Edexcel F3 (Further Pure Mathematics 3) 2024 June

1. The hyperbola \(H\) has

• foci with coordinates \(\left( \pm \frac { 13 } { 2 } , 0 \right)\)
• directrices with equations \(x = \pm \frac { 72 } { 13 }\)
• eccentricity e

Determine

1. the value of \(e\)
2. an equation for \(H\), giving your answer in the form \(p x ^ { 2 } - q y ^ { 2 } = r\), where \(p , q\) and \(r\) are integers.

1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

$$\mathbf { M } = \left( \begin{array} { r r r } 2 & 0 & 3
0 & - 4 & - 3
0 & - 4 & 0 \end{array} \right)$$ Given that \(\mathbf { M }\) has exactly two distinct eigenvalues \(\lambda _ { 1 }\) and \(\lambda _ { 2 }\) where \(\lambda _ { 1 } < \lambda _ { 2 }\)

1. determine a normalised eigenvector corresponding to the eigenvalue \(\lambda _ { 1 }\) The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 4
   - 1
   0 \end{array} \right) + \mu \left( \begin{array} { r } 2
   0
   - 1 \end{array} \right)\), where \(\mu\) is a scalar parameter.
   The transformation \(T\) is represented by \(\mathbf { M }\).
   The line \(l _ { 1 }\) is transformed by \(T\) to the line \(l _ { 2 }\)
2. Determine a vector equation for \(l _ { 2 }\), giving your answer in the form \(\mathbf { r } \times \mathbf { b } = \mathbf { c }\) where \(\mathbf { b }\) and \(\mathbf { c }\) are constant vectors.

1. \(\quad y = \operatorname { arsinh } \left( \sqrt { x ^ { 2 } - 1 } \right) \quad x > 1\)
   1. Prove that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { x ^ { 2 } - 1 } }\)
   $$\mathrm { f } ( x ) = \frac { 1 } { 3 } \operatorname { arsinh } \left( \sqrt { x ^ { 2 } - 1 } \right) - \arctan x \quad x > 1$$
2. Determine the exact values of \(x\) for which \(\mathrm { f } ^ { \prime } ( x ) = 0\)

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

$$\sinh ( A + B ) \equiv \sinh A \cosh B + \cosh A \sinh B$$ (b) Hence express \(10 \sinh x + 8 \cosh x\) in the form \(R \sinh ( x + \alpha )\) where \(R > 0\), giving \(\alpha\) in the form \(\ln p\) where \(p\) is an integer.
(c) Hence solve the equation $$10 \sinh x + 8 \cosh x = 18 \sqrt { 7 }$$ giving your answer in the form \(\ln ( \sqrt { 7 } + q )\) where \(q\) is a rational number to be determined.

5. $$4 x ^ { 2 } + 4 x + 17 \equiv ( 2 x + p ) ^ { 2 } + q$$ where \(p\) and \(q\) are integers.

1. Determine the value of \(p\) and the value of \(q\) Given that $$\frac { 8 x + 5 } { \sqrt { 4 x ^ { 2 } + 4 x + 17 } } \equiv \frac { 1 } { \sqrt { 4 x ^ { 2 } + 4 x + 17 } } + \frac { A x + B } { \sqrt { 4 x ^ { 2 } + 4 x + 17 } }$$ where \(A\) and \(B\) are integers,
2. write down the value of \(A\) and the value of \(B\)
3. Hence use algebraic integration to show that $$\int _ { \frac { 1 } { 3 } } ^ { 1 } \frac { 8 x + 5 } { \sqrt { 4 x ^ { 2 } + 4 x + 17 } } \mathrm {~d} x = k + \frac { 1 } { 2 } \ln k$$ where \(k\) is a rational number to be determined.

1. The ellipse \(E\) has equation

$$\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } = 1$$ The line \(l\) is the normal to \(E\) at the point \(P ( 5 \cos \theta , 3 \sin \theta )\) where \(0 < \theta < \frac { \pi } { 2 }\)

1. Using calculus, show that an equation for \(l\) is $$5 x \sin \theta - 3 y \cos \theta = 16 \sin \theta \cos \theta$$ Given that
   • \(\quad l\) intersects the \(y\)-axis at the point \(Q\)
   • the midpoint of the line segment \(P Q\) is \(M\)
   • determine the exact maximum area of triangle \(O M P\) as \(\theta\) varies, where \(O\) is the origin.
   You must justify your answer.

1. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

\begin{figure}[h] \end{figure} Figure 1 shows the curve with equation $$y = \ln \left( \tanh \frac { x } { 2 } \right) \quad 1 \leqslant x \leqslant 2$$

1. Show that the length, \(s\), of the curve is given by $$s = \int _ { 1 } ^ { 2 } \operatorname { coth } x \mathrm {~d} x$$
2. Hence show that $$s = \ln \left( \mathrm { e } + \frac { 1 } { \mathrm { e } } \right)$$

8. $$I _ { n } = \int _ { 0 } ^ { k } x ^ { n } ( k - x ) ^ { \frac { 1 } { 2 } } \mathrm {~d} x \quad n \geqslant 0$$ where \(k\) is a positive constant.

1. Show that $$I _ { n } = \frac { 2 k n } { 3 + 2 n } I _ { n - 1 } \quad n \geqslant 1$$ Given that $$\int _ { 0 } ^ { k } x ^ { 2 } ( k - x ) ^ { \frac { 1 } { 2 } } \mathrm {~d} x = \frac { 9 \sqrt { 3 } } { 280 }$$
2. use the result in part (a) to determine the exact value of \(k\).

1. The plane \(\Pi _ { 1 }\) has vector equation

$$\mathbf { r } = \left( \begin{array} { l } 5
3
0 \end{array} \right) + s \left( \begin{array} { l } 3
0
1 \end{array} \right) + t \left( \begin{array} { r } 1
- 2
2 \end{array} \right)$$ where \(s\) and \(t\) are scalar parameters.

1. Determine a Cartesian equation for \(\Pi _ { 1 }\) The plane \(\Pi _ { 2 }\) has vector equation \(\mathbf { r } . \left( \begin{array} { r } 5
   - 2
   3 \end{array} \right) = 1\)
2. Determine a vector equation for the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) Give your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors and \(\lambda\) is a scalar parameter. The plane \(\Pi _ { 3 }\) has Cartesian equation \(4 x - 3 y - z = 0\)
3. Use the answer to part (b) to determine the coordinates of the point of intersection of \(\Pi _ { 1 } , \Pi _ { 2 }\) and \(\Pi _ { 3 }\)