79072 If \(A=\left(\begin{array}{cc}0 & \sin \alpha \\ \sin \alpha & 0\end{array}\right)\) and \(\operatorname{det}\left(A^{2}-\frac{1}{2} I\right)=0\), then a possible value of \(\alpha\) is

79073 If \(x, y, z\) are in arithmetic progression with common difference \(d, x \neq 3 d\), and the determinant of the matrix \(\left[\begin{array}{ccc}3 & 4 \sqrt{2} & x \\ 4 & 5 \sqrt{2} & y \\ 5 & k & z\end{array}\right]\) is zero, then the value of \(\mathrm{k}^{\mathbf{2}}\) is

79074 Let \(A=\left[a_{i j}\right]\) be a \(3 \times 3\) matrix, where
\(\mathbf{a}_{\mathrm{ij}}=\left\{\begin{array}{cc}1 & \text { if } \mathbf{i}=\mathbf{j} \\ -\mathbf{x}, & \text { if }|\mathbf{i}-\mathbf{j}|=\mathbf{1} \\ 2 \mathrm{x}+1, & \text { otherwise }\end{array}\right.\)
Let a function \(f: R \rightarrow R\) be defined as \(f(x)=\) det (A). Then, the sum of maximum and minimum values of \(f\) on \(R\) is equal to

79075 Let \(\mathbf{A}=\left(\begin{array}{ccc}{[x+1]} & {[x+2]} & {[x+3]} \\ {[x]} & {[x+3]} & {[x+3]} \\ {[x]} & {[x+2]} & {[x+4]}\end{array}\right)\), where \([t]\) denotes the greatest integer less than or equal to \(t\). If \(\operatorname{det}(A)=192\), then the set of values of \(x\) is the interval

79072 If \(A=\left(\begin{array}{cc}0 & \sin \alpha \\ \sin \alpha & 0\end{array}\right)\) and \(\operatorname{det}\left(A^{2}-\frac{1}{2} I\right)=0\), then a possible value of \(\alpha\) is

79073 If \(x, y, z\) are in arithmetic progression with common difference \(d, x \neq 3 d\), and the determinant of the matrix \(\left[\begin{array}{ccc}3 & 4 \sqrt{2} & x \\ 4 & 5 \sqrt{2} & y \\ 5 & k & z\end{array}\right]\) is zero, then the value of \(\mathrm{k}^{\mathbf{2}}\) is

79074 Let \(A=\left[a_{i j}\right]\) be a \(3 \times 3\) matrix, where
\(\mathbf{a}_{\mathrm{ij}}=\left\{\begin{array}{cc}1 & \text { if } \mathbf{i}=\mathbf{j} \\ -\mathbf{x}, & \text { if }|\mathbf{i}-\mathbf{j}|=\mathbf{1} \\ 2 \mathrm{x}+1, & \text { otherwise }\end{array}\right.\)
Let a function \(f: R \rightarrow R\) be defined as \(f(x)=\) det (A). Then, the sum of maximum and minimum values of \(f\) on \(R\) is equal to

79075 Let \(\mathbf{A}=\left(\begin{array}{ccc}{[x+1]} & {[x+2]} & {[x+3]} \\ {[x]} & {[x+3]} & {[x+3]} \\ {[x]} & {[x+2]} & {[x+4]}\end{array}\right)\), where \([t]\) denotes the greatest integer less than or equal to \(t\). If \(\operatorname{det}(A)=192\), then the set of values of \(x\) is the interval

79072 If \(A=\left(\begin{array}{cc}0 & \sin \alpha \\ \sin \alpha & 0\end{array}\right)\) and \(\operatorname{det}\left(A^{2}-\frac{1}{2} I\right)=0\), then a possible value of \(\alpha\) is

79073 If \(x, y, z\) are in arithmetic progression with common difference \(d, x \neq 3 d\), and the determinant of the matrix \(\left[\begin{array}{ccc}3 & 4 \sqrt{2} & x \\ 4 & 5 \sqrt{2} & y \\ 5 & k & z\end{array}\right]\) is zero, then the value of \(\mathrm{k}^{\mathbf{2}}\) is

79074 Let \(A=\left[a_{i j}\right]\) be a \(3 \times 3\) matrix, where
\(\mathbf{a}_{\mathrm{ij}}=\left\{\begin{array}{cc}1 & \text { if } \mathbf{i}=\mathbf{j} \\ -\mathbf{x}, & \text { if }|\mathbf{i}-\mathbf{j}|=\mathbf{1} \\ 2 \mathrm{x}+1, & \text { otherwise }\end{array}\right.\)
Let a function \(f: R \rightarrow R\) be defined as \(f(x)=\) det (A). Then, the sum of maximum and minimum values of \(f\) on \(R\) is equal to

79075 Let \(\mathbf{A}=\left(\begin{array}{ccc}{[x+1]} & {[x+2]} & {[x+3]} \\ {[x]} & {[x+3]} & {[x+3]} \\ {[x]} & {[x+2]} & {[x+4]}\end{array}\right)\), where \([t]\) denotes the greatest integer less than or equal to \(t\). If \(\operatorname{det}(A)=192\), then the set of values of \(x\) is the interval

79072 If \(A=\left(\begin{array}{cc}0 & \sin \alpha \\ \sin \alpha & 0\end{array}\right)\) and \(\operatorname{det}\left(A^{2}-\frac{1}{2} I\right)=0\), then a possible value of \(\alpha\) is

79073 If \(x, y, z\) are in arithmetic progression with common difference \(d, x \neq 3 d\), and the determinant of the matrix \(\left[\begin{array}{ccc}3 & 4 \sqrt{2} & x \\ 4 & 5 \sqrt{2} & y \\ 5 & k & z\end{array}\right]\) is zero, then the value of \(\mathrm{k}^{\mathbf{2}}\) is

79074 Let \(A=\left[a_{i j}\right]\) be a \(3 \times 3\) matrix, where
\(\mathbf{a}_{\mathrm{ij}}=\left\{\begin{array}{cc}1 & \text { if } \mathbf{i}=\mathbf{j} \\ -\mathbf{x}, & \text { if }|\mathbf{i}-\mathbf{j}|=\mathbf{1} \\ 2 \mathrm{x}+1, & \text { otherwise }\end{array}\right.\)
Let a function \(f: R \rightarrow R\) be defined as \(f(x)=\) det (A). Then, the sum of maximum and minimum values of \(f\) on \(R\) is equal to

79075 Let \(\mathbf{A}=\left(\begin{array}{ccc}{[x+1]} & {[x+2]} & {[x+3]} \\ {[x]} & {[x+3]} & {[x+3]} \\ {[x]} & {[x+2]} & {[x+4]}\end{array}\right)\), where \([t]\) denotes the greatest integer less than or equal to \(t\). If \(\operatorname{det}(A)=192\), then the set of values of \(x\) is the interval