Consider the following series: (i) \( \sum_{n=1}^{\infty} \frac{1}{\sqrt{n}} \) (ii) \( \sum_{n=1}^{\infty} \frac{1}{n(n+1)} \) (iii) \( \sum_{n=1}^{\infty} \frac{1}{n!} \) Choose the correct option.
Consider the matrix \( A \) below: \[ A = \begin{bmatrix} 2 & 3 & 4 & 5 \\ 0 & 6 & 7 & 8 \\ 0 & 0 & \alpha & \beta \\ 0 & 0 & 0 & \gamma \end{bmatrix} \] For which of the following combinations of \( \alpha, \beta, \) and \( \gamma \), is the rank of \( A \) at least three? (i) \( \alpha = 0 \) and \( \beta = \gamma \neq 0 \). (ii) \( \alpha = \beta = \gamma = 0 \). (iii) \( \beta = \gamma = 0 \) and \( \alpha \neq 0 \). (iv) \( \alpha = \beta = \gamma \neq 0 \).
The equation \[ y'' + p(x)y' + q(x)y = r(x) \] is a _________ ordinary differential equation.
Let \[ A = \begin{pmatrix} 1 & 0 & 1 \\ 0 & k & 0 \\ 3 & 0 & -1 \end{pmatrix}. \] If the eigenvalues of \( A \) are -2, 1, and 2, then the value of \( k \) is _. (Answer in integer)
Let \( a_0 = 0 \) and define \( a_n = \frac{1}{2} (1 + a_{n-1}) \) for all positive integers \( n \geq 1 \). The least value of \( n \) for which \( |1 - a_n|<\frac{1}{2^{10}} \) is ______.
The value of \( k \), for which the linear equations \( 2x + 3y = 6 \) and \( 4x + 6y = 3k \) have at least one solution, is ________. (Answer in integer)
Consider the matrix:
\[ A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix} \]
The eigenvalues of the matrix are 0.27 and ____ (rounded off to 2 decimal places).