79085 If \(a \neq p, b \neq q, c \neq r\) and \(a q c=0\), then the
\(\left|\begin{array}{lll}
\mathbf{a} & \mathbf{b} & \mathbf{r} \end{array}\right|\)
value of \(\frac{\mathbf{p}}{\mathbf{p}-\mathbf{a}}+\frac{\mathbf{q}}{\mathbf{q}-\mathbf{b}}+\frac{\mathbf{r}}{\mathbf{r}-\mathbf{c}}\) is

79086 If \(\omega(\neq 1)\) is a cube root of unity, then
\(\left|\begin{array}{ccc} i & 1+i+\omega^{2} & \omega^{2} \\ 1-i & -1 & \omega^{2}-1 \\ -i & -i+\omega-1 & -1 \end{array}\right| \text { is equal to }\)

79088 For \(\left|\begin{array}{ccc}2 & 3 & 5 \\ 1 & 0 & 7 \\ -1 & -2 & 4\end{array}\right|\), the sum of minor and cofactor

79089 If \(2\left|\begin{array}{ll}\sin (\mathrm{A}+\mathrm{B}) & \cos (\mathrm{A}+\mathrm{B}) \\ \cos (\mathrm{A}-\mathrm{B}) & \sin (\mathrm{A}-\mathrm{B})\end{array}\right|+\sqrt{3}=0\), then \(\mathbf{A}=\)

79085 If \(a \neq p, b \neq q, c \neq r\) and \(a q c=0\), then the
\(\left|\begin{array}{lll}
\mathbf{a} & \mathbf{b} & \mathbf{r} \end{array}\right|\)
value of \(\frac{\mathbf{p}}{\mathbf{p}-\mathbf{a}}+\frac{\mathbf{q}}{\mathbf{q}-\mathbf{b}}+\frac{\mathbf{r}}{\mathbf{r}-\mathbf{c}}\) is

79086 If \(\omega(\neq 1)\) is a cube root of unity, then
\(\left|\begin{array}{ccc} i & 1+i+\omega^{2} & \omega^{2} \\ 1-i & -1 & \omega^{2}-1 \\ -i & -i+\omega-1 & -1 \end{array}\right| \text { is equal to }\)

79088 For \(\left|\begin{array}{ccc}2 & 3 & 5 \\ 1 & 0 & 7 \\ -1 & -2 & 4\end{array}\right|\), the sum of minor and cofactor

79089 If \(2\left|\begin{array}{ll}\sin (\mathrm{A}+\mathrm{B}) & \cos (\mathrm{A}+\mathrm{B}) \\ \cos (\mathrm{A}-\mathrm{B}) & \sin (\mathrm{A}-\mathrm{B})\end{array}\right|+\sqrt{3}=0\), then \(\mathbf{A}=\)

79085 If \(a \neq p, b \neq q, c \neq r\) and \(a q c=0\), then the
\(\left|\begin{array}{lll}
\mathbf{a} & \mathbf{b} & \mathbf{r} \end{array}\right|\)
value of \(\frac{\mathbf{p}}{\mathbf{p}-\mathbf{a}}+\frac{\mathbf{q}}{\mathbf{q}-\mathbf{b}}+\frac{\mathbf{r}}{\mathbf{r}-\mathbf{c}}\) is

79086 If \(\omega(\neq 1)\) is a cube root of unity, then
\(\left|\begin{array}{ccc} i & 1+i+\omega^{2} & \omega^{2} \\ 1-i & -1 & \omega^{2}-1 \\ -i & -i+\omega-1 & -1 \end{array}\right| \text { is equal to }\)

79088 For \(\left|\begin{array}{ccc}2 & 3 & 5 \\ 1 & 0 & 7 \\ -1 & -2 & 4\end{array}\right|\), the sum of minor and cofactor

79089 If \(2\left|\begin{array}{ll}\sin (\mathrm{A}+\mathrm{B}) & \cos (\mathrm{A}+\mathrm{B}) \\ \cos (\mathrm{A}-\mathrm{B}) & \sin (\mathrm{A}-\mathrm{B})\end{array}\right|+\sqrt{3}=0\), then \(\mathbf{A}=\)

79085 If \(a \neq p, b \neq q, c \neq r\) and \(a q c=0\), then the
\(\left|\begin{array}{lll}
\mathbf{a} & \mathbf{b} & \mathbf{r} \end{array}\right|\)
value of \(\frac{\mathbf{p}}{\mathbf{p}-\mathbf{a}}+\frac{\mathbf{q}}{\mathbf{q}-\mathbf{b}}+\frac{\mathbf{r}}{\mathbf{r}-\mathbf{c}}\) is

79086 If \(\omega(\neq 1)\) is a cube root of unity, then
\(\left|\begin{array}{ccc} i & 1+i+\omega^{2} & \omega^{2} \\ 1-i & -1 & \omega^{2}-1 \\ -i & -i+\omega-1 & -1 \end{array}\right| \text { is equal to }\)

79088 For \(\left|\begin{array}{ccc}2 & 3 & 5 \\ 1 & 0 & 7 \\ -1 & -2 & 4\end{array}\right|\), the sum of minor and cofactor

79089 If \(2\left|\begin{array}{ll}\sin (\mathrm{A}+\mathrm{B}) & \cos (\mathrm{A}+\mathrm{B}) \\ \cos (\mathrm{A}-\mathrm{B}) & \sin (\mathrm{A}-\mathrm{B})\end{array}\right|+\sqrt{3}=0\), then \(\mathbf{A}=\)