Edexcel FP1 (Further Pure Mathematics 1) 2024 June

1. (a) Given that

$$y = \ln \left( 3 + x ^ { 2 } \right)$$ complete the table with the value of \(y\) corresponding to \(x = 3\), giving your answer to 4 significant figures. In part (b) you must show all stages of your working. \section*{Solutions relying entirely on calculator technology are not acceptable.} (b) Use Simpson's rule with all the values of \(y\) in the completed table to estimate, to 3 significant figures, the value of $$\int _ { 2 } ^ { 5 } \ln \left( 3 + x ^ { 2 } \right) \mathrm { d } x$$ (c) Using your answer to part (b) and making your method clear, estimate the value of $$\int _ { 2 } ^ { 5 } \ln \sqrt { \left( 3 + x ^ { 2 } \right) } \mathrm { d } x$$

1. Use algebra to determine the values of \(x\) for which

$$\left| x ^ { 2 } - 2 x \right| \leqslant x$$

1. Use L'Hospital's rule to show that

$$\lim _ { x \rightarrow 0 } \left( \frac { 1 } { \sin x } - \frac { 1 } { x } \right) = 0$$ (6)

4. $$\left[ \begin{array} { l } \text { The Taylor series expansion of } \mathrm { f } ( x ) \text { about } x = a \text { is given by }
\mathrm { f } ( x ) = \mathrm { f } ( a ) + ( x - a ) \mathrm { f } ^ { \prime } ( a ) + \frac { ( x - a ) ^ { 2 } } { 2 ! } \mathrm { f } ^ { \prime \prime } ( a ) + \ldots + \frac { ( x - a ) ^ { r } } { r ! } \mathrm { f } ^ { ( r ) } ( a ) + \ldots \end{array} \right]$$ The curve with equation \(y = \mathrm { f } ( x )\) satisfies the differential equation $$\cos x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + \sin x = 0$$ Given that \(\left( \frac { \pi } { 4 } , 1 \right)\) is a stationary point of the curve,

1. determine the nature of this stationary point, giving a reason for your answer.
2. Show that \(\frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} x ^ { 3 } } = \sqrt { 2 } - 2\) at this stationary point.
3. Hence determine a series solution for \(y\), in ascending powers of \(\left( x - \frac { \pi } { 4 } \right)\) up to and including the term in \(\left( x - \frac { \pi } { 4 } \right) ^ { 3 }\), giving each coefficient in simplest form.

5. $$y = \mathrm { e } ^ { 3 x } \sin x$$

1. Use Leibnitz's theorem to show that $$\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } } = 28 \mathrm { e } ^ { 3 x } \sin x + 96 \mathrm { e } ^ { 3 x } \cos x$$
2. Hence express \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }\) in the form $$\operatorname { Re } ^ { 3 \mathrm { x } } \sin ( \mathrm { x } + \alpha )$$ where \(R\) and \(\alpha\) are constants to be determined, \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\)

1. The ellipse \(E\) has equation

$$\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } = 1$$ The hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$ where \(a\) and \(b\) are positive constants.
Given that

• the eccentricity of \(H\) is the reciprocal of the eccentricity of \(E\)
• the coordinates of the foci of \(H\) are the same as the coordinates of the foci of \(E\) determine
  1. the value of \(a\)
  2. the value of \(b\)

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

\section*{Solutions relying on calculator technology are not acceptable.}

1. Use the substitution \(t = \tan \left( \frac { \theta } { 2 } \right)\) to show that $$\int \frac { 1 } { 2 \sin \theta + \cos \theta + 2 } d \theta = \int \frac { a } { ( t + b ) ^ { 2 } + c } d t$$ where \(a\), \(b\) and \(c\) are constants to be determined.
2. Hence show that $$\int _ { \frac { \pi } { 2 } } ^ { \frac { 2 \pi } { 3 } } \frac { 1 } { 2 \sin \theta + \cos \theta + 2 } d \theta = \ln \left( \frac { 2 \sqrt { 3 } } { 3 } \right)$$

1. The parabola \(P\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant.

The point \(A \left( a t ^ { 2 } , 2 a t \right)\), where \(t \neq 0\), lies on \(P\).

1. Use calculus to show that an equation of the tangent to \(P\) at \(A\) is $$y t = x + a t ^ { 2 }$$ The point \(B \left( 2 k ^ { 2 } , 4 k \right)\) and the point \(C \left( 2 k ^ { 2 } , - 4 k \right)\), where \(k\) is a constant, lie on \(P\).
   The tangent to \(P\) at \(B\) and the tangent to \(P\) at \(C\) intersect at the point \(D\).
   Given that the area of the triangle \(B C D\) is 432
2. determine the coordinates of \(B\) and the coordinates of \(C\).

1. 1. The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 2
      - 3
      1 \end{array} \right) + \lambda \left( \begin{array} { r } 3
      4
      - 1 \end{array} \right)\)

The line \(l _ { 2 }\) has equation \(\mathbf { r } = \left( \begin{array} { c } 13
5
8 \end{array} \right) + \mu \left( \begin{array} { r } 1
- 2
5 \end{array} \right)\)
where \(\lambda\) and \(\mu\) are scalar parameters.
The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(P\).

1. Determine the coordinates of \(P\). Given that the plane \(\Pi\) contains both \(l _ { 1 }\) and \(l _ { 2 }\)
2. determine a Cartesian equation for \(\Pi\).
   (ii) Determine a Cartesian equation for each of the two lines that
   • pass through \(( 0,0,0 )\)
   • make an angle of \(60 ^ { \circ }\) with the \(x\)-axis
   • make an angle of \(45 ^ { \circ }\) with the \(y\)-axis

1. The motion of a particle \(P\) along the \(x\)-axis is modelled by the differential equation

$$t ^ { 2 } \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } - 2 t ( t + 1 ) \frac { \mathrm { d } x } { \mathrm {~d} t } + 2 ( t + 1 ) x = 8 t ^ { 3 } \mathrm { e } ^ { t }$$ where \(P\) has displacement \(x\) metres from the origin \(O\) at time \(t\) minutes, \(t > 0\)

1. Show that the transformation \(x = t u\) transforms the differential equation (I) into the differential equation $$\frac { \mathrm { d } ^ { 2 } u } { \mathrm {~d} t ^ { 2 } } - 2 \frac { \mathrm {~d} u } { \mathrm {~d} t } = 8 \mathrm { e } ^ { t }$$ Given that \(P\) is at \(O\) when \(t = \ln 3\) and when \(t = \ln 5\)
2. determine the particular solution of the differential equation (I)