I've managed to do part (a) by dotting the original equation by $\underline{\dot r}$ however I'm having troubles with part (b):
We've learnt very few things to deal with these types of equations these are:
Cross it with something Dot it with something then note for some identities such as $\dfrac{d(\underline{r} \times \frac{d\underline{r}}{dt})}{dt} = \underline{r}\times \underline{ \ddot r}$
but I'm having no progress with these to solve (b),
any help please
Since $$\frac{d}{dt} (\dot r -a\times r)= \ddot r -a\times \dot r=0$$ the expression $\dot r -a\times r$ is constant in time. Denote it by $b$; done.
In part (c), the additional assumption delivers $b\perp a$, which makes it possible to write $b$ as the cross-product of $a$ with another vector.
In part (d), you differentiate the quantity which you want to prove constant, and use the ODE: $$\frac{d}{dt}((r+c)\cdot (r+c)) = 2 \dot r \cdot (r+c) =0$$ because $\vec r$ is the cross product of something with $r+c$.
And so on...