GATE CSE 1993

For the below question, one or more OPTIONS are correct
The eigen vector(s) of the matrix
\begin{bmatrix} 0 &0 &\alpha\\ 0 &0 &0\\ 0 &0 &0 \end{bmatrix},\alpha \neq 0
is (are)

The differential equation
\frac{d^2 y}{dx^2}+\frac{dy}{dx}+\sin y =0 is:

Simpson's rule for integration gives exact result when f(x) is a polynomial of degree

Which of the following is (are) valid FORTRAN 77 statement(s)?

Fourier series of the periodic function (period 2 \pi ) defined by
f(x) = \begin{cases} 0, -p \lt x \lt 0\\x, 0 \lt x \lt p \end{cases} \text { is }\\ \frac{\pi}{4} + \Sigma \left [ \frac{1}{\pi n^2} \left(\cos n\pi - 1 \right) \cos nx - \frac{1}{n} \cos n\pi \sin nx \right ]
But putting x = \pi, we get the sum of the series
1 + \frac{1}{3^2} + \frac{1}{5^2} + \frac{1}{7^2} + \cdots \text { is }