square\mathbf{I}} \leqq a + bc \leqq \square.$$

Q1 A quadratic function $y = ax^2 + bx + \frac{3}{a}$ satisfies the following two conditions:
(i) $y$ is maximized at $x = 3$,
(ii) the value of $y$ at $x = 1$ is 2.
We are to find the values of $a$ and $b$.
Using conditions (i) and (ii), we obtain the following relationships between $a$ and $b$:
$$\left\{ \begin{aligned} b & = \mathbf{AB}a \\ \mathbf{C} & = a + b + \frac{\mathbf{D}}{a}. \end{aligned} \right.$$
From these two equalities, we have the equation
$$\mathbf{E}a^2 + \mathbf{F}a - \mathbf{G} = 0$$
and hence
$$a = \mathbf{HI}, \quad b = \mathbf{J}.$$
Thus the maximum value of this function is $\mathbf{K}$.

Let $a$ and $b$ be real numbers, where $a > 0$. Consider the two quadratic functions
$$f ( x ) = 2 x ^ { 2 } - 4 x + 5 , \quad g ( x ) = x ^ { 2 } + a x + b .$$
We are to find the values of $a$ and $b$ when the function $g ( x )$ satisfies the following two conditions.
(i) The minimum value of $g ( x )$ is 8 less than the minimum value of $f ( x )$.
(ii) There exists only one $x$ which satisfies $f ( x ) = g ( x )$.
Since the minimum value of $f ( x )$ is $\mathbf { A }$, from condition (i), we derive the equality
$$b = \frac { a ^ { 2 } } { \mathbf { B } } - \mathbf { C } \text {. }$$
Hence the equation from which we can find the $x$ satisfying $f ( x ) = g ( x )$ is
$$x ^ { 2 } - ( a + \mathbf { D } ) x - \frac { a ^ { 2 } } { \mathbf { E } } + \mathbf { F G } = 0 .$$
Thus, since $a > 0$, from condition (ii) we obtain
$$a = \mathbf { H } , \quad b = \mathbf { I J } .$$
In this case, the $x$ satisfying $f ( x ) = g ( x )$ is $\square \mathbf{ K }$.

Suppose that an integer $x$ and a real number $y$ satisfy both the equation
$$2 ( y + 1 ) = x ( 8 - x ) \tag{1}$$
and the inequality
$$5 x - 4 y + 1 \leqq 0 . \tag{2}$$
We are to find $M$, the maximum value of $y$, and $m$, the minimum value of $y$.
First of all, let us transform (1) into
$$y = - \frac { 1 } { \mathbf { P } } ( x - \mathbf { Q } ) ^ { 2 } + \mathbf{R} .$$
Also, from (1) and (2) we obtain the inequality in $x$
$$2 x ^ { 2 } - \mathbf { S T } x + \mathbf { U V } \leqq 0 . \tag{3}$$
Thus when $x$ is an integer satisfying (3) if we consider the range of values which $y$ can take, we see that $y$ is maximized at $x = \square \mathbf{ V }$ and is minimized at $x = \square \mathbf{ W }$, and hence that
$$M = \mathbf { X } , \quad m = \frac { \mathbf { Y } } { \mathbf{Z} } .$$

Let $a$ and $b$ be real numbers, where $a > 0$. Consider the two quadratic functions
$$f ( x ) = 2 x ^ { 2 } - 4 x + 5 , \quad g ( x ) = x ^ { 2 } + a x + b .$$
We are to find the values of $a$ and $b$ when the function $g ( x )$ satisfies the following two conditions.
(i) The minimum value of $g ( x )$ is 8 less than the minimum value of $f ( x )$.
(ii) There exists only one $x$ which satisfies $f ( x ) = g ( x )$.
Since the minimum value of $f ( x )$ is $\mathbf{A}$, from condition (i), we derive the equality
$$b = \frac { a ^ { 2 } } { \mathbf { B } } - \mathbf { C } \text {. }$$
Hence the equation from which we can find the $x$ satisfying $f ( x ) = g ( x )$ is
$$x ^ { 2 } - ( a + \mathbf { D } ) x - \frac { a ^ { 2 } } { \mathbf { E } } + \mathbf { F G } = 0 .$$
Thus, since $a > 0$, from condition (ii) we obtain
$$a = \mathbf { H } , \quad b = \mathbf { I J } .$$
In this case, the $x$ satisfying $f ( x ) = g ( x )$ is $\mathbf { K }$.

