Edexcel FP1 (Further Pure Mathematics 1) 2023 June

1. (a) Use Simpson's rule with 4 intervals to find an estimate for

$$\int _ { 0 } ^ { 2 } \mathrm { e } ^ { \sin ^ { 2 } x } \mathrm {~d} x$$ Give your answer to 3 significant figures. Given that \(\int _ { 0 } ^ { 2 } \mathrm { e } ^ { \mathrm { sin } ^ { 2 } x } \mathrm {~d} x = 3.855\) to 4 significant figures,
(b) comment on the accuracy of your answer to part (a).

1. The vertical height, \(h \mathrm {~m}\), above horizontal ground, of a passenger on a fairground ride, \(t\) seconds after the ride starts, where \(t \leqslant 5\), is modelled by the differential equation

$$t ^ { 2 } \frac { \mathrm {~d} ^ { 2 } h } { \mathrm {~d} t ^ { 2 } } - 2 t \frac { \mathrm {~d} h } { \mathrm {~d} t } + 2 h = t ^ { 3 }$$

1. Given that \(t = \mathrm { e } ^ { x }\), show that
   1. \(t \frac { \mathrm {~d} h } { \mathrm {~d} t } = \frac { \mathrm { d } h } { \mathrm {~d} x }\)
   2. \(t ^ { 2 } \frac { \mathrm {~d} ^ { 2 } h } { \mathrm {~d} t ^ { 2 } } = \frac { \mathrm { d } ^ { 2 } h } { \mathrm {~d} x ^ { 2 } } - \frac { \mathrm { d } h } { \mathrm {~d} x }\)
2. Hence show that the transformation \(t = \mathrm { e } ^ { x }\) transforms equation (I) into the equation $$\frac { \mathrm { d } ^ { 2 } h } { \mathrm {~d} x ^ { 2 } } - 3 \frac { \mathrm {~d} h } { \mathrm {~d} x } + 2 h = \mathrm { e } ^ { 3 x }$$
3. Hence show that $$h = A t + B t ^ { 2 } + \frac { 1 } { 2 } t ^ { 3 }$$ where \(A\) and \(B\) are constants. Given that when \(t = 1 , h = 2.5\) and when \(t = 2 , \frac { \mathrm {~d} h } { \mathrm {~d} t } = - 1\)
4. determine the height of the passenger above the ground 5 seconds after the start of the ride.

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \frac { x ^ { 2 } - 2 x - 24 } { | x + 6 | }\) and the line with equation \(y = 5 - 4 x\) Use algebra to determine the values of \(x\) for which $$\frac { x ^ { 2 } - 2 x - 24 } { | x + 6 | } < 5 - 4 x$$

1. The ellipse \(E\) has equation

$$\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 9 } = 1$$

1. Determine the exact value of the eccentricity of \(E\) The points \(P ( 4 \cos \theta , 3 \sin \theta )\) and \(Q ( 4 \cos \theta , - 3 \sin \theta )\) lie on \(E\) where \(0 < \theta < \frac { \pi } { 2 }\) The line \(l _ { 1 }\) is the normal to \(E\) at the point \(P\)
2. Use calculus to show that \(l _ { 1 }\) has equation $$4 x \sin \theta - 3 y \cos \theta = 7 \sin \theta \cos \theta$$ The line \(l _ { 2 }\) passes through the origin and the point \(Q\) The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(R\)
3. Determine, in simplest form, the coordinates of \(R\)
4. Hence show that, as \(\theta\) varies, \(R\) lies on an ellipse which has the same eccentricity as ellipse \(E\)

1. (a) Show that the substitution \(t = \tan \left( \frac { x } { 2 } \right)\) transforms the integral

$$\int \frac { 1 } { 2 \sin x - \cos x + 5 } d x$$ into the integral $$\int \frac { 1 } { 3 t ^ { 2 } + 2 t + 2 } \mathrm {~d} t$$ (b) Hence determine $$\int \frac { 1 } { 2 \sin x - \cos x + 5 } d x$$

6. $$y = \ln \left( \mathrm { e } ^ { 2 x } \cos 3 x \right) \quad - \frac { 1 } { 2 } < x < \frac { 1 } { 2 }$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 - 3 \tan 3 x$$
2. Determine \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }\)
3. Hence determine the first 3 non-zero terms in ascending powers of \(x\) of the Maclaurin series expansion of \(\ln \left( \mathrm { e } ^ { 2 x } \cos 3 x \right)\), giving each coefficient in simplest form.
4. Use the Maclaurin series expansion for \(\ln ( 1 + x )\) to write down the first 4 non-zero terms in ascending powers of \(x\) of the Maclaurin series expansion of \(\ln ( 1 + k x )\), where \(k\) is a constant.
5. Hence determine the value of \(k\) for which $$\lim _ { x \rightarrow 0 } \left( \frac { 1 } { x ^ { 2 } } \ln \frac { \mathrm { e } ^ { 2 x } \cos 3 x } { 1 + k x } \right)$$ exists.

1. With respect to a fixed origin \(O\) the point \(A\) has coordinates \(( 3,6,5 )\) and the line \(l\) has equation

$$( \mathbf { r } - ( 12 \mathbf { i } + 30 \mathbf { j } + 39 \mathbf { k } ) ) \times ( 7 \mathbf { i } + 13 \mathbf { j } + 24 \mathbf { k } ) = \mathbf { 0 }$$ The points \(B\) and \(C\) lie on \(l\) such that \(A B = A C = 15\) Given that \(A\) does not lie on \(l\) and that the \(x\) coordinate of \(B\) is negative,

1. determine the coordinates of \(B\) and the coordinates of \(C\)
2. Hence determine a Cartesian equation of the plane containing the points \(A , B\) and \(C\) The point \(D\) has coordinates \(( - 2,1 , \alpha )\), where \(\alpha\) is a constant.
   Given that the volume of the tetrahedron \(A B C D\) is 147
3. determine the possible values of \(\alpha\) Given that \(\alpha > 0\)
4. determine the shortest distance between the line \(l\) and the line passing through the points \(A\) and \(D\), giving your answer to 2 significant figures. \includegraphics[max width=\textwidth, alt={}, center]{c0ac1e1e-16bf-4a06-9eaa-8dcf01177722-24_2267_50_312_1980}