O produto das raízes da equação $2x^2 - 5x + 3 = 0$ é
(A) $\dfrac{2}{3}$ (B) $\dfrac{3}{2}$ (C) $\dfrac{5}{2}$ (D) $2$ (E) $3$

The function $f(x) = x^2 - 4x + 3$ has roots $x_1$ and $x_2$. What is the value of $x_1 + x_2$?
(A) $-4$
(B) $-3$
(C) $3$
(D) $4$
(E) $7$

Consider the polynomial $$p(x) = x^6 + 10x^5 + 11x^4 + 12x^3 + 13x^2 - 12x - 11.$$ Let $z_1, z_2, z_3, z_4, z_5, z_6$ be the six complex roots of $p(x)$. Evaluate $\sum_{i=1}^{6} z_i^2$. [2 points]

For every integer $n > 1$, we define $P _ { n } \in \mathbb { Z } [ X ]$ by $$P _ { n } = X ^ { 4 } - ( 6 + n ) X ^ { 3 } + ( 10 + n ) X ^ { 2 } - ( 6 + n ) X + 1 .$$ We denote by $\alpha _ { n } , \frac { 1 } { \alpha _ { n } } , \gamma _ { n } , \frac { 1 } { \gamma _ { n } }$ the roots of $P _ { n }$ in $\mathbb { C }$ and we set $$t _ { n } = \alpha _ { n } + \frac { 1 } { \alpha _ { n } } , \quad s _ { n } = \gamma _ { n } + \frac { 1 } { \gamma _ { n } } .$$ Show that $t _ { n } + s _ { n } = 6 + n$ and $t _ { n } s _ { n } = 8 + n$.

Let $P ( X )$ be a monic polynomial of degree $d \geqslant 1$ with complex coefficients which we write in the form: $$P ( X ) = a _ { 0 } + a _ { 1 } X + a _ { 2 } X ^ { 2 } + \cdots + a _ { d - 1 } X ^ { d - 1 } + X ^ { d }$$ We assume that $a _ { 0 } \neq 0$. We denote by $\lambda _ { 1 } , \ldots , \lambda _ { d } \in \mathbb { C }$ the roots of $P ( X )$ (with multiplicity). For all integers $n \geqslant 1$, we define: $$N _ { n } = \lambda _ { 1 } ^ { n } + \lambda _ { 2 } ^ { n } + \cdots + \lambda _ { d } ^ { n }$$
8a. Show that if $a _ { 0 } , \ldots , a _ { d - 1 }$ are elements of $\mathbb { Q }$, then $N _ { n } \in \mathbb { Q }$ for all $n \geqslant 1$.
8b. Conversely, show that if $N _ { n } \in \mathbb { Q }$ for all $n \geqslant 1$, then $a _ { 0 } , \ldots , a _ { d - 1 }$ are elements of $\mathbb { Q }$.
8c. Deduce that if $\mu _ { 1 } , \ldots , \mu _ { d }$ are complex numbers and if $P ( X ) = \prod _ { i = 1 } ^ { d } \left( X - \mu _ { i } \right)$, then $P ( X ) \in \mathbb { Q } [ X ]$ if and only if $$\forall n \geqslant 1 , \quad \sum _ { i = 1 } ^ { d } \mu _ { i } ^ { n } \in \mathbb { Q }$$

Let $P ( X )$ be a monic polynomial of degree $d \geqslant 1$ with complex coefficients which we write in the form: $$P ( X ) = a _ { 0 } + a _ { 1 } X + a _ { 2 } X ^ { 2 } + \cdots + a _ { d - 1 } X ^ { d - 1 } + X ^ { d }$$ We assume that $a _ { 0 } \neq 0$. We denote by $\lambda _ { 1 } , \ldots , \lambda _ { d } \in \mathbb { C }$ the roots of $P ( X )$ (with multiplicity). For all integers $n \geqslant 1$, we define: $$N _ { n } = \lambda _ { 1 } ^ { n } + \lambda _ { 2 } ^ { n } + \cdots + \lambda _ { d } ^ { n }$$
8a. Show that if $a _ { 0 } , \ldots , a _ { d - 1 }$ are elements of $\mathbb { Q }$, then $N _ { n } \in \mathbb { Q }$ for all $n \geqslant 1$.
8b. Conversely, show that if $N _ { n } \in \mathbb { Q }$ for all $n \geqslant 1$, then $a _ { 0 } , \ldots , a _ { d - 1 }$ are elements of $\mathbb { Q }$.
8c. Deduce that if $\mu _ { 1 } , \ldots , \mu _ { d }$ are complex numbers and if $P ( X ) = \prod _ { i = 1 } ^ { d } \left( X - \mu _ { i } \right)$, then $P ( X ) \in \mathbb { Q } [ X ]$ if and only if $$\forall n \geqslant 1 , \quad \sum _ { i = 1 } ^ { d } \mu _ { i } ^ { n } \in \mathbb { Q } .$$

If $a^4 + a^3 + a^2 + a + 1 = 0$, find the value of $a^{2m} + a^m + 1/a^m + 1/a^{2m}$ when $m$ is a multiple of 5, and find $a^{4m} + a^{3m} + a^{2m} + a^m$.

