For a sequence $\left\{ a _ { n } \right\}$, the curve $y = x ^ { 2 } - ( n + 1 ) x + a _ { n }$ intersects the $x$-axis, and the curve $y = x ^ { 2 } - n x + a _ { n }$ does not intersect the $x$-axis. What is the value of $\lim _ { n \rightarrow \infty } \frac { a _ { n } } { n ^ { 2 } }$? [3 points]
(1) $\frac { 1 } { 20 }$
(2) $\frac { 1 } { 10 }$
(3) $\frac { 3 } { 20 }$
(4) $\frac { 1 } { 5 }$
(5) $\frac { 1 } { 4 }$

3. Let $\mathrm { p } : x < 3 , \mathrm { q } : - 1 < x < 3$. Then $p$ is a condition for $q$ to hold that is
(A) necessary and sufficient
(B) sufficient but not necessary
(C) necessary but not sufficient
(D) neither sufficient nor necessary

12. For the quadratic function $f ( x ) = a x ^ { 2 } + b x + c$ (where $a$ is a non-zero constant), four students each give a conclusion. Exactly one conclusion is wrong. The wrong conclusion is
A. $-1$ is a zero of $f ( x )$
B. $1$ is an extremum point of $f ( x )$
C. $3$ is an extremum value of $f ( x )$
D. The point $( 2,8 )$ lies on the curve $y = f ( x )$

II. Fill in the Blanks

The smallest value of $k$, for which both the roots of the equation $$x^{2}-8kx+16\left(k^{2}-k+1\right)=0$$ are real, distinct and have values at least 4, is

Let $S$ be the set of all non-zero real numbers $\alpha$ such that the quadratic equation $\alpha x ^ { 2 } - x + \alpha = 0$ has two distinct real roots $x _ { 1 }$ and $x _ { 2 }$ satisfying the inequality $\left| x _ { 1 } - x _ { 2 } \right| < 1$. Which of the following intervals is(are) a subset(s) of $S$ ?
(A) $\left( - \frac { 1 } { 2 } , - \frac { 1 } { \sqrt { 5 } } \right)$
(B) $\left( - \frac { 1 } { \sqrt { 5 } } , 0 \right)$
(C) $\left( 0 , \frac { 1 } { \sqrt { 5 } } \right)$
(D) $\left( \frac { 1 } { \sqrt { 5 } } , \frac { 1 } { 2 } \right)$

If the difference between the roots of the equation $x ^ { 2 } + a x + 1 = 0$ is less than $\sqrt { 5 }$, then the set of possible values of $a$ is
(1) $( - 3,3 )$
(2) $( - 3 , \infty )$
(3) $( 3 , \infty )$
(4) $( - \infty , - 3 )$

The value of k for which the equation $( K - 2 ) x ^ { 2 } + 8 x + K + 4 = 0$ has both roots real, distinct and negative is
(1) 6
(2) 3
(3) 4
(4) 1

The values of ' $a$ ' for which one root of the equation $x ^ { 2 } - ( a + 1 ) x + a ^ { 2 } + a - 8 = 0$ exceeds 2 and the other is lesser than 2 , are given by :
(1) $3 < a < 10$
(2) $a \geq 10$
(3) $- 2 < a < 3$
(4) $a \leq - 2$

Consider the quadratic equation $( c - 5 ) x ^ { 2 } - 2 c x + ( c - 4 ) = 0 , c \neq 5$. Let $S$ be the set of all integral values of $c$ for which one root of the equation lies in the interval $( 0,2 )$ and its other root lies in the interval $( 2,3 )$. Then the number of elements in $S$ is
(1) 11
(2) 12
(3) 18
(4) 10

Consider the two sets: $A = \left\{ m \in R : \right.$ both the roots of $x ^ { 2 } - ( m + 1 ) x + m + 4 = 0$ are real $\}$ and $B = [ - 3,5 )$
Which of the following is not true?
(1) $A - B = ( - \infty , - 3 ) \cup ( 5 , \infty )$
(2) $A \cap B = \{ - 3 \}$
(3) $B - A = ( - 3,5 )$
(4) $A \cup B = R$