Let $a$ and $b$ be real numbers where $0 < b < 7$. Let us consider the maximum value $M$ and the minimum value $m$ of the quadratic function
$$f ( x ) = x ^ { 2 } - 6 x + a$$
over the interval $b \leqq x \leqq 7$.
The function $f ( x )$ can be represented as
$$f ( x ) = ( x - \mathbf { A } ) ^ { 2 } + a - \mathbf { B } .$$
(1) For each of $\mathbf { C }$ ~ $\mathbf { G }$ in the following statements, choose the correct answer from among (0) ~ (9) below.
We are to find $M$ and $m$. There are two cases.
(i) When $0 < b \leqq \mathbf { C }$, then
$$M = \mathbf { D } , \quad m = \mathbf { E } .$$
(ii) When $\mathbf { C } < b < 7$, then
$$M = \mathbf { F } , \quad m = \mathbf { G } .$$
(0) 0
(1) 1
(2) 2
(3) 3
(4) $a - 6$
(5) $a + 7$ (6) $a + 8$ (7) $a - 9$ (8) $b ^ { 2 } - 6 b + a$ (9) $b ^ { 2 } + 6 b + a$
(2) In the case that $M = 13$ and $m = 1$, we have
$$a = \mathbf { H } , \quad b = \mathbf { I } .$$

Let $a$ and $b$ be real numbers where $0 < b < 7$. Let us consider the maximum value $M$ and the minimum value $m$ of the quadratic function
$$f ( x ) = x ^ { 2 } - 6 x + a$$
over the interval $b \leqq x \leqq 7$.
The function $f ( x )$ can be represented as
$$f ( x ) = ( x - \mathbf { A } ) ^ { 2 } + a - \mathbf { B } .$$
(1) For each of $\mathbf { C }$ ~ $\mathbf { G }$ in the following statements, choose the correct answer from among (0) ~ (9) below.
We are to find $M$ and $m$. There are two cases.
(i) When $0 < b \leqq \mathbf { C }$, then
$$M = \mathbf { D } , \quad m = \mathbf { E } .$$
(ii) When $\mathbf { C } < b < 7$, then
$$M = \mathbf { F } , \quad m = \mathbf { G } .$$
(0) 0
(1) 1
(2) 2
(3) 3
(4) $a - 6$
(5) $a + 7$ (6) $a + 8$ (7) $a - 9$ (8) $b ^ { 2 } - 6 b + a$ (9) $b ^ { 2 } + 6 b + a$
(2) In the case that $M = 13$ and $m = 1$, we have
$$a = \mathbf { H } , \quad b = \mathbf { I } .$$

Consider the quadratic function in $x$
$$y = - \frac { 1 } { 8 } x ^ { 2 } + a x + b .$$
When we denote the coordinates of the vertex of the graph of (1) by $( p , q )$, we have
$$p = \mathbf { A } a , \quad q = \mathbf { B } a ^ { 2 } + b .$$
(1) When the vertex ( $p , q$ ) is on the straight line $x + y = 1 , a$ and $b$ satisfy
$$b = \mathbf { C D } a ^ { 2 } - \mathbf { E E } a + \mathbf { F } ,$$
and so $8 a + b$ is maximized at $a = \mathbf { G }$, and the maximum value is $\mathbf { H }$.
(2) When the graph of (1) is tangent to the $x$-axis, the range of values of $a + b$ is
$$a + b \leqq \frac { \mathbf { I } } { \mathbf { J } }$$