The number of pairs $a , b$ of real numbers, such that whenever $\alpha$ is a root of the equation $x ^ { 2 } + a x + b = 0 , \quad \alpha ^ { 2 } - 2$ is also a root of this equation, is :
(1) 6
(2) 8
(3) 4
(4) 2

If the sum of the squares of the reciprocals of the roots $\alpha$ and $\beta$ of the equation $3 x ^ { 2 } + \lambda x - 1 = 0$ is 15 , then $6 \left( \alpha ^ { 3 } + \beta ^ { 3 } \right) ^ { 2 }$ is equal to
(1) 46
(2) 36
(3) 24
(4) 18

If $\alpha , \beta , \gamma , \delta$ are the roots of the equation $x ^ { 4 } + x ^ { 3 } + x ^ { 2 } + x + 1 = 0$, then $\alpha ^ { 2021 } + \beta ^ { 2021 } + \gamma ^ { 2021 } + \delta ^ { 2021 }$ is equal to
(1) 4
(2) 1
(3) - 4
(4) - 1

The sum of the cubes of all the roots of the equation $x ^ { 4 } - 3 x ^ { 3 } - 2 x ^ { 2 } + 3 x + 1 = 0$ is $\_\_\_\_$.

Let $\alpha , \beta$ be the roots of the equation $x ^ { 2 } - 4 \lambda x + 5 = 0$ and $\alpha , \gamma$ be the roots of the equation $x ^ { 2 } - ( 3 \sqrt { 2 } + 2 \sqrt { 3 } ) x + 7 + 3 \lambda \sqrt { 3 } = 0$. If $\beta + \gamma = 3 \sqrt { 2 }$, then $( \alpha + 2 \beta + \gamma ) ^ { 2 }$ is equal to

Let $\alpha , \beta$ be the roots of the equation $x ^ { 2 } - \sqrt { 2 } x + 2 = 0$. Then $\alpha ^ { 14 } + \beta ^ { 14 }$ is equal to
(1) $- 64$
(2) $- 64 \sqrt { 2 }$
(3) $- 128$
(4) $- 128 \sqrt { 2 }$

Let $a \in R$ and let $\alpha , \beta$ be the roots of the equation $x ^ { 2 } + 60 ^ { \frac { 1 } { 4 } } x + a = 0$. If $\alpha ^ { 4 } + \beta ^ { 4 } = - 30$, then the product of all possible values of $a$ is $\_\_\_\_$.

Let $\lambda \neq 0$ be a real number. Let $\alpha , \beta$ be the roots of the equation $14 x ^ { 2 } - 31 x + 3 \lambda = 0$ and $\alpha , \gamma$ be the roots of the equation $35 x ^ { 2 } - 53 x + 4 \lambda = 0$. Then $\frac { 3 \alpha } { \beta }$ and $\frac { 4 \alpha } { \gamma }$ are the roots of the equation :
(1) $7 x ^ { 2 } + 245 x - 250 = 0$
(2) $7 x ^ { 2 } - 245 x + 250 = 0$
(3) $49 x ^ { 2 } - 245 x + 250 = 0$
(4) $49 x ^ { 2 } + 245 x + 250 = 0$

Let $\alpha _ { 1 } , \alpha _ { 2 } , \ldots , \alpha _ { 7 }$ be the roots of the equation $x ^ { 7 } + 3 x ^ { 5 } - 13 x ^ { 3 } - 15 x = 0$ and $\left| \alpha _ { 1 } \right| \geq \left| \alpha _ { 2 } \right| \geq \ldots \geq \left| \alpha _ { 7 } \right|$.
Then, $\alpha _ { 1 } \alpha _ { 2 } - \alpha _ { 3 } \alpha _ { 4 } + \alpha _ { 5 } \alpha _ { 6 }$ is equal to $\_\_\_\_$

Let $\alpha , \beta , \gamma$ be the three roots of the equation $x ^ { 3 } + b x + c = 0$ if $\beta \gamma = 1 = - \alpha$ then $b ^ { 3 } + 2 c ^ { 3 } - 3 \alpha ^ { 3 } - 6 \beta ^ { 3 } - 8 \gamma ^ { 3 }$ is equal to
(1) $\frac { 155 } { 8 }$
(2) 21
(3) $\frac { 169 } { 8 }$
(4) 19

Let $\alpha , \beta$ be the roots of the quadratic equation $x ^ { 2 } + \sqrt { 6 } x + 3 = 0$. Then $\frac { \alpha ^ { 23 } + \beta ^ { 23 } + \alpha ^ { 14 } + \beta ^ { 14 } } { \alpha ^ { 15 } + \beta ^ { 15 } + \alpha ^ { 10 } + \beta ^ { 10 } }$ is equal to
(1) 81
(2) 9
(3) 72
(4) 729

