79143 If \(\left|\begin{array}{lll}b & b^{2} & 1+b^{3} \\ c & c^{2} & 1+c^{3}\end{array}\right|=0\) and the vectors
\(\left(1, a, \mathbf{a}^{2}\right),\left(1, b, b^{2}\right),\left(1, c, c^{2}\right)\) are non-coplaner, then abc \(=\)

79144 If \(\Delta_{1}=\left|\begin{array}{lll}\mathbf{x} & \mathbf{b} & \mathbf{b} \\ \mathbf{a} & \mathbf{x} & \mathbf{b} \\ \mathbf{a} & \mathbf{a} & \mathbf{x}\end{array}\right|\) and \(\Delta_{2}=\left|\begin{array}{ll}\mathbf{x} & \mathbf{b} \\ \mathbf{a} & \mathbf{x}\end{array}\right|\), then \(\frac{\mathbf{d}}{\mathbf{d x}}\left(\Delta_{1}\right)\) is equal to

79145 If \(1, \omega, \omega^{2}\) are the roots of unity, the
\(\Delta=\left|\begin{array}{ccc} 1 & \omega^{n} & \omega^{2 n} \\ \omega^{n} & \omega^{2 n} & 1 \end{array}\right| \text { is equal to }\)

79146 If \(\mathbf{a}+\mathbf{b}+\mathbf{c}=\mathbf{0}\) then determinant
\(\left|\begin{array}{ccc} \mathbf{a}-\mathbf{b}-\mathbf{c} & \mathbf{2 a} & \mathbf{2 a} \\ 2 \mathbf{b} & \mathbf{b}-\mathbf{c}-\mathbf{a} & \mathbf{2 b} \\ \mathbf{2 c} & \mathbf{2 c} & \mathbf{c}-\mathbf{a}-\mathbf{b} \end{array}\right| \text { is equal to, }\)

79143 If \(\left|\begin{array}{lll}b & b^{2} & 1+b^{3} \\ c & c^{2} & 1+c^{3}\end{array}\right|=0\) and the vectors
\(\left(1, a, \mathbf{a}^{2}\right),\left(1, b, b^{2}\right),\left(1, c, c^{2}\right)\) are non-coplaner, then abc \(=\)

79144 If \(\Delta_{1}=\left|\begin{array}{lll}\mathbf{x} & \mathbf{b} & \mathbf{b} \\ \mathbf{a} & \mathbf{x} & \mathbf{b} \\ \mathbf{a} & \mathbf{a} & \mathbf{x}\end{array}\right|\) and \(\Delta_{2}=\left|\begin{array}{ll}\mathbf{x} & \mathbf{b} \\ \mathbf{a} & \mathbf{x}\end{array}\right|\), then \(\frac{\mathbf{d}}{\mathbf{d x}}\left(\Delta_{1}\right)\) is equal to

79145 If \(1, \omega, \omega^{2}\) are the roots of unity, the
\(\Delta=\left|\begin{array}{ccc} 1 & \omega^{n} & \omega^{2 n} \\ \omega^{n} & \omega^{2 n} & 1 \end{array}\right| \text { is equal to }\)

79146 If \(\mathbf{a}+\mathbf{b}+\mathbf{c}=\mathbf{0}\) then determinant
\(\left|\begin{array}{ccc} \mathbf{a}-\mathbf{b}-\mathbf{c} & \mathbf{2 a} & \mathbf{2 a} \\ 2 \mathbf{b} & \mathbf{b}-\mathbf{c}-\mathbf{a} & \mathbf{2 b} \\ \mathbf{2 c} & \mathbf{2 c} & \mathbf{c}-\mathbf{a}-\mathbf{b} \end{array}\right| \text { is equal to, }\)

79143 If \(\left|\begin{array}{lll}b & b^{2} & 1+b^{3} \\ c & c^{2} & 1+c^{3}\end{array}\right|=0\) and the vectors
\(\left(1, a, \mathbf{a}^{2}\right),\left(1, b, b^{2}\right),\left(1, c, c^{2}\right)\) are non-coplaner, then abc \(=\)

79144 If \(\Delta_{1}=\left|\begin{array}{lll}\mathbf{x} & \mathbf{b} & \mathbf{b} \\ \mathbf{a} & \mathbf{x} & \mathbf{b} \\ \mathbf{a} & \mathbf{a} & \mathbf{x}\end{array}\right|\) and \(\Delta_{2}=\left|\begin{array}{ll}\mathbf{x} & \mathbf{b} \\ \mathbf{a} & \mathbf{x}\end{array}\right|\), then \(\frac{\mathbf{d}}{\mathbf{d x}}\left(\Delta_{1}\right)\) is equal to

79145 If \(1, \omega, \omega^{2}\) are the roots of unity, the
\(\Delta=\left|\begin{array}{ccc} 1 & \omega^{n} & \omega^{2 n} \\ \omega^{n} & \omega^{2 n} & 1 \end{array}\right| \text { is equal to }\)

79146 If \(\mathbf{a}+\mathbf{b}+\mathbf{c}=\mathbf{0}\) then determinant
\(\left|\begin{array}{ccc} \mathbf{a}-\mathbf{b}-\mathbf{c} & \mathbf{2 a} & \mathbf{2 a} \\ 2 \mathbf{b} & \mathbf{b}-\mathbf{c}-\mathbf{a} & \mathbf{2 b} \\ \mathbf{2 c} & \mathbf{2 c} & \mathbf{c}-\mathbf{a}-\mathbf{b} \end{array}\right| \text { is equal to, }\)

79143 If \(\left|\begin{array}{lll}b & b^{2} & 1+b^{3} \\ c & c^{2} & 1+c^{3}\end{array}\right|=0\) and the vectors
\(\left(1, a, \mathbf{a}^{2}\right),\left(1, b, b^{2}\right),\left(1, c, c^{2}\right)\) are non-coplaner, then abc \(=\)

79144 If \(\Delta_{1}=\left|\begin{array}{lll}\mathbf{x} & \mathbf{b} & \mathbf{b} \\ \mathbf{a} & \mathbf{x} & \mathbf{b} \\ \mathbf{a} & \mathbf{a} & \mathbf{x}\end{array}\right|\) and \(\Delta_{2}=\left|\begin{array}{ll}\mathbf{x} & \mathbf{b} \\ \mathbf{a} & \mathbf{x}\end{array}\right|\), then \(\frac{\mathbf{d}}{\mathbf{d x}}\left(\Delta_{1}\right)\) is equal to

79145 If \(1, \omega, \omega^{2}\) are the roots of unity, the
\(\Delta=\left|\begin{array}{ccc} 1 & \omega^{n} & \omega^{2 n} \\ \omega^{n} & \omega^{2 n} & 1 \end{array}\right| \text { is equal to }\)

79146 If \(\mathbf{a}+\mathbf{b}+\mathbf{c}=\mathbf{0}\) then determinant
\(\left|\begin{array}{ccc} \mathbf{a}-\mathbf{b}-\mathbf{c} & \mathbf{2 a} & \mathbf{2 a} \\ 2 \mathbf{b} & \mathbf{b}-\mathbf{c}-\mathbf{a} & \mathbf{2 b} \\ \mathbf{2 c} & \mathbf{2 c} & \mathbf{c}-\mathbf{a}-\mathbf{b} \end{array}\right| \text { is equal to, }\)