Edexcel F3 (Further Pure Mathematics 3) 2020 June

1. (a) Use the definition of \(\sinh x\) in terms of exponentials to show that

$$\sinh 3 x \equiv 4 \sinh ^ { 3 } x + 3 \sinh x$$ (b) Hence determine the exact coordinates of the points of intersection of the curve with equation \(y = \sinh 3 x\) and the curve with equation \(y = 19 \sinh x\), giving your answers as simplified logarithms where necessary.

2. Determine

1. \(\int \frac { 1 } { 3 x ^ { 2 } + 12 x + 24 } \mathrm {~d} x\)
2. \(\int \frac { 1 } { \sqrt { 27 - 6 x - x ^ { 2 } } } \mathrm {~d} x\)

3. $$\mathbf { M } = \left( \begin{array} { c c c } 3 & - 4 & k \\ 1 & - 2 & k \\ 1 & - 5 & 5 \end{array} \right) \text { where } k \text { is a constant }$$ 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 a normalised eigenvector corresponding to the eigenvalue 3
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4.

1. Show that, for \(n \geqslant 2\)
2. Hence find the functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) such that $$\int x ^ { 4 } \cos x \mathrm {~d} x = \mathrm { f } ( x ) \sin x + \mathrm { g } ( x ) \cos x + c$$ where \(c\) is an arbitrary constant. $$I _ { n } = \int x ^ { n } \cos x \mathrm {~d} x$$
   1. Show that, for \(n \geqslant 2\) $$I _ { n } = x ^ { n } \sin x + n x ^ { n - 1 } \cos x - n ( n - 1 ) I _ { n - 2 }$$
   2. Hence find the functions \(\mathrm { f } ( x )\) and \(\mathrm { g } ( x )\) such that

5. The hyperbola \(H\) has equation \(\frac { x ^ { 2 } } { 25 } - \frac { y ^ { 2 } } { 4 } = 1\) The line \(l\) has equation \(y = m x + c\), where \(m\) and \(c\) are constants. Given that \(l\) is a tangent to \(H\),

1. show that \(25 m ^ { 2 } = 4 + c ^ { 2 }\)
2. Hence find the equations of the tangents to \(H\) that pass through the point ( 1,2 ).
3. Find the coordinates of the point of contact each of these tangents makes with \(H\).

6. $$\mathbf { A } = \left( \begin{array} { r r r } 1 & - 1 & 1 \\ 1 & 1 & 1 \\ 1 & 2 & a \end{array} \right) \quad a \neq 1$$

1. Find \(\mathbf { A } ^ { - 1 }\) in terms of \(a\).
   . The straight line \(l _ { 1 }\) is mapped onto the straight line \(l _ { 2 }\) by the transformation represented by the matrix \(\mathbf { B }\). $$\mathbf { B } = \left( \begin{array} { r r r } 1 & - 1 & 1 \\ 1 & 1 & 1 \\ 1 & 2 & 4 \end{array} \right)$$ The equation of \(l _ { 2 }\) is $$( \mathbf { r } - ( 12 \mathbf { i } + 4 \mathbf { j } + 6 \mathbf { k } ) ) \times ( - 6 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } ) = \mathbf { 0 }$$
2. Find a vector equation for the line \(l _ { 1 }\)

7. The curve \(C\) has parametric equations $$x = \cosh t + t , \quad y = \cosh t - t \quad 0 \leqslant t \leqslant \ln 3$$

1. Show that $$\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } = 2 \cosh ^ { 2 } t$$ The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis. The area of the curved surface generated is given by \(S\).
2. Show that $$S = 2 \pi \sqrt { 2 } \int _ { 0 } ^ { \ln 3 } \left( \cosh ^ { 2 } t - t \cosh t \right) d t$$
3. Hence find the value of \(S\), giving your answer in the form $$\frac { \pi \sqrt { 2 } } { 9 } ( a + b \ln 3 )$$ where \(a\) and \(b\) are constants to be determined.

8. The plane \(\Pi _ { 1 }\) has equation $$x - 5 y + 3 z = 11$$ The plane \(\Pi _ { 2 }\) has equation $$3 x - 2 y + 2 z = 7$$ The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) intersect in the line \(l\).

1. Find a vector equation for \(l\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\) where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors and \(\lambda\) is a scalar parameter. The point \(P ( 2,0,3 )\) lies on \(\Pi _ { 1 }\) The line \(m\), which passes through \(P\), is parallel to \(l\). The point \(Q ( 3,2,1 )\) lies on \(\Pi _ { 2 }\) The line \(n\), which passes through \(Q\), is also parallel to \(l\).
2. Find, in exact simplified form, the shortest distance between \(m\) and \(n\).
   

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