For what value of $t$ does $f(x)=x^3+t x^2+(2-t) x-3$ have only real roots?

If $f(x)=a x^2+b x+c$ and $f\left(\frac{1}{n}\right)=\frac{n+1}{n^2}$ for all $n \in N$, then what is the value of $\lim \limits_{x \rightarrow 0} f^{\prime}(x)$ ?

Consider the differential equation $f^{\prime \prime}+p f^{\prime}+q f=0$ where $p$ and $q$ are constants, $p^2=4 q$ and $q \neq 0$. If y is a nonzero solution of the equation, then which of the following statements is correct?