Edexcel FP3 (Further Pure Mathematics 3) 2018 June

1. (a) Starting from the definitions of \(\sinh x\) and \(\cosh x\) in terms of exponentials, show that, for \(x \in \mathbb { R }\)

$$\tanh x = \frac { \mathrm { e } ^ { 2 x } - 1 } { \mathrm { e } ^ { 2 x } + 1 }$$ (b) Hence, given that \(- 1 < \theta < 1\), prove that $$\operatorname { artanh } \theta = \frac { 1 } { 2 } \ln \left( \frac { 1 + \theta } { 1 - \theta } \right)$$ uestion 1 continued
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2. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve with equation $$y = 5 \cosh x - 6 \sinh x$$ The curve crosses the \(x\)-axis at the point \(A\).

1. Find the exact value of the \(x\) coordinate of the point \(A\), giving your answer as a natural logarithm.
2. Show that $$( 5 \cosh x - 6 \sinh x ) ^ { 2 } \equiv a \cosh 2 x + b \sinh 2 x + c$$ where \(a , b\) and \(c\) are constants to be found. The finite region \(R\), bounded by the curve and the coordinate axes, is shown shaded in Figure 1. The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
3. Use calculus to find the volume of the solid generated, giving your answer as an exact multiple of \(\pi\).

3. \(\mathbf { M } = \left( \begin{array} { r r r } 3 & k & 2
- 1 & 0 & 1
1 & k & 1 \end{array} \right)\), where \(k\) is a constant Given that 3 is an eigenvalue of \(\mathbf { M }\),

1. find the value of \(k\).
2. Hence find the other two eigenvalues of \(\mathbf { M }\).
3. Find an eigenvector corresponding to the eigenvalue 3
   3. \(\quad \mathbf { M } = \left( \begin{array} { r c c } 3 & k & 2
   - 1 & 0 & 1
   1 & k & 1 \end{array} \right)\), where \(k\) is a constant Given that 3 is an eigenvalue of \(\mathbf { M }\), (a) find the value of \(k\).

4. The curve \(C\) has equation $$y = \operatorname { arsinh } x + x \sqrt { x ^ { 2 } + 1 } , \quad 0 \leqslant x \leqslant 1$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \sqrt { x ^ { 2 } + 1 }\)
2. Hence show that the length of the curve \(C\) is given by $$\int _ { 0 } ^ { 1 } \sqrt { 4 x ^ { 2 } + 5 } d x$$
3. Using the substitution \(x = \frac { \sqrt { 5 } } { 2 } \sinh u\), find the exact length of the curve \(C\), giving your answer in the form \(a + b \ln c\), where \(a , b\) and \(c\) are constants to be found.

5. Given that $$I _ { n } = \int x ^ { n } \sqrt { ( x + 8 ) } \mathrm { d } x , \quad n \geqslant 0 , x \geqslant 0$$

1. show that, for \(n \geqslant 1\) $$I _ { n } = \frac { p x ^ { n } ( x + 8 ) ^ { \frac { 3 } { 2 } } } { 2 n + 3 } - \frac { q n } { 2 n + 3 } I _ { n - 1 }$$ where \(p\) and \(q\) are constants to be found.
2. Use part (a) to find the exact value of $$\int _ { 0 } ^ { 10 } x ^ { 2 } \sqrt { ( x + 8 ) } d x$$ giving your answer in the form \(k \sqrt { 2 }\), where \(k\) is rational.

6. The line \(l _ { 1 }\) has equation $$\mathbf { r } = \mathbf { i } + 2 \mathbf { k } + \lambda ( 2 \mathbf { i } + 3 \mathbf { j } - \mathbf { k } )$$ where \(\lambda\) is a scalar parameter. The line \(l _ { 2 }\) has equation $$\frac { x + 1 } { 1 } = \frac { y - 4 } { 1 } = \frac { z - 1 } { 3 }$$

1. Prove that the lines \(l _ { 1 }\) and \(l _ { 2 }\) are skew.
2. Find the shortest distance between the lines \(l _ { 1 }\) and \(l _ { 2 }\) The plane \(\Pi\) contains \(l _ { 1 }\) and intersects \(l _ { 2 }\) at the point \(( 3,8,13 )\).
3. Find a cartesian equation for the plane \(\Pi\).

7. The ellipse \(E\) has foci at the points \(( \pm 3,0 )\) and has directrices with equations \(x = \pm \frac { 25 } { 3 }\)

1. Find a cartesian equation for the ellipse \(E\). The straight line \(l\) has equation \(y = m x + c\), where \(m\) and \(c\) are positive constants.
2. Show that the \(x\) coordinates of any points of intersection of \(l\) and \(E\) satisfy the equation $$\left( 16 + 25 m ^ { 2 } \right) x ^ { 2 } + 50 m c x + 25 \left( c ^ { 2 } - 16 \right) = 0$$ Given that the line \(l\) is a tangent to \(E\),
3. show that \(c ^ { 2 } = p m ^ { 2 } + q\), where \(p\) and \(q\) are constants to be found. The line \(l\) intersects the \(x\)-axis at the point \(A\) and intersects the \(y\)-axis at the point \(B\).
4. Show that the area of triangle \(O A B\), where \(O\) is the origin, is $$\frac { 25 m ^ { 2 } + 16 } { 2 m }$$
5. Find the minimum area of triangle \(O A B\).

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