Question 161
A equação $x^2 - 5x + 6 = 0$ tem como raízes
(A) $x = 1$ e $x = 6$ (B) $x = 2$ e $x = 3$ (C) $x = -2$ e $x = -3$ (D) $x = 1$ e $x = -6$ (E) $x = -1$ e $x = 6$

Question 173
Um número inteiro positivo de dois algarismos é tal que a soma de seus algarismos é 9 e o produto de seus algarismos é 18. Esse número é
(A) 27 (B) 36 (C) 45 (D) 63 (E) 72

QUESTION 146
The function $f(x) = 2x^2 - 3x + 1$ has roots
(A) $x = 1$ and $x = \frac{1}{2}$
(B) $x = -1$ and $x = \frac{1}{2}$
(C) $x = 1$ and $x = -\frac{1}{2}$
(D) $x = -1$ and $x = -\frac{1}{2}$
(E) $x = 2$ and $x = \frac{1}{2}$

QUESTION 168
The equation $x^2 - 5x + 6 = 0$ has roots
(A) $x = 1$ and $x = 6$
(B) $x = 2$ and $x = 3$
(C) $x = -2$ and $x = -3$
(D) $x = 1$ and $x = -6$
(E) $x = -1$ and $x = 6$

8. If $a$ and $b$ are two distinct zeros of the function $f ( x ) = x ^ { 2 } - p x + q$ ($p > 0$, $q > 0$), and the three numbers $a$, $b$, and $-2$ can be arranged to form an arithmetic sequence, and can also be arranged to form a geometric sequence, then the value of $p + q$ equals
A. 6
B. 7
C. 8
D. 9

Suppose $a , b$ denote the distinct real roots of the quadratic polynomial $x ^ { 2 } + 20 x - 2020$ and suppose $c , d$ denote the distinct complex roots of the quadratic polynomial $x ^ { 2 } - 20 x + 2020$. Then the value of
$$a c ( a - c ) + a d ( a - d ) + b c ( b - c ) + b d ( b - d )$$
is
(A) 0
(B) 8000
(C) 8080
(D) 16000

If $\frac { 1 } { \sqrt { \alpha } } , \frac { 1 } { \sqrt { \beta } }$ are the roots of the equation $a x ^ { 2 } + b x + 1 = 0 , ( a \neq 0 , a , b \in R )$, then the equation $x \left( x + b ^ { 3 } \right) + \left( a ^ { 3 } - 3 a b x \right) = 0$ has roots:
(1) $\sqrt { \alpha \beta }$ and $\alpha \beta$
(2) $\alpha ^ { - \frac { 3 } { 2 } }$ and $\beta ^ { - \frac { 3 } { 2 } }$
(3) $\alpha \beta ^ { \frac { 1 } { 2 } }$ and $\alpha ^ { \frac { 1 } { 2 } } \beta$
(4) $\alpha ^ { \frac { 3 } { 2 } }$ and $\beta ^ { \frac { 3 } { 2 } }$

If equations $a x ^ { 2 } + b x + c = 0 , ( a , b , c \in R , a \neq 0 )$ and $2 x ^ { 2 } + 3 x + 4 = 0$ have a common root, then $a : b : c$ equals :
(1) $2 : 3 : 4$
(2) $4 : 3 : 2$
(3) $1 : 2 : 3$
(4) $3 : 2 : 1$

Let $p , q$ and $r$ be real numbers ( $p \neq q , r \neq 0$ ), such that the roots of the equation $\frac { 1 } { x + p } + \frac { 1 } { x + q } = \frac { 1 } { r }$ are equal in magnitude but opposite in sign, then the sum of squares of these roots is equal to
(1) $p ^ { 2 } + q ^ { 2 }$
(2) $\frac { p ^ { 2 } + q ^ { 2 } } { 2 }$
(3) $2 \left( p ^ { 2 } + q ^ { 2 } \right)$
(4) $p ^ { 2 } + q ^ { 2 } + r ^ { 2 }$

Let $a , b \in R , a \neq 0$ be such that the equation, $a x ^ { 2 } - 2 b x + 5 = 0$ has a repeated root $\alpha$, which is also a root of the equation, $x ^ { 2 } - 2 b x - 10 = 0$. If $\beta$ is the other root of this equation, then $\alpha ^ { 2 } + \beta ^ { 2 }$ is equal to:
(1) 25
(2) 26
(3) 28
(4) 24

Let $\lambda \neq 0$ be in $R$. If $\alpha$ and $\beta$ are the roots of the equation, $x ^ { 2 } - x + 2 \lambda = 0$ and $\alpha$ and $\gamma$ are the roots of the equation, $3 x ^ { 2 } - 10 x + 27 \lambda = 0$, then $\frac { \beta \gamma } { \lambda }$ is equal to:
(1) 27
(2) 18
(3) 9
(4) 36

The product of the roots of the equation $9 x ^ { 2 } - 18 | x | + 5 = 0$ is :
(1) $\frac { 5 } { 9 }$
(2) $\frac { 25 } { 81 }$
(3) $\frac { 5 } { 27 }$
(4) $\frac { 25 } { 9 }$

If $\alpha$ and $\beta$ are the roots of the equation $2\mathrm{x}(2\mathrm{x}+1)=1$, then $\beta$ is equal to:
(1) $2\alpha(\alpha+1)$
(2) $-2\alpha(\alpha+1)$
(3) $2\alpha(\alpha-1)$
(4) $2\alpha^{2}$

Let $p$ and $q$ be two positive numbers such that $p + q = 2$ and $p ^ { 4 } + q ^ { 4 } = 272$. Then $p$ and $q$ are roots of the equation:
(1) $x ^ { 2 } - 2 x + 2 = 0$
(2) $x ^ { 2 } - 2 x + 8 = 0$
(3) $x ^ { 2 } - 2 x + 136 = 0$
(4) $x ^ { 2 } - 2 x + 16 = 0$

