The curve \(C\) has equation
\(y=\frac{x^{2}}{x+\lambda}\)
where \(\lambda\) is a non-zero constant. Obtain the equation of each of the asymptotes of \(C\).
In separate diagrams, sketch \(C\) for the cases \(\lambda\gt 0\) and \(\lambda\lt 0\). In both cases the coordinates of the turning points must be indicated.
Show that if \(y\) depends on \(x\) and \(x=\mathrm{e}^{u}\) then
\(x^{2} \frac{\mathrm{~d}^{2} y}{\mathrm{~d} x^{2}}=\frac{\mathrm{d}^{2} y}{\mathrm{~d} u^{2}}-\frac{\mathrm{d} y}{\mathrm{~d} u} .\)
Given that \(y\) satisfies the differential equation
\(x^{2} \frac{\mathrm{~d}^{2} y}{\mathrm{~d} x^{2}}+5 x \frac{\mathrm{~d} y}{\mathrm{~d} x}+3 y=30 x^{2}\)
use the substitution \(x=\mathrm{e}^{u}\) to show that
\(\frac{\mathrm{d}^{2} y}{\mathrm{~d} u^{2}}+4 \frac{\mathrm{~d} y}{\mathrm{~d} u}+3 y=30 \mathrm{e}^{2 u}\)
Hence find the general solution for \(y\) in terms of \(x\).
(a) Starting from the definitions of sech and tanh in terms of exponentials, prove that
\(1-\operatorname{sech}^{2} t=\tanh ^{2} t\)
The curve \(C\) has parametric equations
\(x=\frac{1}{2} \tanh ^{2} t+\ln \operatorname{sech} t, \quad y=1+\tanh ^{4} t, \quad \text { for } t\gt 0 .\)
(b) Show that \(\frac{\mathrm{d} y}{\mathrm{~d} x}=-4 \operatorname{sech}^{2} t\).
(c) Find the coordinates of the point on \(C\) with \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=-\frac{9}{2}\), giving your answer in the form \((a+\ln b, c)\) where \(a, b\) and \(c\) are rational numbers.
The curve \(C\) has parametric equations
\(x=\frac{1}{2} \mathrm{e}^{2 t}-\frac{1}{3} t^{3}-\frac{1}{2}, \quad y=2 \mathrm{e}^{t}(t-1), \quad \text { for } 0 \leqslant t \leqslant 1 .\)
Find the exact length of \(C\).
It is given that
\(x=1+\frac{1}{t} \quad \text { and } \quad y=\cos ^{-1} t \quad \text { for } 0\lt t\lt 1 .\)
(a) Show that \(\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{t^{2}}{\sqrt{1-t^{2}}}\).
(b) Show that \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=-t^{a}\left(1-t^{2}\right)^{b}\left(2-t^{2}\right)\), where \(a\) and \(b\) are constants to be determined.
It is given that
\(x=1+\frac{1}{t} \quad \text { and } \quad y=t \mathrm{e}^{t} .\)
(a) Show that \(\frac{\mathrm{d} y}{\mathrm{~d} x}=-\mathrm{e}^{t}\left(t^{3}+t^{2}\right)\).
(b) Find \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\) in terms of \(t\).
The curve \(C\) has parametric equations
\(x=\frac{2}{3} t^{\frac{3}{2}}-2 t^{\frac{1}{2}}, \quad y=2 t+5, \quad \text { for } 0\lt t \leqslant 3 .\)
(a) Find the exact length of \(C\).
(b) Find the set of values of \(t\) for which \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\gt 0\).
It is given that
\(x=-t+\tan ^{-1} t \quad \text { and } \quad y=t+\sinh ^{-1} t\)
(a) Show that \(\frac{\mathrm{d} y}{\mathrm{~d} x}=-\frac{t^{2}+1+\sqrt{t^{2}+1}}{t^{2}}\).
(b) Find the value of \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\) when \(t=\frac{3}{4}\).
The curve \(C\) has parametric equations
\(x=\mathrm{e}^{t}-\frac{1}{3} t^{3}, \quad y=4 \mathrm{e}^{\frac{1}{2} t}(t-2), \quad \text { for } 0 \leqslant t \leqslant 2\)
Find, in terms of e , the length of \(C\).
The curve \(C\) has parametric equations
\[x=\frac{1}{2} t^{2}-\ln t, \quad y=2 t+1, \quad \text { for } \frac{1}{2} \leqslant t \leqslant 2 .\]
(a) Find the exact length of \(C\).
(b) Find \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\) in terms of \(t\), simplifying your answer.
The curve \(C\) has parametric equations
\[x=3 t+2 t^{-1}+a t^{3}, \quad y=4 t-\frac{3}{2} t^{-1}+b t^{3}, \quad \text { for } 1 \leqslant t \leqslant 2\]
where \(a\) and \(b\) are constants.
(a) It is given that \(a=\frac{2}{3}\) and \(b=-\frac{1}{2}\).
Show that \(\left(\frac{\mathrm{d} x}{\mathrm{~d} t}\right)^{2}+\left(\frac{\mathrm{d} y}{\mathrm{~d} t}\right)^{2}=\frac{25}{4}\left(t^{2}+t^{-2}\right)^{2}\) and find the exact length of \(C\).
