Edexcel F3 (Further Pure Mathematics 3) 2018 June

1. Solve the equation

$$15 \operatorname { sech } ^ { 2 } x + 7 \tanh x = 13$$ Give your answers in terms of simplified natural logarithms.

2. $$\mathbf { A } = \left( \begin{array} { l l } 3 & 2
2 & 6 \end{array} \right)$$

1. Find the eigenvalues and corresponding normalised eigenvectors of the matrix \(\mathbf { A }\).
2. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { P } ^ { \mathrm { T } } \mathbf { A P } = \mathbf { D }\).

3. Given that $$y = \arctan \left( \frac { \sin x } { \cos x - 1 } \right) \quad x \neq 2 n \pi , \quad n \in \mathbb { Z }$$ Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = k$$ where \(k\) is a constant to be found.
\(\_\_\_\_\) "

4. The hyperbola \(H\) has equation $$\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$ The line \(l\) is a normal to \(H\) at the point \(P ( a \sec \theta , b \tan \theta ) , 0 < \theta < \frac { \pi } { 2 }\)

1. Using calculus, show that an equation for \(l\) is $$a x \sin \theta + b y = \left( a ^ { 2 } + b ^ { 2 } \right) \tan \theta$$ The line \(l\) meets the \(x\)-axis at the point \(Q\), and the point \(M\) is the midpoint of \(P Q\).
2. Find the coordinates of \(M\).
3. Hence find the cartesian equation of the locus of \(M\) as \(\theta\) varies, giving your answer in the form \(y ^ { 2 } = \mathrm { f } ( x )\).

5. $$\mathbf { M } = \left( \begin{array} { r r r } 4 & - 5 & 0
k & 2 & 0
- 3 & - 5 & k \end{array} \right) \text {, where } k \text { is a real constant, } k \neq 0 , k \neq - \frac { 8 } { 5 }$$

1. Find, in terms of \(k\), the inverse of the matrix \(\mathbf { M }\). A transformation \(T : \mathbb { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 3 }\) is represented by the matrix $$\left( \begin{array} { r r r } 4 & - 5 & 0
   - 1 & 2 & 0
   - 3 & - 5 & - 1 \end{array} \right)$$ The transformation \(T\) maps the plane \(\Pi _ { 1 }\) onto the plane \(\Pi _ { 2 }\)
   Given that the plane \(\Pi _ { 2 }\) has equation \(2 x - z = 4\)
2. find a cartesian equation of the plane \(\Pi _ { 1 }\)

6. The curve \(C\) has parametric equations $$x = \theta - \tanh \theta , \quad y = \operatorname { sech } \theta , \quad 0 \leqslant \theta \leqslant \ln 3$$

1. Find
   1. \(\frac { \mathrm { d } x } { \mathrm {~d} \theta }\)
   2. \(\frac { \mathrm { d } y } { \mathrm {~d} \theta }\) The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
2. Find the exact area of the curved surface formed, giving your answer as a multiple of \(\pi\).

7. The plane \(\Pi _ { 1 }\) has equation \(x + y + z = 3\) and the plane \(\Pi _ { 2 }\) has equation \(2 x + 3 y - z = 4\) The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) intersect in the line \(L\).

1. Find a cartesian equation for the line \(L\). The plane \(\Pi _ { 3 }\) has equation $$\text { r. } \left( \begin{array} { r } 5
   - 4
   4 \end{array} \right) = 12$$ The line \(L\) meets the plane \(\Pi _ { 3 }\) at the point \(A\).
2. Find the coordinates of \(A\).
3. Find the acute angle between \(\overrightarrow { O A }\) and the line \(L\), where \(O\) is the origin. Give your answer in degrees to one decimal place.

8. $$I _ { n } = \int \frac { x ^ { n } } { \sqrt { \left( x ^ { 2 } + k ^ { 2 } \right) } } \mathrm { d } x \quad \text { where } k \text { is a constant and } n \in \mathbb { Z } ^ { + }$$

1. Show that, for \(n \geqslant 2\) $$I _ { n } = \frac { x ^ { n - 1 } } { n } \left( x ^ { 2 } + k ^ { 2 } \right) ^ { \frac { 1 } { 2 } } - \frac { ( n - 1 ) } { n } k ^ { 2 } I _ { n - 2 }$$
2. Hence find the exact value of $$\int _ { 0 } ^ { 1 } \frac { x ^ { 5 } } { \sqrt { \left( x ^ { 2 } + 1 \right) } } \mathrm { d } x$$