If $\mathrm{z}=x+\mathrm{i} y$ is a complex number, then the equation $\left|\frac{z+i}{z-i}\right|=\sqrt{3}$ represents the

If $\int \frac{2 x^2+3}{\left(x^2-1\right)\left(x^2-4\right)} \mathrm{d} x=\log \left[\left(\frac{x-2}{x+2}\right)^{\mathrm{a}} \cdot\left(\frac{x+1}{x-1}\right)^{\mathrm{b}}\right]+\mathrm{c}$, (where c is the constant of integration) then the value of $a+b$ is equal to

For $N \in \mathbb{N}, \frac{\mathrm{~d}^{\mathrm{n}}}{\mathrm{d} x^{\mathrm{n}}}(\log x)=$

In a single toss of a fair die, the odds against the event that number 4 or 5 turns up is