If $$\tan \alpha = {m \over {m + 1}}\,$$ and $$\tan \beta = {2 \over {2m + 1}},$$ find the possible values of $$\left( {\alpha + \beta } \right).$$

Express $${1 \over {1 - \cos \,\theta + 2i\sin \theta }}$$ in the form x + iy.

A horizontal uniform rope of length L, resting on a frictionless horizontal surface, is pulled at one end by force F. What is the tension in the rope at a distance $$l$$ from the end where the force is applied?

A car accelerates from rest at a constant rate $$\alpha $$ for some time after which it decelerates at a constant rate $$\beta $$ to come to rest. If the total time lapse is t seconds, evaluate.

(i) maximum velocity reached, and

(ii) the total distance travelled.