Edexcel F3 (Further Pure Mathematics 3) 2024 January

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

Solutions relying entirely on calculator technology are not acceptable.

1. Show that $$\int _ { 4 } ^ { 4 \sqrt { 3 } } \frac { 8 } { 16 + x ^ { 2 } } d x = p \pi$$ where \(p\) is a rational number to be determined.
2. Determine the exact value of \(k\) for which $$\int _ { \frac { 3 } { 4 } } ^ { k } \frac { 2 } { \sqrt { 9 - 4 x ^ { 2 } } } d x = \frac { \pi } { 12 }$$

2. $$\mathbf { T } = \left( \begin{array} { l l l } 2 & 3 & 7 \\ 3 & 2 & 6 \\ a & 4 & b \end{array} \right) \quad \mathbf { U } = \left( \begin{array} { r r r } 6 & - 1 & - 4 \\ 15 & c & - 9 \\ - 8 & a & 5 \end{array} \right)$$ where \(a\), \(b\) and \(c\) are constants.
Given that \(\mathbf { T U } = \mathbf { I }\)

1. determine the value of \(a\), the value of \(b\) and the value of \(c\) The transformation represented by the matrix \(\mathbf { T }\) transforms the line \(l _ { 1 }\) to the line \(l _ { 2 }\) Given that \(l _ { 2 }\) has equation $$\frac { x - 1 } { 3 } = \frac { y } { - 4 } = z + 2$$
2. determine a Cartesian equation for \(l _ { 1 }\)

1. The ellipse \(E\) has equation

$$\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$ where \(b\) is a constant and \(0 < b < 7\) The eccentricity of the ellipse is \(e\)

1. Write down, in terms of \(e\) only,
   1. the coordinates of the foci of \(E\)
   2. the equations of the directrices of \(E\) Given that
      • the point \(P ( x , y )\) lies on \(E\) where \(x > 0\)
2. the point \(S\) is the focus of \(E\) on the positive \(x\)-axis
3. the line \(l\) is the directrix of \(E\) which crosses the positive \(x\)-axis
4. the point \(M\) lies on \(l\) such that the line through \(P\) and \(M\) is parallel to the \(x\)-axis
5. determine an expression for
   1. \(P S ^ { 2 }\) in terms of \(e , x\) and \(y\)
   2. \(P M ^ { 2 }\) in terms of \(e\) and \(x\)
6. Hence show that
$$b ^ { 2 } = 49 \left( 1 - e ^ { 2 } \right)$$ Given that \(E\) crosses the \(y\)-axis at the points with coordinates \(( 0 , \pm 4 \sqrt { 3 } )\)
7. determine the value of \(e\) Given that the \(x\) coordinate of \(P\) is \(\frac { 7 } { 2 }\)
8. determine the area of triangle \(O P M\), where \(O\) is the origin.

4. $$\mathbf { M } = \left( \begin{array} { r r r } 0 & - 1 & 3 \\ - 1 & 4 & - 1 \\ 3 & - 1 & 0 \end{array} \right)$$ Given that \(\left( \begin{array} { r } 1 \\ - 2 \\ 1 \end{array} \right)\) is an eigenvector of \(\mathbf { M }\)

1. determine its corresponding eigenvalue. Given that - 3 is an eigenvalue of \(\mathbf { M }\)
2. determine a corresponding eigenvector. Hence, given that \(\left( \begin{array} { l } 1 \\ 1 \\ 1 \end{array} \right)\) is also an eigenvector of \(\mathbf { M }\)
3. determine a diagonal matrix \(\mathbf { D }\) and an orthogonal matrix \(\mathbf { P }\) such that \(\mathbf { D } = \mathbf { P } ^ { \mathrm { T } } \mathbf { M P }\)

1. (a) Use the definitions of hyperbolic functions in terms of exponentials to prove that

$$\begin{gathered} 1 - \operatorname { sech } ^ { 2 } x \equiv \tanh ^ { 2 } x \\ I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 3 } \ln 2 } \tanh ^ { n } 3 x \mathrm {~d} x \quad n \in \mathbb { Z } \quad n \geqslant 0 \end{gathered}$$ (b) Show that $$I _ { n } = I _ { n - 2 } - \frac { p ^ { n - 1 } } { 3 ( n - 1 ) } \quad n \geqslant 2$$ where \(p\) is a rational number to be determined.
(c) Hence determine the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 3 } \ln 2 } \tanh ^ { 5 } 3 x \mathrm {~d} x$$ giving your answer in the form \(a \ln b + c\) where \(a , b\) and \(c\) are rational numbers to be found.

1. The points \(A , B\) and \(C\) have coordinates ( \(3,2,2\) ), ( \(- 1,1,3\) ) and ( \(- 2,4,2\) ) respectively. The plane \(\Pi _ { 1 }\) contains the points \(A , B\) and \(C\)
   1. Determine a Cartesian equation of \(\Pi _ { 1 }\)
   Given that
   • point \(D\) has coordinates \(( - 1,1 , - 2 )\)
   • line \(l\) passes through \(D\) and is perpendicular to \(\Pi _ { 1 }\)
   • plane \(\Pi _ { 2 }\) has equation \(\mathbf { r } . ( 14 \mathbf { i } - \mathbf { j } - 17 \mathbf { k } ) = - 66\)
   • \(I\) meets \(\Pi _ { 2 }\) at the point \(E\)
   • show that \(D E = p \sqrt { 22 }\) where \(p\) is a rational number to be determined.
   The point \(F\) has coordinates ( \(4,3 , q\) ) where \(q\) is a constant.
   Given that \(A , B , C\) and \(F\) are the vertices of a tetrahedron of volume 12
2. determine the possible values of \(q\)

7.

1. Show that \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{469976eb-f1a9-4bdc-8f52-64ab23856109-26_1088_691_251_676} \captionsetup{labelformat=empty} \caption{Figure 1}
   \end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = \arccos ( \operatorname { sech } x ) + \operatorname { coth } x \quad x > 0$$ The point \(P\) is a minimum turning point of \(C\)
2. Show that the \(x\) coordinate of \(P\) is \(\ln ( q + \sqrt { q } )\) where \(q = \frac { 1 } { 2 } ( 1 + \sqrt { k } )\) and \(k\) is an integer to be determined.

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 2 shows a sketch of part of the curve \(C\) with equation \(y ^ { 2 } = 8 x\) and part of the line \(l\) with equation \(x = 18\) The region \(R\), shown shaded in Figure 2, is bounded by \(C\) and \(l\)

1. Show that the perimeter of \(R\) is given by $$\alpha + 2 \int _ { 0 } ^ { \beta } \sqrt { 1 + \frac { y ^ { 2 } } { 16 } } d y$$ where \(\alpha\) and \(\beta\) are positive constants to be determined.
2. Use the substitution \(y = 4 \sinh u\) and algebraic integration to determine the exact perimeter of \(R\), giving your answer in simplest form.