Consider the set of non-zero relative integers between $-30$ and $30$; this set can be written as follows: $\{-30; -29; -28; \ldots -1; 1; \ldots; 28; 29; 30\}$. It contains 60 elements. We choose from this set successively and without replacement a relative integer $a$ then a relative integer $c$.

1. How many different pairs $(a; c)$ can we obtain?

Consider the event $M$: ``the equation $ax^2 + 2x + c = 0$ has two distinct real solutions'', where $a$ and $c$ are the relative integers previously chosen.

1. Show that event $M$ occurs if and only if $ac < 1$.
2. Explain why the opposite event $\bar{M}$ contains 1740 outcomes.
3. What is the probability of event $M$? Round the result to $10^{-2}$.

The equation $x ^ { 2 } + b x + c = 0$ has nonzero real coefficients satisfying $b ^ { 2 } > 4 c$. Moreover, exactly one of $b$ and $c$ is irrational. Consider the solutions $p$ and $q$ of this equation.
(A) Both $p$ and $q$ must be rational.
(B) Both $p$ and $q$ must be irrational.
(C) One of $p$ and $q$ is rational and the other irrational.
(D) We cannot conclude anything about rationality of $p$ and $q$ unless we know $b$ and $c$.

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

A number $p$ is randomly chosen from the interval $[ 0,5 ]$. The probability that the equation $x ^ { 2 } + 2 p x + 3 p - 2 = 0$ has two negative roots is $\_\_\_\_$ .

If $c$ is a real number with $0 < c < 1$, then show that the values taken by the function $y = \frac{x^2 + 2x + c}{x^2 + 4x + 3c}$, as $x$ varies over real numbers, range over all real numbers.

a) Show that if $a$ and $b$ are irrational numbers that are roots of a quadratic with rational coefficients, then $(a-b)^2$ is not a perfect square of any rational number.
b) i) If $a = r \pm \sqrt{s}$ is a quadratic surd, find a rational $x$ such that $a + x$ is irrational but $a_n = (r + (r^2 - s)) \pm \sqrt{s} \notin \mathbb{Q}$. If $a$ is not a surd, take $x = -a$.
ii) Find $y$ such that the required condition holds.

The set of values of $m$ for which $mx^2 - 6mx + 5m + 1 > 0$ for all real $x$ is
(A) $m < \frac{1}{4}$
(B) $m \geq 0$
(C) $0 \leq m \leq \frac{1}{4}$
(D) $0 \leq m < \frac{1}{4}$

The set of values of $m$ for which $m x ^ { 2 } - 6 m x + 5 m + 1 > 0$ for all real $x$ is
(a) $m < \frac { 1 } { 4 }$
(b) $m \geq 0$
(c) $0 \leq m \leq \frac { 1 } { 4 }$
(d) $0 \leq m < \frac { 1 } { 4 }$.

The set of values of $m$ for which $m x ^ { 2 } - 6 m x + 5 m + 1 > 0$ for all real $x$ is
(a) $m < \frac { 1 } { 4 }$
(b) $m \geq 0$
(c) $0 \leq m \leq \frac { 1 } { 4 }$
(d) $0 \leq m < \frac { 1 } { 4 }$.

The set of values of $m$ for which $mx^2 - 6mx + 5m + 1 > 0$ for all real $x$ is
(A) $m < \frac{1}{4}$
(B) $m \geq 0$
(C) $0 \leq m \leq \frac{1}{4}$
(D) $0 \leq m < \frac{1}{4}$

The set of values of $m$ for which $m x ^ { 2 } - 6 m x + 5 m + 1 > 0$ for all real $x$ is
(A) $m < \frac { 1 } { 4 }$
(B) $m \geq 0$
(C) $0 \leq m \leq \frac { 1 } { 4 }$
(D) $0 \leq m < \frac { 1 } { 4 }$

Consider a quadratic equation $ax^2 + 2bx + c = 0$, where $a, b$ and $c$ are positive real numbers. If the equation has no real roots, then which of the following is true?
(A) $a, b, c$ cannot be in AP or HP, but can be in GP.
(B) $a, b, c$ cannot be in GP or HP, but can be in AP.
(C) $a, b, c$ cannot be in AP or GP, but can be in HP.
(D) $a, b, c$ cannot be in AP, GP or HP.

