106-- $\alpha$ and $\beta$ are roots of the equation $x^2 + 6x + a = 0$. If $0 < \alpha < \beta < 0$ and $12\sqrt{2} + 85 = 12\sqrt{2} + 85$, and $3\alpha^2 + 2\beta^2 = 12\sqrt{2} + 85$, what is the value of $a$?
(1) $1$ (2) $\dfrac{13}{4}$ (3) $\dfrac{21}{5}$ (4) $2$

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

If the equations $x^2 + 2x + 3 = 0$ and $ax^2 + bx + c = 0, a, b, c \in R$, have a common root, then $a : b : c$ is:
(1) $1 : 3 : 2$
(2) $3 : 1 : 2$
(3) $1 : 2 : 3$
(4) $3 : 2 : 1$

Let $p , q \in Q$. If $2 - \sqrt { 3 }$ is a root of the quadratic equation $x ^ { 2 } + p x + q = 0$, then
(1) $p ^ { 2 } - 4 q + 12 = 0$
(2) $q ^ { 2 } + 4 p + 14 = 0$
(3) $p ^ { 2 } - 4 q - 12 = 0$
(4) $q ^ { 2 } - 4 p - 16 = 0$

Let $\alpha , \beta$ be the roots of the equation $x ^ { 2 } - \sqrt { 2 } x + \sqrt { 6 } = 0$ and $\frac { 1 } { \alpha ^ { 2 } } + 1 , \frac { 1 } { \beta ^ { 2 } } + 1$ be the roots of the equation $x ^ { 2 } + a x + b = 0$. Then the roots of the equation $x ^ { 2 } - ( a + b - 2 ) x + ( a + b + 2 ) = 0$ are :
(1) non-real complex numbers
(2) real and both negative
(3) real and both positive
(4) real and exactly one of them is positive

Q1 Let $a$ and $b$ be rational numbers and let $p$ be a real number. Consider the quadratic equation
$$x ^ { 2 } + a x + b = 0 \tag{1}$$
which has a solution $x = \frac { \sqrt { 5 } + 3 } { \sqrt { 5 } + 2 }$, and consider the inequality
$$x + 1 < 2 x + p + 3 . \tag{2}$$
(1) First, we are to find the values of $a$ and $b$.
When we rationalize the denominator of $x = \frac { \sqrt { 5 } + 3 } { \sqrt { 5 } + 2 }$, we have
$$x = \sqrt { \mathbf { A } } - \mathbf { B }$$
Since this is a solution of equation (1), by substituting this in (1) we have
$$- a + b + \mathbf { C } + ( a - \mathbf { D } ) \sqrt { \mathbf { E } } = 0 .$$
Hence we see that
$$a = \mathbf { F } \text { and } b = \mathbf { G H } .$$
(2) Next, we are to find the smallest integer $p$ such that both solutions of equation (1) satisfy inequality (2).
When we solve inequality (2), we have
$$x > - p - 1 .$$
Since both solutions of equation (1) satisfy this, we see that
$$p > \sqrt { \mathbf { J } } - \mathbf { K } .$$
Hence the smallest integer $p$ is $\mathbf { L }$.

9. The roots of the equation $2 x ^ { 2 } - 11 x + c = 0$ differ by 2 . The value of $c$ is
A $\frac { 105 } { 8 }$
B $\frac { 113 } { 8 }$
C $\frac { 117 } { 8 }$
D $\frac { 119 } { 8 }$

Let k be a positive real number. If the roots of the equation
$$2 x ^ { 2 } + k x - 1 = 0$$
have a difference of 2, what is k?
A) 1
B) 2
C) $\sqrt { 2 }$
D) $2 \sqrt { 2 }$
E) $\sqrt { 3 }$

Let a be a real number. One root of the equation
$$a x ^ { 2 } - 18 x + 18 = 0$$
is 2 times the other. Accordingly, what is a?
A) 2
B) 3
C) 4
D) 5
E) 6

Let $m$ and $n$ be two non-zero and distinct real numbers,
$$x ^ { 2 } + ( m + 1 ) x + n - m = 0$$
one of the roots of the equation is the number $m - n$.
Accordingly, what is the ratio $\frac { \mathbf { n } } { \mathbf { m } }$?
A) 2 B) 3 C) 4 D) 5 E) 6

For the equation $x^2 - 2x + c = 0$, the discriminant is also a root of this equation. What is the product of the possible values of the real number $c$?
A) 1
B) 2
C) 4
D) $\frac{1}{2}$
E) $\frac{1}{4}$