Further Mathematics 9231: Integration Topical Past Paper Questions

It is given that $ v = xy^{3} $ and $$ xy^{2}\frac{\mathrm{d}^{2}y}{\mathrm{d}x^{2}}+2xy\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^{2}+(3x+2)y^{2}\frac{\mathrm{d}y}{\mathrm{d}x}+(x+1)y^{3}=x+5. $$ (a) Show that $$ \frac{1}{3}\frac{\mathrm{d}^{2}v}{\mathrm{d}x^{2}}+\frac{\mathrm…

The diagram shows the curve with equation $ y = \frac{1}{2}(3^x) $ for $ 0 \leq x \leq 1 $, together with a set of N rectangles each of width $ \frac{1}{N} $. (a) By considering the sum of the areas of these rectangles, show that $ \int_{0}^{1} \frac{1}{2}(3^{x})dx [A] (c) By sim…

The random variable X has probability density function f given by $$ \mathrm{f}(x)=\left\{\begin{aligned}&\frac{1}{21}(x-1)^{2}\quad&2\leqslant x\leqslant5,\\ &0\quad&otherwise.\end{aligned}\right. $$ (a) Find the cumulative distribution function of X. [3] The random variable Y i…