79028 An equilateral triangle has each side equal to ' \(a\) '. If the coordinates of its vertices are \(\left(x_{1}, y_{1}\right)\), \(\left(x_{2}, y_{2}\right),\left(x_{3}, y_{3}\right)\), then what is the square of the determinant \(\left|\begin{array}{lll}\mathbf{x}_{1} & \mathbf{y}_{1} & 1 \\ \mathbf{x}_{2} & \mathbf{y}_{2} & 1 \\ \mathbf{x}_{3} & \mathbf{y}_{3} & 1\end{array}\right|\) equal to ?

79029 If the system of equations \(\alpha x+y+z=5, x+2 y\) \(+3 z=4, x+3 y+5 z=\beta\), has infinitely many solutions, then the ordered pair \((\alpha, \beta)\) is equal to :

79030 Let \(A=\left(\begin{array}{cc}4 & -2 \\ \alpha & \beta\end{array}\right)\) If \(A^{2}+\gamma A+18 I=0\), then \(\operatorname{det}\) (A) is equal to

79031 Let \(A\) be a \(2 \times 2\) matrix with real entries such that \(A^{T}=\alpha A+I\), where \(\alpha \in \mathbb{R}-\{-1,1\}\). If det \(\left(A^{2}-A\right)=4\), then the sum of all possible values of \(\alpha\) is equal to

79028 An equilateral triangle has each side equal to ' \(a\) '. If the coordinates of its vertices are \(\left(x_{1}, y_{1}\right)\), \(\left(x_{2}, y_{2}\right),\left(x_{3}, y_{3}\right)\), then what is the square of the determinant \(\left|\begin{array}{lll}\mathbf{x}_{1} & \mathbf{y}_{1} & 1 \\ \mathbf{x}_{2} & \mathbf{y}_{2} & 1 \\ \mathbf{x}_{3} & \mathbf{y}_{3} & 1\end{array}\right|\) equal to ?

79029 If the system of equations \(\alpha x+y+z=5, x+2 y\) \(+3 z=4, x+3 y+5 z=\beta\), has infinitely many solutions, then the ordered pair \((\alpha, \beta)\) is equal to :

79030 Let \(A=\left(\begin{array}{cc}4 & -2 \\ \alpha & \beta\end{array}\right)\) If \(A^{2}+\gamma A+18 I=0\), then \(\operatorname{det}\) (A) is equal to

79031 Let \(A\) be a \(2 \times 2\) matrix with real entries such that \(A^{T}=\alpha A+I\), where \(\alpha \in \mathbb{R}-\{-1,1\}\). If det \(\left(A^{2}-A\right)=4\), then the sum of all possible values of \(\alpha\) is equal to

79028 An equilateral triangle has each side equal to ' \(a\) '. If the coordinates of its vertices are \(\left(x_{1}, y_{1}\right)\), \(\left(x_{2}, y_{2}\right),\left(x_{3}, y_{3}\right)\), then what is the square of the determinant \(\left|\begin{array}{lll}\mathbf{x}_{1} & \mathbf{y}_{1} & 1 \\ \mathbf{x}_{2} & \mathbf{y}_{2} & 1 \\ \mathbf{x}_{3} & \mathbf{y}_{3} & 1\end{array}\right|\) equal to ?

79029 If the system of equations \(\alpha x+y+z=5, x+2 y\) \(+3 z=4, x+3 y+5 z=\beta\), has infinitely many solutions, then the ordered pair \((\alpha, \beta)\) is equal to :

79030 Let \(A=\left(\begin{array}{cc}4 & -2 \\ \alpha & \beta\end{array}\right)\) If \(A^{2}+\gamma A+18 I=0\), then \(\operatorname{det}\) (A) is equal to

79031 Let \(A\) be a \(2 \times 2\) matrix with real entries such that \(A^{T}=\alpha A+I\), where \(\alpha \in \mathbb{R}-\{-1,1\}\). If det \(\left(A^{2}-A\right)=4\), then the sum of all possible values of \(\alpha\) is equal to

79028 An equilateral triangle has each side equal to ' \(a\) '. If the coordinates of its vertices are \(\left(x_{1}, y_{1}\right)\), \(\left(x_{2}, y_{2}\right),\left(x_{3}, y_{3}\right)\), then what is the square of the determinant \(\left|\begin{array}{lll}\mathbf{x}_{1} & \mathbf{y}_{1} & 1 \\ \mathbf{x}_{2} & \mathbf{y}_{2} & 1 \\ \mathbf{x}_{3} & \mathbf{y}_{3} & 1\end{array}\right|\) equal to ?

79029 If the system of equations \(\alpha x+y+z=5, x+2 y\) \(+3 z=4, x+3 y+5 z=\beta\), has infinitely many solutions, then the ordered pair \((\alpha, \beta)\) is equal to :

79030 Let \(A=\left(\begin{array}{cc}4 & -2 \\ \alpha & \beta\end{array}\right)\) If \(A^{2}+\gamma A+18 I=0\), then \(\operatorname{det}\) (A) is equal to

79031 Let \(A\) be a \(2 \times 2\) matrix with real entries such that \(A^{T}=\alpha A+I\), where \(\alpha \in \mathbb{R}-\{-1,1\}\). If det \(\left(A^{2}-A\right)=4\), then the sum of all possible values of \(\alpha\) is equal to