For a real number $a$, let $f ( a )$ be the number of elements in the set $$\left\{ x \mid a x ^ { 2 } + 2 ( a - 2 ) x - ( a - 2 ) = 0 , x \text { is a real number } \right\}$$ Which of the following statements in are correct? [3 points]
Remarks ᄀ. $\lim _ { a \rightarrow 0 } f ( a ) = f ( 0 )$ ㄴ. There are 2 real numbers $c$ such that $\lim _ { a \rightarrow c + 0 } f ( a ) \neq \lim _ { a \rightarrow c - 0 } f ( a )$. ㄷ. The function $f ( a )$ is discontinuous at 3 points.
(1) ᄂ
(2) ᄃ
(3) ᄀ, ᄂ
(4) ㄴ,ㄷ
(5) ᄀ, ᄂ, ᄃ

Find the product of all real roots of the irrational equation $\sqrt { x ^ { 2 } - 7 x + 15 } = x ^ { 2 } - 7 x + 9$. [3 points]

For the irrational equation $$\sqrt { 4 x ^ { 2 } - 5 x + 7 } - 4 x ^ { 2 } + 5 x = 1$$ What is the product of all real roots? [3 points]
(1) $- \frac { 1 } { 2 }$
(2) $- \frac { 3 } { 2 }$
(3) $- \frac { 5 } { 2 }$
(4) $- \frac { 7 } { 2 }$
(5) $- \frac { 9 } { 2 }$

Find the product of all real roots of the irrational equation $\sqrt { 2 x ^ { 2 } - 6 x } = x ^ { 2 } - 3 x - 4$, and call it $k$. Find the value of $k ^ { 2 }$. [3 points]

For the function $f ( x ) = x ( x + 1 ) ( x - 4 )$, answer the following. For the matrix $A = \left( \begin{array} { l l } 2 & 1 \\ 0 & 3 \end{array} \right)$, what is the sum of all constant values $a$ that satisfy $A \binom { 0 } { f ( a ) } = \binom { 0 } { 0 }$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For the irrational equation $x ^ { 2 } - 6 x - \sqrt { x ^ { 2 } - 6 x - 1 } = 3$, let $k$ be the product of all real roots. Find the value of $k ^ { 2 }$. [3 points]

A quadratic function $f ( x )$ with leading coefficient 1 satisfies $$\lim _ { x \rightarrow a } \frac { f ( x ) - ( x - a ) } { f ( x ) + ( x - a ) } = \frac { 3 } { 5 }$$ When the two roots of the equation $f ( x ) = 0$ are $\alpha$ and $\beta$, what is the value of $| \alpha - \beta |$? (Here, $a$ is a constant.) [4 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For two conditions on real number $x$: $$\begin{aligned} & p : x = a , \\ & q : 3 x ^ { 2 } - a x - 32 = 0 \end{aligned}$$ What is the value of positive $a$ such that $p$ is a sufficient condition for $q$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

8. If $a$ and $b$ are two distinct zeros of the function $f ( x ) = x ^ { 2 } - p x + q$ ($p > 0$, $q > 0$), and the three numbers $a$, $b$, and $-2$ can be arranged to form an arithmetic sequence, and can also be arranged to form a geometric sequence, then the value of $p + q$ equals
A. 6
B. 7
C. 8
D. 9

Let $a , b > 0 , a + b = 5$. The maximum value of $\sqrt { a + 1 } + \sqrt { b + 3 }$ is $\_\_\_\_$ .

20. (15 points) Let the function $f ( x ) = x ^ { 2 } + a x + b , ( a , b \in R )$ .
(1) When $b = \frac { a ^ { 2 } } { 4 } + 1$ , find the expression for the minimum value $g ( a )$ of the function $f ( x )$ on $[ - 1,1 ]$ ;
(2) Given that the function $f ( x )$ has a zero on $[ - 1,1 ]$ , $0 \leq b - 2 a \leq 1$ , find the range of values for $b$ .

