Edexcel FP3 (Further Pure Mathematics 3) 2017 June

1. Given that \(y = \operatorname { arsinh } ( \tanh x )\), show that

$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \operatorname { sech } ^ { 2 } x } { \sqrt { 1 + \tanh ^ { 2 } x } }$$ \section*{-} \includegraphics[max width=\textwidth, alt={}, center]{64dc962a-1788-49ac-a4db-af1241b552a0-03_51_51_276_2012}
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2. The ellipse \(E\) has equation $$\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 25 } = 1$$ The line \(l\) is the normal to \(E\) at the point \(P ( 6 \cos \theta , 5 \sin \theta )\), where \(0 < \theta < \frac { \pi } { 2 }\)

1. Use calculus to show that an equation of \(l\) is $$6 x \sin \theta - 5 y \cos \theta = 11 \sin \theta \cos \theta$$ The line \(l\) meets the \(x\)-axis at the point \(Q\). The point \(R\) is the foot of the perpendicular from \(P\) to the \(x\)-axis.
2. Show that \(\frac { O Q } { O R } = e ^ { 2 }\), where \(e\) is the eccentricity of the ellipse \(E\).

3. (a) Using the definition for \(\cosh x\) in terms of exponentials, show that $$\cosh 2 x \equiv 2 \cosh ^ { 2 } x - 1$$ (b) Find the exact values of \(x\) for which $$29 \cosh x - 3 \cosh 2 x = 38$$ giving your answers in terms of natural logarithms.

4. Use the substitution \(x + 2 = u ^ { 2 }\), where \(u > 0\), to show that $$\int _ { - 1 } ^ { 7 } \frac { ( x + 2 ) ^ { \frac { 1 } { 2 } } } { x + 5 } \mathrm {~d} x = a + b \pi \sqrt { 3 }$$ where \(a\) and \(b\) are rational numbers to be found. \includegraphics[max width=\textwidth, alt={}, center]{image-not-found}
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5. The plane \(\Pi _ { 1 }\) has equation \(x - 2 y - 3 z = 5\) and the plane \(\Pi _ { 2 }\) has equation \(6 x + y - 4 z = 7\)

1. Find, to the nearest degree, the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) The point \(P\) has coordinates \(( 2,3 , - 1 )\). The line \(l\) is perpendicular to \(\Pi _ { 1 }\) and passes through the point \(P\). The line \(l\) intersects \(\Pi _ { 2 }\) at the point \(Q\).
2. Find the coordinates of \(Q\). The plane \(\Pi _ { 3 }\) passes through the point \(Q\) and is perpendicular to \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\)
3. Find an equation of the plane \(\Pi _ { 3 }\) in the form \(\mathbf { r } . \mathbf { n } = p\)

6. The matrix \(\mathbf { M }\) is given by $$\mathbf { M } = \left( \begin{array} { r r r } 1 & k & 0
2 & - 2 & 1
- 4 & 1 & - 1 \end{array} \right) , k \in \mathbb { R } , k \neq \frac { 1 } { 2 }$$

1. Show that \(\operatorname { det } \mathbf { M } = 1 - 2 k\).
2. Find \(\mathbf { M } ^ { - 1 }\) in terms of \(k\). The straight line \(l _ { 1 }\) is mapped onto the straight line \(l _ { 2 }\) by the transformation represented by the matrix $$\left( \begin{array} { r r r } 1 & 0 & 0
   2 & - 2 & 1
   - 4 & 1 & - 1 \end{array} \right)$$ Given that \(l _ { 2 }\) has cartesian equation $$\frac { x - 1 } { 5 } = \frac { y + 2 } { 2 } = \frac { z - 3 } { 1 }$$
3. find a cartesian equation of the line \(l _ { 1 }\)

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

1. Show that, for \(n \geqslant 2\), $$I _ { n } = \frac { 3 a ^ { n - 1 } } { n b ^ { n } } + \frac { n - 1 } { n } I _ { n - 2 }$$ where \(a\) and \(b\) are integers to be found.
2. Hence, or otherwise, find the exact value of $$\int _ { 0 } ^ { \ln 2 } \cosh ^ { 4 } x \mathrm {~d} x$$

8. The curve \(C\) has equation $$y = \ln \left( \frac { \mathrm { e } ^ { x } + 1 } { \mathrm { e } ^ { x } - 1 } \right) , \quad \ln 2 \leqslant x \leqslant \ln 3$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 2 \mathrm { e } ^ { x } } { \mathrm { e } ^ { 2 x } - 1 }$$
2. Find the length of the curve \(C\), giving your answer in the form \(\ln a\), where \(a\) is a rational number.
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