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

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.

30. If $\mathrm { b } > \mathrm { a }$, then the equation $( \mathrm { x } - \mathrm { a } ) ( \mathrm { x } - \mathrm { b } ) - 1 = 0$ has:
(A) both roots in (a, b)
(B) both roots in ( $- ¥$, a)
(C) both roots in $( b , + ¥ )$
(D) one root in ( $- ¥$, a) and the other in ( $b , + \neq$ )

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 $\_\_\_\_$.

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

The number of real roots of the equation, $e ^ { 4 x } + e ^ { 3 x } - 4 e ^ { 2 x } + e ^ { x } + 1 = 0$ is:
(1) 1
(2) 3
(3) 2
(4) 4

Let $[ \mathrm { t } ]$ denote the greatest integer $\leq \mathrm { t }$. Then the equation in $\mathrm { x } , [ \mathrm { x } ] ^ { 2 } + 2 [ \mathrm { x } + 2 ] - 7 = 0$ has :
(1) exactly two solutions
(2) exactly four integral solutions
(3) no integral solution
(4) infinitely many solutions

Let $f ( x ) = 2 x ^ { 2 } - x - 1$ and $S = \{ n \in \mathbb { Z } : | f ( n ) | \leq 800 \}$. Then, the value of $\sum _ { n \in S } f ( n )$ is equal to $\_\_\_\_$ .

The equation $x ^ { 2 } - 4 x + [ x ] + 3 = x [ x ]$, where $[ x ]$ denotes the greatest integer function, has:
(1) exactly two solutions in $( - \infty , \infty )$
(2) no solution
(3) a unique solution in $( - \infty$, 1)
(4) a unique solution in $( - \infty , \infty )$

Let $m$ and $n$ be the numbers of real roots of the quadratic equations $x ^ { 2 } - 12 x + [ x ] + 31 = 0$ and $x ^ { 2 } - 5 | x + 2 | - 4 = 0$ respectively, where $[ x ]$ denotes the greatest integer $\leq x$. Then $m ^ { 2 } + m n + n ^ { 2 }$ is equal to

Q1 Consider the equation
$$(x-1)^2 = |3x-5|.$$
(1) Among all solutions of equation (1), the solutions satisfying $x \geqq \frac{5}{3}$ are $x = \mathbf{A}$ and $x = \mathbf{B}$, where $\mathbf{A} < \mathbf{B}$.
(2) Equation (1) has a total of $\mathbf{C}$ solutions. When the minimum one is denoted by $\alpha$, the integer $m$ satisfying $m-1 < \alpha \leqq m$ is $\mathbf{DE}$.

Q1 Consider the equation
$$(x-1)^2 = |3x-5|.$$
(1) Among all solutions of equation (1), the solutions satisfying $x \geqq \frac{5}{3}$ are $x = \mathbf{A}$ and $x = \mathbf{B}$, where $\mathbf{A} < \mathbf{B}$.
(2) Equation (1) has a total of $\mathbf{C}$ solutions. When the minimum one is denoted by $\alpha$, the integer $m$ satisfying $m-1 < \alpha \leqq m$ is $\mathbf{DE}$.

Let $a$ be a constant. For the two functions in $x$
$$\begin{aligned} & f ( x ) = 2 x ^ { 2 } + x + a - 2 \\ & g ( x ) = - 4 x - 5 \end{aligned}$$
we are to find the real values of $x$ for which $f ( x ) = g ( x )$ and also find the values of the two functions there.
(1) For each of $\mathbf { N } , \mathbf { O }$ and $\mathbf { P }$ in the following statements, choose the appropriate condition from (0) $\sim$ (8) below.
When $\mathbf { N }$, there are two real values of $x$ for which $f ( x ) = g ( x )$. When $\mathbf { O }$, there is only one real value of $x$ for which $f ( x ) = g ( x )$. When $\mathbf{P}$, there is no real value of $x$ for which $f ( x ) = g ( x )$.
(0) $a > \frac { 1 } { 8 }$
(1) $a = \frac { 17 } { 8 }$
(2) $a = \frac { 1 } { 6 }$
(3) $a < \frac { 1 } { 6 }$
(4) $a < \frac { 17 } { 8 }$
(5) $a < \frac { 1 } { 8 }$ (6) $a > \frac { 1 } { 6 }$ (7) $a = \frac { 1 } { 8 }$ (8) $a > \frac { 17 } { 8 }$
(2) When N, the values of $x$ for which $f ( x ) = g ( x )$ are $\frac { - \mathrm { Q } \pm \sqrt { \mathrm { R } - \mathbf { S } a } } { \mathbf{T} }$, and the values of the functions there are $\mp \sqrt { \mathbf { U } - \mathbf { V } a }$.
When O, the value of $x$ for which $f ( x ) = g ( x )$ is $- \frac { \mathrm { W } } { \mathrm { X} }$, and the value of the functions there is $\mathbf{Y}$.
(3) Consider the case where $f ( x ) = g ( x )$ and the absolute value of these functions there is greater than or equal to 3. The condition for this case is that $a \leqq - \mathbf { Z }$.

2. (a) Factorise the expression $x ^ { 2 } + x - 6$.
(b) For which values of the real constant $a$ does the equation
$$x ^ { 2 } + x - a = 0$$
have at least one real solution? Write down these solutions in terms of $a$.
(c) Show that, for any value of the real constant $b$, the equation
$$x ^ { 3 } - ( b + 1 ) x + b = 0$$
has $x = 1$ as a solution. Find all values of $b$ for which this equation has exactly two distinct solutions.

Let $n$ and $k$ be positive integers. The value of $n _ { k }$ is defined as
- If $n$ is divisible by $k$, then $n _ { k } = \frac { n } { k }$ - If $n$ is not divisible by $k$, then $n _ { k } = 0$
Example: $$\begin{aligned} & 10 _ { 2 } = 5 \\ & 10 _ { 3 } = 0 \end{aligned}$$
Accordingly,
$$n _ { 2 } + n _ { 3 } = 10$$
what is the sum of the $n$ numbers that satisfy the equality?
A) 24 B) 28 C) 32 D) 36 E) 40