$\operatorname { cosec } 18 ^ { \circ }$ is a root of the equation:
(1) $x ^ { 2 } - 2 x - 4 = 0$
(2) $4 x ^ { 2 } + 2 x - 1 = 0$
(3) $x ^ { 2 } + 2 x - 4 = 0$
(4) $x ^ { 2 } - 2 x + 4 = 0$

Let $a, b \in R$ be such that the equation $ax^2 - 2bx + 15 = 0$ has repeated root $\alpha$ and if $\alpha$ and $\beta$ are the roots of the equation $x^2 - 2bx + 21 = 0$, then $\alpha^2 + \beta^2$ is equal to:
(1) 37
(2) 58
(3) 68
(4) 92

$\alpha = \sin 36 ^ { \circ }$ is a root of which of the following equation
(1) $16 x ^ { 4 } - 20 x ^ { 2 } + 5 = 0$
(2) $16 x ^ { 4 } + 20 x ^ { 2 } + 5 = 0$
(3) $10 x ^ { 4 } - 10 x ^ { 2 } - 5 = 0$
(4) $16 x ^ { 4 } - 10 x ^ { 2 } + 5 = 0$

If for some $p , q , r \in R$, all have positive sign, one of the roots of the equation $\left( p ^ { 2 } + q ^ { 2 } \right) x ^ { 2 } - 2 q ( p + r ) x + q ^ { 2 } + r ^ { 2 } = 0$ is also a root of the equation $x ^ { 2 } + 2 x - 8 = 0$, then $\frac { q ^ { 2 } + r ^ { 2 } } { p ^ { 2 } }$ is equal to $\_\_\_\_$.

If $\alpha , \beta$ are the roots of the equation $x ^ { 2 } - \left( 5 + 3 ^ { \sqrt { \log _ { 3 } 5 } } - 5 ^ { \sqrt { \log _ { 5 } 3 } } \right) x + 3 \left( 3 ^ { \left( \log _ { 3 } 5 \right) ^ { \frac { 1 } { 3 } } } - 5 ^ { \left( \log _ { 5 } 3 \right) ^ { \frac { 2 } { 3 } } } - 1 \right) = 0$ then the equation, whose roots are $\alpha + \frac { 1 } { \beta }$ and $\beta + \frac { 1 } { \alpha }$,
(1) $3 x ^ { 2 } - 20 x - 12 = 0$
(2) $3 x ^ { 2 } - 10 x - 4 = 0$
(3) $3 x ^ { 2 } - 10 x + 2 = 0$
(4) $3 x ^ { 2 } - 20 x + 16 = 0$

If the value of real number $\alpha > 0$ for which $x^{2} - 5\alpha x + 1 = 0$ and $x^{2} - \alpha x - 5 = 0$ have a common real roots is $\frac{3}{\sqrt{2\beta}}$ then $\beta$ is equal to $\_\_\_\_$

Let $x = \frac { m } { n } \left( m , n \right.$ are co-prime natural numbers) be a solution of the equation $\cos \left( 2 \sin ^ { - 1 } x \right) = \frac { 1 } { 9 }$ and let $\alpha , \beta ( \alpha > \beta )$ be the roots of the equation $m x ^ { 2 } - n x - m + n = 0$. Then the point $( \alpha , \beta )$ lies on the line
(1) $3 x + 2 y = 2$
(2) $5 x - 8 y = - 9$
(3) $3 x - 2 y = - 2$
(4) $5 x + 8 y = 9$

Let $\int _ { \alpha } ^ { \log _ { e } 4 } \frac { \mathrm {~d} x } { \sqrt { \mathrm { e } ^ { x } - 1 } } = \frac { \pi } { 6 }$. Then $\mathrm { e } ^ { \alpha }$ and $\mathrm { e } ^ { - \alpha }$ are the roots of the equation : (1) $x ^ { 2 } + 2 x - 8 = 0$ (2) $x ^ { 2 } - 2 x - 8 = 0$ (3) $2 x ^ { 2 } - 5 x + 2 = 0$ (4) $2 x ^ { 2 } - 5 x - 2 = 0$

Let $\alpha _ { \theta }$ and $\beta _ { \theta }$ be the distinct roots of $2 x ^ { 2 } + ( \cos \theta ) x - 1 = 0 , \theta \in ( 0,2 \pi )$. If m and M are the minimum and the maximum values of $\alpha _ { \theta } ^ { 4 } + \beta _ { \theta } ^ { 4 }$, then $16 ( M + m )$ equals :
(1) 24
(2) 25
(3) 17
(4) 27

Let $a, b, c$ be nonzero real numbers, and the two roots of the quadratic equation $ax^2 + bx + c = 0$ both lie between 1 and 3. Select the equation whose two roots must lie between 4 and 5.
(1) $a(x-2)^2 + b(x-2) + c = 0$
(2) $a(x+2)^2 + b(x+2) + c = 0$
(3) $a(2x-7)^2 + b(2x-7) + c = 0$
(4) $a\left(\frac{x+7}{2}\right)^2 + b\left(\frac{x+7}{2}\right) + c = 0$
(5) $a(3x-11)^2 + b(3x-11) + c = 0$

$$(3x-1)(x+1)+(3x-1)(x-2)=0$$
What is the sum of the real numbers $x$ that satisfy the equation?
A) $\frac{2}{3}$
B) $\frac{3}{4}$
C) $\frac{3}{5}$
D) $\frac{5}{6}$
E) $\frac{7}{6}$