Mathematics A H230: Proof Topical Past Paper Questions

In this question you must show detailed reasoning. Given that $ \int_{4}^{a}\left(\frac{4}{\sqrt{x}}+3\right)dx=7 $, find the value of a. [7]

(a) Prove that the following statement is not true. $ p $ is a positive integer $ \Rightarrow 2^p \geqslant p^2 $ [1] (b) Prove that the following statement is true. $ m $ and $ n $ are consecutive positive odd numbers $ \Rightarrow mn + 1 $ is the square of an even number [4]

In each of the following cases choose one of the statements $$ P\Rightarrow\mathcal{Q}\qquad P\Leftarrow\mathcal{Q}\qquad P\Leftrightarrow\mathcal{Q} $$ to describe the relationship between $P$ and $Q$. P: $ y = 3x^{5} - 4x^{2} + 12x $ Q: $ \frac{dy}{dx}=15x^{4}-8x+12 $ (ii) $P:…