The Great Pyramid of Egypt is one of the ancient wonders of the world. Its shape can be viewed as a regular square pyramid. The area of a square with side length equal to the height of the pyramid equals the area of one lateral triangular face of the pyramid. The ratio of the height of a lateral triangular face to the side length of the base square is
A. $\frac { \sqrt { 5 } - 1 } { 4 }$
B. $\frac { \sqrt { 5 } - 1 } { 2 }$
C. $\frac { \sqrt { 5 } + 1 } { 4 }$
D. $\frac { \sqrt { 5 } + 1 } { 2 }$

The Great Pyramid of Khufu in Egypt is one of the ancient wonders of the world. Its shape can be viewed as a regular square pyramid. The area of a square with side length equal to the height of the pyramid equals the area of one lateral triangular face of the pyramid. Then the ratio of the height of the lateral triangle to the base of the square is
A. $\frac { \sqrt { 5 } - 1 } { 4 }$
B. $\frac { \sqrt { 5 } - 1 } { 2 }$
C. $\frac { \sqrt { 5 } + 1 } { 4 }$
D. $\frac { \sqrt { 5 } + 1 } { 2 }$

Let $a , b$ and $c$ be three real numbers. Then the equation $\frac { 1 } { x - a } + \frac { 1 } { x - b } + \frac { 1 } { x - c } = 0$
(A) always have real roots.
(B) can have real or complex roots depending on the values of $a , b$ and $c$.
(C) always have real and equal roots.
(D) always have real roots, which are not necessarily equal.

Let $a, b, c$ be positive integers such that $\frac{b}{a}$ is an integer. If $a, b, c$ are in geometric progression and the arithmetic mean of $a, b, c$ is $b + 2$, then the value of $$\frac{a^2 + a - 14}{a + 1}$$ is

The number of 5 digit numbers which are divisible by 4, with digits from the set $\{ 1,2,3,4,5 \}$ and the repetition of digits is allowed, is $\_\_\_\_$.

Suppose $a , b$ denote the distinct real roots of the quadratic polynomial $x ^ { 2 } + 20 x - 2020$ and suppose $c , d$ denote the distinct complex roots of the quadratic polynomial $x ^ { 2 } - 20 x + 2020$. Then the value of
$$a c ( a - c ) + a d ( a - d ) + b c ( b - c ) + b d ( b - d )$$
is
(A) 0
(B) 8000
(C) 8080
(D) 16000

If $a , b , c \in \mathrm { R }$ and 1 is a root of equation $a x ^ { 2 } + b x + c = 0$, then the curve $y = 4 a x ^ { 2 } + 3 b x + 2 c , a \neq 0$ intersect $x$-axis at
(1) two distinct points whose coordinates are always rational numbers
(2) no point
(3) exactly two distinct points
(4) exactly one point

If the sum of the square of the roots of the equation $x^{2} - (\sin\alpha - 2)x - (1+\sin\alpha) = 0$ is least, then $\alpha$ is equal to
(1) $\frac{\pi}{6}$
(2) $\frac{\pi}{4}$
(3) $\frac{\pi}{3}$
(4) $\frac{\pi}{2}$

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 $\frac { 1 } { \sqrt { \alpha } } , \frac { 1 } { \sqrt { \beta } }$ are the roots of the equation $a x ^ { 2 } + b x + 1 = 0 , ( a \neq 0 , a , b \in R )$, then the equation $x \left( x + b ^ { 3 } \right) + \left( a ^ { 3 } - 3 a b x \right) = 0$ has roots:
(1) $\sqrt { \alpha \beta }$ and $\alpha \beta$
(2) $\alpha ^ { - \frac { 3 } { 2 } }$ and $\beta ^ { - \frac { 3 } { 2 } }$
(3) $\alpha \beta ^ { \frac { 1 } { 2 } }$ and $\alpha ^ { \frac { 1 } { 2 } } \beta$
(4) $\alpha ^ { \frac { 3 } { 2 } }$ and $\beta ^ { \frac { 3 } { 2 } }$

The equation $\sqrt { 3 x ^ { 2 } + x + 5 } = x - 3$, where $x$ is real, has
(1) no solution
(2) exactly four solutions
(3) exactly one solution
(4) exactly two solutions

If equations $a x ^ { 2 } + b x + c = 0 , ( a , b , c \in R , a \neq 0 )$ and $2 x ^ { 2 } + 3 x + 4 = 0$ have a common root, then $a : b : c$ equals :
(1) $2 : 3 : 4$
(2) $4 : 3 : 2$
(3) $1 : 2 : 3$
(4) $3 : 2 : 1$

The sum of all real values of $x$ satisfying the equation $(x^2 - 5x + 5)^{x^2 + 4x - 60} = 1$ is: (1) 3 (2) $-4$ (3) 6 (4) 5

If, for a positive integer $n$, the quadratic equation,
$$x(x + 1) + (x + 1)(x + 2) + \ldots + (x + \overline{n-1})(x + n) = 10n$$
has two consecutive integral solutions, then $n$ is equal to:
(1) 12
(2) 9
(3) 10
(4) 11