Edexcel FP3 (Further Pure Mathematics 3)

1. Find the exact values of x for which

$$4 \tanh ^ { 2 } x - 2 \operatorname { sech } ^ { 2 } x = 3 ,$$ giving your answers in the form \(\pm \ln \mathrm { a }\), where a is real.

2. \begin{figure}[h] \end{figure} Figure 1 shows part of the curve with equation \(y = 2 \cosh \left( \frac { 1 } { 2 } x \right)\). The points \(A\) and \(B\) lie on the curve and have \(x\)-coordinates \(- \ln 2\) and \(\ln 2\) respectively. The arc of the curve joining \(A\) and \(B\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Find the exact area of the curved surface area formed.
(Total 7 marks)

3. Using the substitution \(\mathrm { x } = \frac { 3 } { \sinh \theta }\), or otherwise, find the exact value of $$\int _ { 4 } ^ { 3 \sqrt { } 3 } \frac { 1 } { x \sqrt { } \left( x ^ { 2 } + 9 \right) } d x$$ giving your answer in the form a ln b , where a and b are rational numbers.
(Total 8 marks)

4. \(y = \arctan ( \sqrt { } x ) , \quad x > 0,0 < y < \frac { \pi } { 2 }\).

1. Find the value of \(\frac { \mathrm { dy } } { \mathrm { dx } }\) at \(\mathrm { x } = \frac { 1 } { 4 }\).
2. Show that \(2 x ( 1 + x ) \frac { d ^ { 2 } y } { d x ^ { 2 } } + ( 1 + 3 x ) \frac { d y } { d x } = 0\).

5. $$\mathrm { I } _ { \mathrm { n } } = \int _ { 0 } ^ { \frac { \pi } { 2 } } \sin ^ { \mathrm { n } } x \mathrm { dx } , \mathrm { n } \geqslant 0$$

1. Show that \(I _ { n } = \frac { n - 1 } { n } I _ { n - 2 }\), for \(n \geqslant 2\)
2. Using the result in part (a), find the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 2 } } x \sin ^ { 5 } x \cos x d x$$

1. Referred to a fixed origin O , the points \(\mathrm { P } , \mathrm { Q }\) and R have coordinates \(( \mathbf { i } - 3 \mathbf { j } + \mathbf { k } ) , ( - 2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k } )\) and \(( 3 \mathbf { j } - 5 \mathbf { k } )\) respectively. The plane \(\Pi _ { 1 }\) passes through \(\mathrm { P } , \mathrm { Q }\) and R . Find
   1. \(\overrightarrow { \mathrm { PQ } } \times \overrightarrow { \mathrm { QR } }\),
   2. a cartesian equation of \(\Pi _ { 1 }\).
   The plane \(\Pi _ { 2 }\) has equation \(\mathbf { r }\). ( \(\mathbf { i } + \mathbf { j } - \mathbf { k }\) ) \(= 6\). The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) intersect in the line I .
2. Find a vector equation of I, giving your answer in the form ( \(\mathbf { r } - \mathbf { a }\) ) \(\times \mathbf { b } = \mathbf { 0 }\).

7. \(\quad \mathbf { A } = \left( \begin{array} { c c c } 2 & \mathrm { k } & 0
1 & 1 & 0
0 & - 2 & 1 \end{array} \right)\), where k is a constant. Given that \(\left( \begin{array} { c } 9
3
- 2 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\),

1. show that \(\mathrm { k } = 6\),
2. find the eigenvalues of \(\mathbf { A }\). A transformation \(\mathrm { T } : \mathbb { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 3 }\) is represented by the matrix \(\mathbf { A }\).
   The point P has coordinates \(( \mathrm { t } - 2 , \mathrm { t } , 2 \mathrm { t } )\) where t is a parameter.
3. Show that, for any value of \(t\), the transformation \(T\) maps \(P\) onto a point on the line with equation \(x - 4 y - 4 = 0\)
   (5)

8. The point \(\mathrm { P } ( 5 \sec \mathrm { u } , 3 \tan \mathrm { u } )\) lies on the hyperbola H with equation \(\frac { \mathrm { x } ^ { 2 } } { 25 } - \frac { \mathrm { y } ^ { 2 } } { 9 } = 1\). The tangent to \(H\) at \(P\) intersects the asymptote of \(H\) with equation \(y = \frac { 3 } { 5 } x\) at the point \(R\) and the asymptote with equation \(\mathrm { y } = - \frac { 3 } { 5 } \mathrm { x }\) at the point S .

1. Use differentiation to show that an equation of the tangent to H at P is $$3 x = 5 y \sin u + 15 \cos u$$
2. Prove that P is the mid-point of RS.