Edexcel F3 (Further Pure Mathematics 3) 2018 Specimen

1. The curve \(C\) has equation

$$y = 9 \cosh x + 3 \sinh x + 7 x$$ Use differentiation to find the exact \(x\) coordinate of the stationary point of \(C\), giving your answer as a natural logarithm.

2. An ellipse has equation $$\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 4 } = 1$$ The point \(P\) lies on the ellipse and has coordinates \(( 5 \cos \theta , 2 \sin \theta ) , 0 < \theta < \frac { \pi } { 2 }\) The line \(L\) is a normal to the ellipse at the point \(P\).

1. Show that an equation for \(L\) is $$5 x \sin \theta - 2 y \cos \theta = 21 \sin \theta \cos \theta$$ Given that the line \(L\) crosses the \(y\)-axis at the point \(Q\) and that \(M\) is the midpoint of \(P Q\),
2. find the exact area of triangle \(O P M\), where \(O\) is the origin, giving your answer as a multiple of \(\sin 2 \theta\)
   

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3. Without using a calculator, find

1. \(\int _ { - 2 } ^ { 1 } \frac { 1 } { x ^ { 2 } + 4 x + 13 } \mathrm {~d} x\), giving your answer as a multiple of \(\pi\),
2. \(\int _ { - 1 } ^ { 4 } \frac { 1 } { \sqrt { 4 x ^ { 2 } - 12 x + 34 } } \mathrm {~d} x\), giving your answer in the form \(p \ln ( q + r \sqrt { 2 } )\),
   where \(p , q\) and \(r\) are rational numbers to be found.
   

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1. Given that \(y = \operatorname { artanh } ( \cos x )\)
   1. show that
   $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \operatorname { cosec } x$$
2. Hence find the exact value of $$\int _ { 0 } ^ { \frac { \pi } { 6 } } \cos x \operatorname { artanh } ( \cos x ) d x$$ giving your answer in the form \(a \ln ( b + c \sqrt { 3 } ) + d \pi\), where \(a , b , c\) and \(d\) are rational numbers to be found.

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1. The coordinates of the points \(A , B\) and \(C\) relative to a fixed origin \(O\) are \(( 1,2,3 )\),

The point \(D\) has coordinates \(( k , 4,14 )\) where \(k\) is a positive constant.
Given that the volume of the tetrahedron \(A B C D\) is 6 cubic units,
(b) find the value of \(k\). \section*{\(( - 1,3,4 )\) and \(( 2,1,6 )\) respectively. The plane \(\Pi\) contains the points \(A , B\) and \(C\).
(a) Find a cartesian equation of the plane \(\Pi\).
6. \(( - 1,3,4 )\) and \(( 2,1,6 )\) respectively. The plane (a) Find a cartesian equation of the plane \(\Pi\).}

1. The curve \(C\) has parametric equations

$$x = 3 t ^ { 4 } , \quad y = 4 t ^ { 3 } , \quad 0 \leqslant t \leqslant 1$$ The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis. The area of the curved surface generated is \(S\).

1. Show that $$S = k \pi \int _ { 0 } ^ { 1 } t ^ { 5 } \left( t ^ { 2 } + 1 \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} t$$ where \(k\) is a constant to be found.
2. Use the substitution \(u ^ { 2 } = t ^ { 2 } + 1\) to find the value of \(S\), giving your answer in the form \(p \pi ( 11 \sqrt { 2 } - 4 )\) where \(p\) is a rational number to be found.