The rank of matrix \(\begin{bmatrix} k & -1 & 0 \\[0.3em] 0 & k & -1 \\[0.3em] -1 & 0 & k \end{bmatrix}\) is 2, for \( k = \)
If \(A = \begin{bmatrix} 4 & 2 \\[0.3em] -3 & 3 \end{bmatrix}\), then \(A^{-1} =\)
If \(P(A) = \frac{7}{11}\), \( P(B) = \frac{6}{11} \), and \( P(A \cup B) = \frac{8}{11} \), then \( P(A|B) = \) ?}
The eigenvalues of \(\begin{bmatrix} 0 & -i \\[0.3em]i & 0 \end{bmatrix}\)are ____ .
The system of equations \( 2x + 3y + 5z = 9 \); \( 7x + 3y - 2z = 8 \); \( 2x + 3y + \lambda z = h \) have a unique solution ____ .
The solution of the differential equation \( \frac{dx}{dt} = x^2 \) with \( x(0) = 1 \) will tend to infinity as ____ .
If \(f(z) = \frac{1}{2} \log_e(x^2 + y^2) + i \tan^{-1} \left( \frac{y}{x} \right)\) be an analytic function, then \( \alpha \) is ___ .
The rank of the matrix\(\begin{bmatrix} 1 & 1 & 1 \\[0.3em] a & a^2 & a^3 \end{bmatrix}\) is ____ .
The mean of the density function is \(f(x) = \lambda e^{-\lambda x}, x > 0\) is ____ .
For the function \(f(x) = x^2 e^{-x},\) the maximum occurs when \( x \) is equal to ____ .