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

103. For which values of $m$, the equation $0 = (2m-1)x^2 + 6x + m - 2 = 0$ has two real roots?
(1) $-2 < m < 2.5$ (2) $-2 < m < 3.5$
(3) $-1 < m < 3.5$ (4) $-1 < m < 2.5$

If $x ^ { 2 } + ( a - b ) x + ( 1 - a - b ) = 0$ where $a , b$ Î $R$ then find the values of $a$ for which equation has unequal real roots for all values of $b$.

14. If $f ( x ) = x ^ { 2 } + 2 b x + 2 c ^ { 2 }$ and $g ( x ) = - x ^ { 2 } - 2 c x + b ^ { 2 }$ such that min $f ( x ) > \max g ( x )$, then the relation between $b$ and $c$, is :
(a) no real value of $b$ and $c$
(b) $0 <$ c $<$ b $\sqrt { } 2$
(c) $| \mathrm { c } | < | \mathrm { b } | \sqrt { } 2$
(d) $| c | > | b | \sqrt { } 2$

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 $\_\_\_\_$

Q82. 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 $\_\_\_\_$

Let $\alpha , \beta$ be the roots of quadratic equation $12 \mathrm { x } ^ { 2 } - 20 \mathrm { x } + 3 \lambda = 0$, $\lambda \in \mathbf { z }$. If $1 / 2 \leq | \beta - \alpha | \leq 3 / 2$ then the sum of all possible valued of $\lambda$ is $\_\_\_\_$ -

Let the equation $\mathrm { x } ^ { 4 } - \mathrm { ax } ^ { 2 } + 9 = 0$ have four real and distinct roots.
Then the least integral value of $a$ is
(A) 5
(B) 7
(C) 6
(D) 8

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 $\mathbf { G }$.

2. (a) For what values of the constant $k$ does the quadratic equation
$$x ^ { 2 } - 2 x - 1 = k$$
have:
(i) no real solutions;
(ii) one real solution;
(iii) two real solutions.
(b) Showing your working, express $\left( x ^ { 2 } - 2 x - 1 \right) ^ { 2 }$ as a polynomial of degree 4 in $x$.
(c) Show that the quartic equation
$$x ^ { 4 } - 4 x ^ { 3 } + 2 x ^ { 2 } + 4 x + 1 = h$$
has exactly two real solutions if either $h = 0$ or $h > 4$. Show that there is no value of $h$ such that the above quartic equation has just one real solution.

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

Consider the quadratic $f ( x ) = x ^ { 2 } - 2 p x + q$ and the statement:
$\left( ^ { * } \right) f ( x ) = 0$ has two real roots whose difference is greater than 2 and less than 4.
Which one of the following statements is true if and only if (*) is true?

Which one of the following is a sufficient condition for the equation $x ^ { 3 } - 3 x ^ { 2 } + a = 0$, where $a$ is a constant, to have exactly one real root?
A $a > 0$
B $a \leqslant 0$
C $\quad a \geqslant 4$
D $a < 4$
$\mathbf { E } \quad | a | > 4$
$\mathbf { F } \quad | a | \leqslant 4$
G $\quad a = \frac { 9 } { 4 }$
$\mathbf { H } \quad | a | = \frac { 3 } { 2 }$

Find the complete set of values of the constant $c$ for which the cubic equation
$$2 x ^ { 3 } - 3 x ^ { 2 } - 12 x + c = 0$$
has three distinct real solutions.
A $- 20 < c < 7$
B $- 7 < c < 20$
C $c > 7$
D $c > - 7$
E $c < 20$
F $c < - 20$