The set of all real values of $\lambda$ for which the quadratic equation $\left( \lambda ^ { 2 } + 1 \right) x ^ { 2 } - 4 \lambda x + 2 = 0$ always have exactly one root in the interval $( 0,1 )$ is :
(1) $( - 3 , - 1 )$
(2) $( 0,2 )$
(3) $( 1,3 ]$
(4) $( 2,4 ]$

The set of all values of $k > - 1$, for which the equation $\left( 3 x ^ { 2 } + 4 x + 3 \right) ^ { 2 } - ( k + 1 ) \left( 3 x ^ { 2 } + 4 x + 3 \right) \left( 3 x ^ { 2 } + 4 x + 2 \right) + k \left( 3 x ^ { 2 } + 4 x + 2 \right) ^ { 2 } = 0$ has real roots, is: (1) $\left[ - \frac { 1 } { 2 } , 1 \right)$ (2) $\left( 1 , \frac { 5 } { 2 } \right]$ (3) $\left( \frac { 1 } { 2 } , \frac { 3 } { 2 } \right] - \{ 1 \}$ (4) $[ 2,3 )$

Let $S = \left\{ \sin ^ { 2 } 2 \theta : \left( \sin ^ { 4 } \theta + \cos ^ { 4 } \theta \right) x ^ { 2 } + ( \sin 2 \theta ) x + \left( \sin ^ { 6 } \theta + \cos ^ { 6 } \theta \right) = 0 \right.$ has real roots $\}$. If $\alpha$ and $\beta$ be the smallest and largest elements of the set $S$, respectively, then $3 \left( ( \alpha - 2 ) ^ { 2 } + ( \beta - 1 ) ^ { 2 } \right)$ equals $\_\_\_\_$

For each of A $\sim$ D in questions (1)$\sim$(4) below, choose the appropriate answer from among (0) $\sim$ (3) of each question. For $\mathbf { E } \sim \mathbf { G }$ in question (5), put the correct number.
Suppose that $a , b$ and $c$ are integers, and $a > 0$. Also, suppose that the graph of a quadratic function $y = a x ^ { 2 } - 2 b x + c$ intersects the $x$-axis and all points of intersection are in the interval $0 < x < 1$.
(1) The relationship between $a$ and $b$ is A. (0) $a > b$
(1) $a < b$
(2) $a = b$
(3) indeterminate
(2) The conditions on $b$ and $c$ are $\mathbf { B }$. (0) $b < 0 , c < 0$
(1) $b < 0 , c > 0$
(2) $b > 0 , c < 0$
(3) $b > 0 , c > 0$
(3) The relationship between $2 b$ and $a + c$ is $\mathbf { C }$. (0) $2 b > a + c$
(1) $2 b < a + c$
(2) $2 b = a + c$
(3) indeterminate
(4) The relationship between $b$ and $c$ is $\mathbf { D }$. (0) $b > c$
(1) $b < c$
(2) $b = c$
(3) indeterminate
(5) The smallest integer which $a$ can take is $\mathbf { E }$. In this case, the value of $b$ is $\mathbf { F }$, and the value of $c$ is $\mathbf { G }$.

Let $b$ and $c$ be real numbers. The quadratic equation $x ^ { 2 } + b x + c = 0$ has real roots, but the quadratic equation $x ^ { 2 } + ( b + 2 ) x + c = 0$ has no real roots. Select the correct options.
(1) $c < 0$
(2) $b < 0$
(3) $x ^ { 2 } + ( b + 1 ) x + c = 0$ has real roots
(4) $x ^ { 2 } + ( b + 2 ) x - c = 0$ has real roots
(5) $x ^ { 2 } + ( b - 2 ) x + c = 0$ has real roots