Q1 For A $\sim$ K in the following sentences, choose the correct answer from among choices (0) $\sim$ (9) below. (1) Consider the quadratic function
$$y = a x ^ { 2 } + b x + c$$
whose graph is as shown in the figure at the right.
Then $a , b$ and $c$ satisfy the following expressions:
(i) $a \mathbf { A } 0 , b \mathbf { B } 0 , c \mathbf { C } 0$;
(ii) $a + b + c \mathbf { D } 0$;
(iii) $a - b + c \mathbf { E } 0$;
(iv) $4 a + 2 b + c \mathbf { F } 0$;
(v) $b ^ { 2 } - 4 a c \mathbf { G } 0$.
(2) Given the condition that $a , b$ and $c$ satisfy (i) and (ii) in (1), consider the case where the value of $a ^ { 2 } - 8 b - 8 c$ is minimized.
We see that $a = \mathbf { H }$. When we express $y = a x ^ { 2 } + b x + c$ in terms of $b$, we have
$$y = \mathbf { H } x ^ { 2 } + b x - b + \mathbf { I } \text {. }$$
Also, we see that the range of the values of $b$ is $\mathbf { J } < b < \mathbf { K }$. (0) 0
(1) 1
(2) 2
(3) 3
(4) 4
(5) - 2 (6) - 4 (7) $>$ (8) $=$ (9) $<$

(Course 2) Q1 For A $\sim$ K in the following sentences, choose the correct answer from among choices (0) $\sim$ (9) below. (1) Consider the quadratic function
$$y = a x ^ { 2 } + b x + c$$
whose graph is as shown in the figure at the right.
Then $a , b$ and $c$ satisfy the following expressions:
(i) $a \mathbf { A } 0 , b \mathbf { B } 0 , c \mathbf { C } 0$;
(ii) $a + b + c \mathbf { D } 0$;
(iii) $a - b + c \mathbf { E } 0$;
(iv) $4 a + 2 b + c \mathbf { F } 0$;
(v) $b ^ { 2 } - 4 a c \mathbf { G } 0$.
(2) Given the condition that $a , b$ and $c$ satisfy (i) and (ii) in (1), consider the case where the value of $a ^ { 2 } - 8 b - 8 c$ is minimized.
We see that $a = \mathbf { H }$. When we express $y = a x ^ { 2 } + b x + c$ in terms of $b$, we have
$$y = \mathbf { H } x ^ { 2 } + b x - b + \mathbf { I } \text {. }$$
Also, we see that the range of the values of $b$ is $\mathbf { J } < b < \mathbf { K }$. (0) 0
(1) 1
(2) 2
(3) 3
(4) 4
(5) - 2 (6) - 4 (7) $>$ (8) $=$ (9) $<$

$\mathrm { f } ( x )$ is a polynomial function defined for all real $x$.
Which of the following is a necessary condition for the inequality
$$\frac { \mathrm { f } ( a ) + \mathrm { f } ( b ) } { 2 } \geq \mathrm { f } \left( \frac { a + b } { 2 } \right)$$
to be true for all real numbers $a$ and $b$ with $a < b$ ?

Here is an attempt to solve the inequality $x ^ { 4 } - 2 x ^ { 2 } - 3 < 0$ by completing the square:
$$x ^ { 4 } - 2 x ^ { 2 } - 3 < 0$$
I if and only if $x ^ { 4 } - 2 x ^ { 2 } + 1 < 4$ II if and only if $\left( x ^ { 2 } - 1 \right) ^ { 2 } < 4$ III if and only if $- 2 < x ^ { 2 } - 1 < 2$ IV if and only if $x ^ { 2 } - 1 < 2$ V if and only if $x ^ { 2 } < 3$ VI if and only if $- \sqrt { 3 } < x < \sqrt { 3 }$
Which of the following statements is true? A The argument is completely correct. B The first error occurs in line I. C The first error occurs in line II. D The first error occurs in line III. E The first error occurs in line IV. F The first error occurs in line V. G The first error occurs in line VI.

Where $a, b$ and $c$ are real numbers,
$$y = ax^2 + bx + c$$
the parabola intersects the line $y = 1$ at points B and C, and intersects the line $y = 6$ at only point A. The locations of points A, B and C in the rectangular coordinate plane are shown in the figure below.
Accordingly, what are the signs of the numbers $a$, $b$ and $c$ respectively?
A) +, -, -
B) +, +, -
C) -, +, +
D) -, -, +
E) -, -, -