CAIE
FP1
2010
June
Q11 EITHER
Challenging +1.3
The variables \(z\) and \(x\) are related by the differential equation
$$3 z ^ { 2 } \frac { \mathrm {~d} ^ { 2 } z } { \mathrm {~d} x ^ { 2 } } + 6 z ^ { 2 } \frac { \mathrm {~d} z } { \mathrm {~d} x } + 6 z \left( \frac { \mathrm {~d} z } { \mathrm {~d} x } \right) ^ { 2 } + 5 z ^ { 3 } = 5 x + 2$$
Use the substitution \(y = z ^ { 3 }\) to show that \(y\) and \(x\) are related by the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = 5 x + 2$$
Given that \(z = 1\) and \(\frac { \mathrm { d } z } { \mathrm {~d} x } = - \frac { 2 } { 3 }\) when \(x = 0\), find \(z\) in terms of \(x\).
Deduce that, for large positive values of \(x , z \approx x ^ { \frac { 1 } { 3 } }\).
CAIE
FP1
2010
June
Q3
Standard +0.8
3 The sequence \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\) is such that \(x _ { 1 } = 3\) and
$$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 2 } + 4 x _ { n } - 2 } { 2 x _ { n } + 3 }$$
for \(n = 1,2,3 , \ldots\). Prove by induction that \(x _ { n } > 2\) for all \(n\).
CAIE
FP1
2010
June
Q8
Challenging +1.2
8 Obtain the general solution of the differential equation
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 5 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = 10 \sin 3 x - 20 \cos 3 x$$
Show that, for large positive \(x\) and independently of the initial conditions,
$$y \approx R \sin ( 3 x + \phi )$$
where the constants \(R\) and \(\phi\), such that \(R > 0\) and \(0 < \phi < 2 \pi\), are to be determined correct to 2 decimal places.
CAIE
FP1
2011
June
Q2
Standard +0.8
2 The roots of the equation
$$x ^ { 3 } + p x ^ { 2 } + q x + r = 0$$
are \(\frac { \beta } { k } , \beta , k \beta\), where \(p , q , r , k\) and \(\beta\) are non-zero real constants. Show that \(\beta = - \frac { q } { p }\).
Deduce that \(r p ^ { 3 } = q ^ { 3 }\).
CAIE
FP1
2011
June
Q10
Challenging +1.2
10 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 \(a\).