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 $\alpha , \beta$ 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 $\alpha$ 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 $\_\_\_\_$.

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