(b) It is given instead that \(a=b=0\).
Find the value of \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\) when \(t=1\).
The curve \(C\) has parametric equations
\(x=\sin t, \quad y=\sin 2 t, \quad \text { for } 0 \leqslant t \leqslant \pi .\)
Find \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\) in terms of \(t\).
Hence, or otherwise, find the coordinates of the stationary points on \(C\) and determine their nature.
Let
\(I_{n}=\int_{0}^{\frac{1}{2} \pi} \cos ^{n} x \mathrm{~d} x\)
where \(n \geqslant 0\). Show that, for all \(n \geqslant 2\),
\(I_{n}=\frac{n-1}{n} I_{n-2} .\)
A curve has parametric equations \(x=a \sin ^{3} t\) and \(y=a \cos ^{3} t\), where \(a\) is a constant and \(0 \leqslant t \leqslant \frac{1}{2} \pi\). Show that the mean value \(m\) of \(y\) over the interval \(0 \leqslant x \leqslant a\) is given by
\(m=3 a \int_{0}^{\frac{1}{2} \pi}\left(\cos ^{4} t-\cos ^{6} t\right) \mathrm{d} t\)
Find the exact value of \(m\), in terms of \(a\).
[Question 11 is printed on the next page.]
The plane \(\Pi_{1}\) has parametric equation
\(\mathbf{r}=2 \mathbf{i}-3 \mathbf{j}+\mathbf{k}+\lambda(\mathbf{i}-2 \mathbf{j}-\mathbf{k})+\mu(\mathbf{i}+2 \mathbf{j}-2 \mathbf{k}) .\)
Find a cartesian equation of \(\Pi_{1}\).
The plane \(\Pi_{2}\) has cartesian equation \(3 x-2 y-3 z=4\). Find the acute angle between \(\Pi_{1}\) and \(\Pi_{2}\).
Find a vector equation of the line of intersection of \(\Pi_{1}\) and \(\Pi_{2}\).
(a) Starting from the definitions of tanh and sech in terms of exponentials, prove that
\(\tanh ^{2} t+\operatorname{sech}^{2} t=1\)
(b) The curve \(C\) has parametric equations
\(x=\ln (\cosh t), \quad y=\tan ^{-1}(\sinh t), \quad \text { for } 0 \leqslant t \leqslant 1\)
Find the length of \(C\).
The curve \(C\) has equation
\(9 y^{2}-3 \sinh ^{-1}(x y)=1-3 \ln 3 .\)
(a) Show that, at the point \(\left(4, \frac{1}{3}\right)\) on \(C, \frac{\mathrm{~d} y}{\mathrm{~d} x}=-\frac{1}{2}\).
(b) Find the value of \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\) at the point \(\left(4, \frac{1}{3}\right)\).
(a) Starting from the definitions of tanh and sech in terms of exponentials, prove that
\(1-\tanh ^{2} u=\operatorname{sech}^{2} u .\)
(b) Show that \(\frac{\mathrm{d}}{\mathrm{d} t}\left(\operatorname{sech}^{-1} t\right)=-\frac{1}{t \sqrt{1-t^{2}}}\).
It is given that
\(x=\tanh ^{-1} t \quad \text { and } \quad y=t \operatorname{sech}^{-1} t, \quad \text { for } 0<t<1 .\)
(c) Show that \(\frac{\mathrm{d} y}{\mathrm{~d} x}=-\sqrt{1-t^{2}}+\left(1-t^{2}\right) \operatorname{sech}^{-1} t\).
(d) Find \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}\) in terms of \(t\).
The curve \(C\) has parametric equations
\(x=\cosh t, \quad y=\sinh t, \quad \text { for } 0\lt t \leqslant \frac{3}{5} .\)
The length of \(C\) is denoted by \(s\).
(a) Show that \(s=\int_{0}^{\frac{3}{5}} \sqrt{\cosh 2 t} \mathrm{~d} t\).
(b) By finding the Maclaurin's series for \(\sqrt{\cosh 2 t}\) up to and including the term in \(t^{2}\), deduce an approximation to \(s\).
(a) Show that \((\cosh x+\sinh x)^{\frac{1}{2}}=\mathrm{e}^{\frac{1}{2} x}\).
(b) Find the particular solution of the differential equation
\(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+\frac{\mathrm{d} y}{\mathrm{~d} x}+3 y=5(\cosh x+\sinh x)^{\frac{1}{2}}\)
given that, when \(x=0, y=1\) and \(\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{4}{3}\).
It is given that \(y=\cosh u\), where \(u\gt 0\), and
\(\sqrt{\cosh ^{2} u-1}\left(\frac{\mathrm{~d}^{2} u}{\mathrm{~d} x^{2}}+\frac{\mathrm{d} u}{\mathrm{~d} x}\right)+\cosh u\left(\frac{\mathrm{~d} u}{\mathrm{~d} x}\right)^{2}-2 \cosh u=4 \mathrm{e}^{-x}\)
(a) Show that
\(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}+\frac{\mathrm{d} y}{\mathrm{~d} x}-2 y=4 \mathrm{e}^{-x}\)
(b) Find \(u\) in terms of \(x\), given that, when \(x=0, u=\ln 3\) and \(\frac{\mathrm{d} u}{\mathrm{~d} x}=3\).