Let $s, sr, sr^2, sr^3$ be the roots of $x^4 + ax^3 + bx^2 + cx + d = 0$ (roots in geometric progression). Show that $c^2 = a^2 d$.

(a) Let $n \geq 1$ be an integer. Prove that $X ^ { n } + Y ^ { n } + Z ^ { n }$ can be written as a polynomial with integer coefficients in the variables $\alpha = X + Y + Z$, $\beta = X Y + Y Z + Z X$ and $\gamma = X Y Z$.
(b) Let $G _ { n } = x ^ { n } \sin ( n A ) + y ^ { n } \sin ( n B ) + z ^ { n } \sin ( n C )$, where $x , y , z , A , B , C$ are real numbers such that $A + B + C$ is an integral multiple of $\pi$. Using (a) or otherwise, show that if $G _ { 1 } = G _ { 2 } = 0$, then $G _ { n } = 0$ for all positive integers $n$.

Let $\alpha, \beta$ be the roots of the equation $x^2 - px + r = 0$ and $\frac{\alpha}{2}, 2\beta$ be the roots of the equation $x^2 - qx + r = 0$. Then the value of $r$ is
(A) $\frac{2}{9}(p-q)(2q-p)$
(B) $\frac{2}{9}(q-p)(2p-q)$
(C) $\frac{2}{9}(q-2p)(2q-p)$
(D) $\frac{2}{9}(2p-q)(2q-p)$

Let $p$ and $q$ be real numbers such that $p \neq 0 , p ^ { 3 } \neq q$ and $p ^ { 3 } \neq - q$. If $\alpha$ and $\beta$ are nonzero complex numbers satisfying $\alpha + \beta = - p$ and $\alpha ^ { 3 } + \beta ^ { 3 } = q$, then a quadratic equation having $\frac { \alpha } { \beta }$ and $\frac { \beta } { \alpha }$ as its roots is
A) $\left( p ^ { 3 } + q \right) x ^ { 2 } - \left( p ^ { 3 } + 2 q \right) x + \left( p ^ { 3 } + q \right) = 0$
B) $\left( p ^ { 3 } + q \right) x ^ { 2 } - \left( p ^ { 3 } - 2 q \right) x + \left( p ^ { 3 } + q \right) = 0$
C) $\left( \mathrm { p } ^ { 3 } - \mathrm { q } \right) \mathrm { x } ^ { 2 } - \left( 5 \mathrm { p } ^ { 3 } - 2 \mathrm { q } \right) \mathrm { x } + \left( \mathrm { p } ^ { 3 } - \mathrm { q } \right) = 0$
D) $\left( p ^ { 3 } - q \right) x ^ { 2 } - \left( 5 p ^ { 3 } + 2 q \right) x + \left( p ^ { 3 } - q \right) = 0$

Let $f ( x ) = x ^ { 4 } + a x ^ { 3 } + b x ^ { 2 } + c$ be a polynomial with real coefficients such that $f ( 1 ) = - 9$. Suppose that $i \sqrt { 3 }$ is a root of the equation $4 x ^ { 3 } + 3 a x ^ { 2 } + 2 b x = 0$, where $i = \sqrt { - 1 }$. If $\alpha _ { 1 } , \alpha _ { 2 } , \alpha _ { 3 }$, and $\alpha _ { 4 }$ are all the roots of the equation $f ( x ) = 0$, then $\left| \alpha _ { 1 } \right| ^ { 2 } + \left| \alpha _ { 2 } \right| ^ { 2 } + \left| \alpha _ { 3 } \right| ^ { 2 } + \left| \alpha _ { 4 } \right| ^ { 2 }$ is equal to $\_\_\_\_$ .