For which values of $\theta$, with $0 < \theta < \pi / 2$, does the quadratic polynomial in $t$ given by $t ^ { 2 } + 4 t \cos \theta + \cot \theta$ have repeated roots?
(A) $\frac { \pi } { 6 }$ or $\frac { 5 \pi } { 18 }$
(B) $\frac { \pi } { 6 }$ or $\frac { 5 \pi } { 12 }$
(C) $\frac { \pi } { 12 }$ or $\frac { 5 \pi } { 18 }$
(D) $\frac { \pi } { 12 }$ or $\frac { 5 \pi } { 12 }$

Let $a$ be a fixed real number. Consider the equation
$$(x + 2)^{2}(x + 7)^{2} + a = 0, \quad x \in \mathbb{R},$$
where $\mathbb{R}$ is the set of real numbers. For what values of $a$, will the equation have exactly one double-root?

If $a , b , c$ are distinct odd natural numbers, then the number of rational roots of the polynomial $a x ^ { 2 } + b x + c$
(A) must be 0 .
(B) must be 1 .
(C) must be 2 .
(D) cannot be determined from the given data.

Suppose $a , b , c \in \mathbb { R }$ and $$f ( x ) = a x ^ { 2 } + b x + c , x \in \mathbb { R } .$$ If $0 \leq f ( x ) \leq ( x - 1 ) ^ { 2 }$ for all $x$, and $f ( 3 ) = 2$, then
(A) $a = \frac { 1 } { 2 } , b = - 1 , c = \frac { 1 } { 2 }$.
(B) $a = \frac { 1 } { 3 } , b = - \frac { 1 } { 3 } , c = 0$.
(C) $a = \frac { 2 } { 3 } , b = - \frac { 5 } { 3 } , c = 1$.
(D) $a = \frac { 3 } { 4 } , b = - 2 , c = \frac { 5 } { 4 }$.

Let $a , b , c , p , q$ be real numbers. Suppose $\alpha , \beta$ are the roots of the equation $x ^ { 2 } + 2 p x + q = 0$ and $\alpha , \frac { 1 } { \beta }$ are the roots of the equation $a x ^ { 2 } + 2 b x + c = 0$, where $\beta ^ { 2 } \notin \{ - 1,0,1 \}$. STATEMENT-1 : $\left( p ^ { 2 } - q \right) \left( b ^ { 2 } - a c \right) \geq 0$ and STATEMENT-2 : $b \neq p a$ or $c \neq q a$
(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
(B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
(C) STATEMENT-1 is True, STATEMENT-2 is False
(D) STATEMENT-1 is False, STATEMENT-2 is True

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) $\left( \frac { 1 } { \sqrt { 5 } } , \frac { 1 } { 2 } \right)$

Let $\mathbb { R } ^ { 2 }$ denote $\mathbb { R } \times \mathbb { R }$. Let
$$S = \left\{ ( a , b , c ) : a , b , c \in \mathbb { R } \text { and } a x ^ { 2 } + 2 b x y + c y ^ { 2 } > 0 \text { for all } ( x , y ) \in \mathbb { R } ^ { 2 } - \{ ( 0,0 ) \} \right\}$$
Then which of the following statements is (are) TRUE?
(A) $\left( 2 , \frac { 7 } { 2 } , 6 \right) \in S$
(B) If $\left( 3 , b , \frac { 1 } { 12 } \right) \in S$, then $| 2 b | < 1$.
(C) For any given $( a , b , c ) \in S$, the system of linear equations
$$\begin{aligned} & a x + b y = 1 \\ & b x + c y = - 1 \end{aligned}$$
has a unique solution.
(D) For any given $( a , b , c ) \in S$, the system of linear equations
$$\begin{aligned} & ( a + 1 ) x + b y = 0 \\ & b x + ( c + 1 ) y = 0 \end{aligned}$$
has a unique solution.

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

Let $p , q , r \in R$ and $r > p > 0$. If the quadratic equation $p x ^ { 2 } + q x + r = 0$ has two complex roots $\alpha$ and $\beta$, then $| \alpha | + | \beta |$ is
(1) equal to 1
(2) less than 2 but not equal to 1
(3) greater than 2
(4) equal to 2