Edexcel F3 (Further Pure Mathematics 3) 2022 January

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1. Use the definitions of hyperbolic functions in terms of exponentials to prove that $$8 \cosh ^ { 4 } x = \cosh 4 x + p \cosh 2 x + q$$ where \(p\) and \(q\) are constants to be determined.
2. Hence, or otherwise, solve the equation $$\cosh 4 x - 17 \cosh 2 x + 9 = 0$$ giving your answers in exact simplified form in terms of natural logarithms.

2. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) with parametric equations $$x = \ln ( \sec \theta + \tan \theta ) - \sin \theta \quad y = \cos \theta \quad 0 \leqslant \theta \leqslant \frac { \pi } { 4 }$$ The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis and is used to form a solid of revolution \(S\). Using calculus, show that the total surface area of \(S\) is given by $$\frac { \pi } { 2 } ( p + q \sqrt { 2 } )$$ where \(p\) and \(q\) are integers to be determined.

3. (a) Given that \(y = \operatorname { arsech } \left( \frac { x } { 2 } \right)\), where \(0 < x \leqslant 2\), show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { p } { x \sqrt { q - x ^ { 2 } } }$$ where \(p\) and \(q\) are constants to be determined. In part (b) solutions based entirely on calculator technology are not acceptable. $$\mathrm { f } ( x ) = \operatorname { artanh } ( x ) + \operatorname { arsech } \left( \frac { x } { 2 } \right) \quad 0 < x \leqslant 1$$ (b) Determine, in simplest form, the exact value of \(x\) for which \(\mathrm { f } ^ { \prime } ( x ) = 0\)

4. $$\mathbf { M } = \left( \begin{array} { l l l } 6 & k & 2 \\ k & 5 & 0 \\ 2 & 0 & 7 \end{array} \right)$$ where \(k\) is a constant. Given that 3 is an eigenvalue of \(\mathbf { M }\),

1. determine the possible values of \(k\). Given that \(k < 0\)
2. determine the other eigenvalues of \(\mathbf { M }\).
3. Determine a normalised eigenvector corresponding to the eigenvalue 3

5. Determine

1. \(\int \frac { 1 } { \sqrt { x ^ { 2 } - 3 x + 5 } } \mathrm {~d} x\)
2. \(\int \frac { 1 } { \sqrt { 63 + 4 x - 4 x ^ { 2 } } } \mathrm {~d} x\)

6. $$I _ { n } = \int \mathrm { e } ^ { x } \sin ^ { n } x \mathrm {~d} x \quad n \in \mathbb { Z } \quad n \geqslant 0$$

1. Show that $$I _ { n } = \frac { \mathrm { e } ^ { x } \sin ^ { n - 1 } x } { n ^ { 2 } + 1 } ( \sin x - n \cos x ) + \frac { n ( n - 1 ) } { n ^ { 2 } + 1 } I _ { n - 2 } \quad n \geqslant 2$$
2. Hence find the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 2 } } e ^ { x } \sin ^ { 4 } x d x$$ giving your answer in the form \(A \mathrm { e } ^ { \frac { \pi } { 2 } } + B\) where \(A\) and \(B\) are rational numbers to be determined.

7. The line \(l _ { 1 }\) has equation $$\frac { x - 3 } { 4 } = \frac { y - 5 } { - 2 } = \frac { z - 4 } { 7 }$$ The plane \(\Pi\) has equation $$2 x + 4 y - z = 1$$ The line \(l _ { 1 }\) intersects the plane \(\Pi\) at the point \(P\)

1. Determine the coordinates of \(P\) The acute angle between \(l _ { 1 }\) and \(\Pi\) is \(\theta\) degrees.
2. Determine, to one decimal place, the value of \(\theta\) The line \(l _ { 2 }\) lies in \(\Pi\) and passes through \(P\) Given that the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\) is also \(\theta\) degrees,
3. determine a vector equation for \(l _ { 2 }\)

8. The ellipse \(E\) has equation $$\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$$

1. Determine the eccentricity of \(E\)
2. Hence, for this ellipse, determine
   1. the coordinates of the foci,
   2. the equations of the directrices. The point \(P\) lies on \(E\) and has coordinates \(( 3 \cos \theta , 2 \sin \theta )\). The line \(l _ { 1 }\) is the tangent to \(E\) at the point \(P\)
3. Using calculus, show that an equation for \(l _ { 1 }\) is $$2 x \cos \theta + 3 y \sin \theta = 6$$ The line \(l _ { 2 }\) passes through the origin and is perpendicular to \(l _ { 1 }\) The line \(l _ { 1 }\) intersects the line \(l _ { 2 }\) at the point \(Q\)
4. Determine the coordinates of \(Q\)
5. Show that, as \(\theta\) varies, the point \(Q\) lies on the curve with equation $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = \alpha x ^ { 2 } + \beta y ^ { 2 }$$ where \(\alpha\) and \(\beta\) are constants to be determined.
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