If $p$ and $q$ are non-zero real numbers and $\alpha ^ { 3 } + \beta ^ { 3 } = - p , \alpha \beta = q$, then a quadratic equation whose roots are $\frac { \alpha ^ { 2 } } { \beta } , \frac { \beta ^ { 2 } } { \alpha }$ is :
(1) $p x ^ { 2 } - q x + p ^ { 2 } = 0$
(2) $q x ^ { 2 } + p x + q ^ { 2 } = 0$
(3) $p x ^ { 2 } + q x + p ^ { 2 } = 0$
(4) $q x ^ { 2 } - p x + q ^ { 2 } = 0$

Let $\alpha$ and $\beta$ be the roots of equation $px^2 + qx + r = 0$, $p \neq 0$. If $p$, $q$, $r$ are in A.P. and $\frac{1}{\alpha} + \frac{1}{\beta} = 4$, then the value of $|\alpha - \beta|$ is:
(1) $\frac{\sqrt{61}}{9}$
(2) $\frac{2\sqrt{17}}{9}$
(3) $\frac{\sqrt{34}}{9}$
(4) $\frac{2\sqrt{13}}{9}$

If $\lambda \in \mathrm { R }$ is such that the sum of the cubes of the roots of the equation, $x ^ { 2 } + ( 2 - \lambda ) x + ( 10 - \lambda ) = 0$ is minimum, then the magnitude of the difference of the roots of this equation is
(1) 20
(2) $2 \sqrt { 5 }$
(3) $2 \sqrt { 7 }$
(4) $4 \sqrt { 2 }$

The value of $\lambda$ such that sum of the squares of the roots of the quadratic equation, $x ^ { 2 } + ( 3 - \lambda ) x + 2 = \lambda$ has the least value is:
(1) 2
(2) $\frac { 4 } { 9 }$
(3) $\frac { 15 } { 8 }$
(4) 1

If $\lambda$ be the ratio of the roots of the quadratic equation in $x , 3 m ^ { 2 } x ^ { 2 } + m ( m - 4 ) x + 2 = 0$, then the least value of $m$ for which $\lambda + \frac { 1 } { \lambda } = 1$, is :
(1) $2 - \sqrt { 3 }$
(2) $- 2 + \sqrt { } \overline { 2 }$
(3) $4 - 2 \sqrt { 3 }$
(4) $4 - 3 \sqrt { 2 }$

If $\alpha$ and $\beta$ be the roots of the equation $x^2 - 2x + 2 = 0$, then the least value of $n$ for which $\frac{\alpha^n}{\beta} = 1$ is
(1) 5
(2) 4
(3) 2
(4) 3

Let $\alpha$ and $\beta$ be the roots of the equation $x ^ { 2 } - x - 1 = 0$. If $p _ { k } = ( \alpha ) ^ { k } + ( \beta ) ^ { k } , k \geq 1$, then which one of the following statements is not true?
(1) $p _ { 3 } = p _ { 5 } - p _ { 4 }$
(2) $p _ { 5 } = 11$
(3) $\left( p _ { 1 } + p _ { 2 } + p _ { 3 } + p _ { 4 } + p _ { 5 } \right) = 26$
(4) $p _ { 5 } = p _ { 2 } \cdot p _ { 3 }$

Let $\alpha$ and $\beta$ be the roots of the equation, $5x^{2} + 6x - 2 = 0$. If $S_{n} = \alpha^{n} + \beta^{n}, n = 1,2,3,\ldots$, then
(1) $6S_{6} + 5S_{5} = 2S_{4}$
(2) $5S_{6} + 6S_{5} + 2S_{4} = 0$
(3) $5S_{6} + 6S_{5} = 2S_{4}$
(4) $6S_{6} + 5S_{5} + 2S_{4} = 0$

If $\alpha$ and $\beta$ are the roots of the equation $x ^ { 2 } + p x + 2 = 0$ and $\frac { 1 } { \alpha }$ and $\frac { 1 } { \beta }$ are the roots of the equation $2 x ^ { 2 } + 2 q x + 1 = 0$, then $\left( \alpha - \frac { 1 } { \alpha } \right) \left( \beta - \frac { 1 } { \beta } \right) \left( \alpha + \frac { 1 } { \beta } \right) \left( \beta + \frac { 1 } { \alpha } \right)$ is equal to:
(1) $\frac { 9 } { 4 } \left( 9 + q ^ { 2 } \right)$
(2) $\frac { 9 } { 4 } \left( 9 - q ^ { 2 } \right)$
(3) $\frac { 9 } { 4 } \left( 9 + p ^ { 2 } \right)$
(4) $\frac { 9 } { 4 } \left( 9 - p ^ { 2 } \right)$

Let $\alpha$ and $\beta$ be the roots of $x ^ { 2 } - 3 x + p = 0$ and $\gamma$ and $\delta$ be the roots of $x ^ { 2 } - 6 x + q = 0$. If $\alpha , \beta , \gamma , \delta$ from a geometric progression. Then ratio $( 2 q + p ) : ( 2 q - p )$ is
(1) $3 : 1$
(2) $9 : 7$
(3) $5 : 3$
(4) $33 : 31$

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

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 $\_\_\_\_$.

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