Edexcel F3 (Further Pure Mathematics 3) 2023 June

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

$$7 \cosh x + 3 \sinh x = 2 \mathrm { e } ^ { x } + 7$$ Give your answers as simplified natural logarithms.

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

Solutions relying entirely on calculator technology are not acceptable. $$\mathbf { M } = \left( \begin{array} { r r r } 2 & 0 & 0
0 & 1 & 4
3 & - 2 & - 3 \end{array} \right)$$

1. Determine \(\mathbf { M } ^ { - 1 }\) The transformation represented by \(\mathbf { M }\) maps the plane \(\Pi _ { 1 }\) to the plane \(\Pi _ { 2 }\) The point \(( x , y , z )\) on \(\Pi _ { 1 }\) maps to the point \(( u , v , w )\) on \(\Pi _ { 2 }\)
2. Determine \(x , y\) and \(z\) in terms of \(u , v\) and \(w\) as appropriate. The plane \(\Pi _ { 1 }\) has equation $$3 x - 7 y + 2 z = - 3$$
3. Find a Cartesian equation for \(\Pi _ { 2 }\) Give your answer in the form \(a u + b v + c w = d\) where \(a , b , c\) and \(d\) are integers to be determined.

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with equation $$y = \frac { 1 } { 2 } ( \tan x + \cot x ) \quad \frac { \pi } { 6 } \leqslant x \leqslant \frac { \pi } { 3 }$$

1. Show that the length of \(C\) is given by $$\frac { 1 } { 2 } \int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 3 } } \left( \tan ^ { 2 } x + \cot ^ { 2 } x \right) d x$$
2. Hence determine the exact length of \(C\), giving your answer in simplest form.

1. The plane \(\Pi _ { 1 }\) contains the point \(A ( 2,4 , - 5 )\) and is normal to the vector \(\left( \begin{array} { r } - 1
   3
   3 \end{array} \right)\)

The plane \(\Pi _ { 2 }\) contains the point \(B ( 3,6 , - 2 )\) and is normal to the vector \(\left( \begin{array} { r } 2
0
- 5 \end{array} \right)\)
The line \(l\) is the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\)

1. Determine a vector equation for \(l\). The points \(C\) and \(D\) both lie on \(l\).
   Given that \(C\) and \(D\) are 5 units apart,
2. determine the exact volume of the tetrahedron \(A B C D\).

5. $$\mathbf { M } = \left( \begin{array} { r r r } 1 & 2 & k
- 1 & - 3 & 4
2 & 6 & - 8 \end{array} \right) \quad \text { where } k \text { is a constant }$$ Given that \(\mathbf { M }\) has a repeated eigenvalue, determine

1. the possible values of \(k\),
2. all corresponding eigenvalues of \(\mathbf { M }\) for each value of \(k\).

1. The ellipse \(E\) has equation \(\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 9 } = 1\)

The point \(P ( 4 \cos \theta , 3 \sin \theta )\) lies on \(E\).

1. Use calculus to show that an equation of the tangent to \(E\) at \(P\) is $$3 x \cos \theta + 4 y \sin \theta = 12$$
2. Determine an equation for the normal to \(E\) at \(P\). The tangent to \(E\) at \(P\) meets the \(x\)-axis at the point \(A\).
   The normal to \(E\) at \(P\) meets the \(y\)-axis at the point \(B\).
3. Show that the locus of the midpoint of \(A\) and \(B\) as \(\theta\) varies has equation $$x ^ { 2 } \left( p - q y ^ { 2 } \right) = r$$ where \(p , q\) and \(r\) are integers to be determined.

7. $$I _ { n } = \int \cosh ^ { n } 2 x \mathrm {~d} x \quad n \geqslant 0$$

1. Show that, for \(n \geqslant 2\) $$I _ { n } = \frac { \cosh ^ { n - 1 } 2 x \sinh 2 x } { 2 n } + \frac { n - 1 } { n } I _ { n - 2 }$$
2. Hence determine $$\int ( 1 + \cosh 2 x ) ^ { 3 } d x$$ collecting any like terms in your answer.

1. (a) Differentiate \(x \operatorname { arcosh } 5 x\) with respect to \(x\)
   (b) Hence, or otherwise, show that

$$\int _ { \frac { 1 } { 4 } } ^ { \frac { 3 } { 5 } } \operatorname { arcosh } 5 x \mathrm {~d} x = \frac { 3 } { 20 } - \frac { 2 \sqrt { 2 } } { 5 } + \ln ( p + q \sqrt { 2 } ) ^ { k } - \frac { 1 } { 4 } \ln r$$ where \(p , q , r\) and \(k\) are rational numbers to be determined.