By sketching a suitable pair of graphs, show that the equation $e^{2x+1}=14-x^3$ has exactly one real root. Show by calculation that this root lies between $0.5$ and $1$.

The equation $\frac{\ln x^2-6}{2}=x-5$ has a root $\alpha ,$ such that $3<\alpha <4$. Use the iterative formula based on the equation $x=\ln x+2$ with starting value $x=3$ to find the value of $\alpha$ correct to $3$ decimal places.

It is given that ${\int }_0^a\left(\frac{1}{3x+5}+2e^{6x}\right)dx=0.6,$ where the constant $a>0.$

Show that $a=\frac{1}{6}\ln \left(2.8+\ln \left(\frac{5}{3a+5}\right)\right).$

It is given that ${\int }_0^a\left(\frac{1}{3x+5}+2e^{6x}\right)dx=0.6,$ where the constant $a>0.$

Use an iterative formula based on the equation in part a, with a starting value of $a=0.2,$ to find the value of $a$ correct to $3$ decimal places. Give the result of each iteration to an appropriate number of decimal places.

A curve has parametric equations $x=\frac{1}{t},y=\sqrt{t+1}$

The point $P$ on the curve has parameter $p.$ It is given that the gradient of the curve at $P$ is $-0.4.$

Show that $p=\frac{2\sqrt{5}}{5}(p+1{)}^{\frac{1}{4}}.$

The diagram shows part of the curve $y={\cos }^{3}x,$ where $x$ is in radians. The shaded region between the curve, the axes and the line $x=\alpha$ is denoted by $R.$ The area of $R$ is equal to $0.3.$

Using the substitution $u=\sin x,$ find ${\int }_0^{\alpha }{\cos }^{3}x \mathrm{d}x.$ Hence show that $\sin \alpha =\frac{0.9}{3-{\sin }^2\alpha }.$

It is given that ${\int }_1^a\ln x \mathrm{d}x=5,$ where $a$ is a constant greater than $1 .$

Show that $a=\frac{4+a}{\ln a}.$