trigonometry Questions

Let $\alpha ,\beta$ be such that $\pi <\alpha -\beta <3\pi .$ If $\sin \alpha +\sin \beta =-\frac{21}{65}$ and $\cos \alpha +\cos \beta =-\frac{27}{65},$ then the value of $\cos \frac{\alpha -\beta }{2}$ is

If $u=\sqrt{a^2{\cos }^2\theta +b^2{\sin }^2\theta }+\sqrt{a^2{\sin }^2\theta +b^2{\cos }^2\theta },$ then the difference between the maximum and minimum values of $u^2$ is given by

The sides of a triangle are $\sin \alpha ,\cos \alpha$ and $\sqrt{1+\sin \alpha \cos \alpha }$ for some $0<\alpha <\frac{\pi }{2}.$ Then the greatest angle of the triangle is

A person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank of the river is $60^{\circ }$ and when he retires $40$ meter away from the tree the angle of elevation becomes $30^{\circ }.$ The breadth of the river is

If $f:R\to S,$ defined by $f(x)=\sin x-\sqrt{3}\cos x+1,$ is onto, then the interval of $S$ is