Further Mathematics 7367: Further Algebra and Functions Topical Past Paper Questions

Use the method of differences to show that $$ \sum_{r=2}^{n}\frac{1}{(r-1)r(r+1)}=\frac{1}{4}-\frac{1}{2n}+\frac{1}{2(n+1)} $$ Find the smallest integer $n$ such that [5 marks] $$ \sum_{r=2}^{n}\frac{1}{(r-1)r(r+1)}>0.24999 $$

The quadratic equation $ z^{2}-5z+8=0 $ has roots $ \alpha $ and $ \beta $ Write down the value of $ \alpha + \beta $ and the value of $ \alpha\beta $ [2 marks] Without finding the value of $ \alpha $ or the value of $ \beta $, show that $ \alpha^4 + \beta^4 = -47 $ [4 marks] ) F…

The curve $C_{1}$ has equation $$ \frac{x^{2}}{25}-\frac{y^{2}}{4}=1 $$ The curve $C_{2}$ has equation (a) $$ x^{2}-25y^{2}-6x-200y-416=0 $$ Find a sequence of transformations that maps the graph of $ C_1 $ onto the graph of $ C_2 $ [4 marks] Find the equations of the asymptotes…