Mathematics 7357: Proof Topical Past Paper Questions

It is given that $$ \mathrm{f}(x)=5x^{3}+x $$ Use differentiation from first principles to prove that $$ \mathrm{f}^{\prime}(x)=15x^{2}+1 $$ [5 marks]

) Expand and simplify $ (a - b)^{2} $ [1 mark] (b) Peter thinks that the sum of any rational number and its reciprocal is always greater than 2 Peter checks two examples: $$ \frac{2}{3}+\frac{3}{2}=2.1\dot{6} $$ $$ 2+\frac{1}{2}=2.5 $$ Use a counter example to show that Peter is…

Assume that $a$ and $b$ are integers such that $$ a^{2}-4b-2=0 $$ Prove that a is even. Prove that a is even. [2 marks] (b) Hence, prove that $ 2b + 1 $ is even and explain why this is a contradiction. [3 marks] (c) Explain what can be deduced about the solutions of the equation…