Let $\alpha$ be a root of the equation $( a - c ) x ^ { 2 } + ( b - a ) x + ( c - b ) = 0$ where $a , \quad b , \quad c$ are distinct real numbers such that the matrix $\begin{pmatrix} \alpha ^ { 2 } & \alpha & 1 \end{pmatrix}$ is singular. Then the value of $\frac { ( a - c ) ^ { 2 } } { ( b - a )( c - b ) } + \frac { ( b - a ) ^ { 2 } } { ( a - c )( c - b ) } + \frac { ( c - b ) ^ { 2 } } { ( a - c )( b - a ) }$ is
(1) 6
(2) 3
(3) 9
(4) 12

Let $\alpha$ and $\beta$ be the roots of $x^2 - \sqrt{6}x + 3 = 0$. If $\alpha^n + \beta^n$ is an integer for $n \geq 1$, then the greatest value of $n$ for which $\alpha^n + \beta^n$ is NOT an integer is $\_\_\_\_$.

If $z$ is a complex number, then the number of common roots of the equation $z^{1985} + z^{100} + 1 = 0$ and $z^3 + 2z^2 + 2z + 1 = 0$, is equal to:
(1) 1
(2) 2
(3) 0
(4) 3

Let the polynomial function $f ( x ) = x ^ { 3 } + a x ^ { 2 } + b x + c$, where $a , b , c$ are all rational numbers. Select the correct options.
(1) The graph of $y = f ( x )$ and the parabola $y = x ^ { 2 } + 100$ may have no intersection points
(2) If $f ( 0 ) f ( 1 ) < 0 < f ( 0 ) f ( 2 )$, then the equation $f ( x ) = 0$ must have three distinct real roots
(3) If $1 + 3 i$ is a complex root of the equation $f ( x ) = 0$, then the equation $f ( x ) = 0$ has a rational root
(4) There exist rational numbers $a , b , c$ such that $f ( 1 ) , f ( 2 ) , f ( 3 ) , f ( 4 )$ form an arithmetic sequence in order
(5) There exist rational numbers $a , b , c$ such that $f ( 1 ) , f ( 2 ) , f ( 3 ) , f ( 4 )$ form a geometric sequence in order

A game company will hold a lottery activity. The company announces that each lottery draw requires using one token, and the winning probability for each draw is $\frac{1}{10}$. A certain person decides to save a certain number of tokens and start drawing after the activity begins, stopping only when all tokens are used. Select the correct options.
(1) The expected value of the number of draws needed for the person to win once is 10
(2) The probability that the person wins at least once in two draws is 0.2
(3) The probability that the person does not win in 10 draws is less than the probability of winning in 1 draw
(4) The person must save at least 22 tokens to guarantee a winning probability greater than 0.9
(5) If the person saves sufficiently many tokens, the winning probability can be guaranteed to be 1

The leading coefficient is 1, and the fourth-degree polynomial $\mathbf { P } ( \mathbf { x } )$ with real coefficients has roots $-i$ and $2i$. What is $\mathbf { P } ( \mathbf { 0 } )$?
A) 2
B) 4
C) 6
D) 7
E) 8

$$\begin{aligned} & P ( x ) = x ^ { 2 } - 2 x + m \\ & Q ( x ) = x ^ { 2 } + 3 x + n \end{aligned}$$
polynomials are given. These two polynomials have a common root and the roots of the polynomial $P(x)$ are equal, so what is the sum $m + n$?
A) $-5$
B) $-3$
C) 2
D) 4
E) 5

Let b and c be non-zero real numbers such that the roots of the equation
$$x ^ { 2 } + b x + c = 0$$
are b and c. Accordingly, what is the product $b \cdot c$?
A) $- 6$
B) $- 5$
C) $- 4$
D) $- 3$
E) $- 2$

The sum of the roots of the equation $x ^ { 2 } - a x + 1 = 0$, which has two real roots, is a root of the equation $$x ^ { 2 } + 6 x + a = 0$$ Accordingly, what is a?\ A) - 3\ B) - 4\ C) - 5\ D) - 6\ E) - 7

A 4th degree polynomial $P ( x )$ with real coefficients and leading coefficient 1 satisfies
$$P ( x ) = P ( - x )$$
for every real number $x$.
$$P ( 2 ) = P ( 3 ) = 0$$
Given that, what is $\mathbf { P ( 1 ) }$?
A) 12 B) 18 C) 24 D) 30 E) 36